then (iv) is equivalent to $U$ being just projective in $\mathcal{L}$.
However, the situation is different with homogeneity – we do not have
cofinality and uniqueness. If $\mathcal{L}$ is a _thin_ category (i.e. a
category where the hom-sets are degenerate), then every object is homogeneous
in $\mathcal{L}$. In our example, if we considered the MU-category
$\mathcal{I}∪\\{*\\}$ with a degenerate space and constant surjections added,
then the point $*$ would be homogeneous in $σ\mathcal{I}∪\\{*\\}$ as well
(besides the pseudo-arc). The reason the pseudo-arc is the only homogeneous
object in $\langle\mathcal{I},σ\mathcal{I}\rangle$ is simply because every
$σ\mathcal{I}$-object is cofinal in $\langle\mathcal{I},σ\mathcal{I}\rangle$
trivially.

######  Remark 4.17.

At a first look it may seem that the conditions defining the free completion
are just technical conditions for the proof of Theorem 4.15 to work, but it
can be shown that a free completion for $\mathcal{K}$ always exists and that
it is unique up to an MU-equivalence. The conditions are also meaningful on
their own. (L1) says that all necessary limits exist, while (L2) makes sure
that all $\mathcal{L}$-objects are “close to” $\mathcal{K}$. Altogether, they
say that $\langle\mathcal{K},\mathcal{L}\rangle$ is a “completion”. (F1) is a
factorization existence condition, while (F2) guarantees that the
factorizations are “continuously unique”. Altogether, they say that
$\mathcal{K}$-objects are “small” with respect to limits of sequences.

If $\mathcal{L}$ is discrete at every $\mathcal{K}$-object, then the
factorization conditions simplify: (F1) says that for every $\mathcal{L}$-map
$h\colon Y\leftarrow X_{∞}$ to a $\mathcal{K}$-object there is a
$\mathcal{K}$-map $g\colon Y\leftarrow X_{n}$ such that $h=g∘f_{n,∞}$, and
(F2) says that if $g∘f_{n,∞}=g^{\prime}∘f_{n,∞}$ for some $\mathcal{K}$-maps
$g,g^{\prime}$, then already $g∘f_{n,n^{\prime}}=g∘f_{n,n^{\prime}}$ for some
$n^{\prime}≥n$: the equality is witnessed before reaching the limit, so the
factorization of $h$ is essentially unique. The discrete combination of (F1)
and (F2) is closely related to the notion of _finitely presentable object_ [1,
Definition 1.1], but here we are in the projective setting, and
$\mathcal{K}$-objects are finitely presentable only with respect to
$\mathcal{K}$-sequences (as opposed to any directed diagram in $\mathcal{L}$).
The factorization conditions are also closely related to Morita’s conditions
(see [34, 7.1]) used in shape theory.

Any limit $\langle X_{∞},f_{*,∞}\rangle$ separates maps to the limit object,
i.e. for $\mathcal{L}$-maps $h≠h^{\prime}\colon X_{∞}\leftarrow Y$ there is
$n∈ω$ such that $f_{n,∞}∘h≠f_{n,∞}∘h^{\prime}$. The condition (C) says that
this happens MU-continuously.

###  4.2 Obtaining a free completion and the pseudo-arc

Next we look at how the conditions defining the free completion can be
fulfilled, and then we conclude the story of the pseudo-arc as a Fraïssé
limit. The proof of the following observation is clear and left to the reader.

######  Observation 4.18.

The properties (L1), (F2), (C) of $\langle\mathcal{K},\mathcal{L}\rangle$ are
hereditary in $\mathcal{K}$, and (F1) is hereditary in $\mathcal{K}$ with
respect to full subcategories. More precisely, let
$\mathcal{K}^{\prime}⊆\mathcal{K}⊆\mathcal{L}$ be MU-categories.

  1. (i)

If $\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (L1), then
$\langle\mathcal{K}^{\prime},\mathcal{L}\rangle$ satisfies (L1).

  2. (ii)

If $\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F1) and
$\mathcal{K}^{\prime}⊆\mathcal{K}$ is full, then
$\langle\mathcal{K}^{\prime},\mathcal{L}\rangle$ satisfies (F1).

  3. (iii)

If $\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F2), then
$\langle\mathcal{K}^{\prime},\mathcal{L}\rangle$ satisfies (F2).

  4. (iv)

If $\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (C), then
$\langle\mathcal{K}^{\prime},\mathcal{L}\rangle$ satisfies (C).

######  Construction 4.19 ($σ$-closure).

Given MU-categories $\mathcal{K}⊆\mathcal{L}$ we want to define the _$σ$
-closure_ $σ\mathcal{K}$ of $\mathcal{K}$ in $\mathcal{L}$. This will be the
“relevant part” of $\mathcal{L}$ containing $\mathcal{K}$ such that
$\langle\mathcal{K},σ\mathcal{K}\rangle$ has the best chance to be a free
completion. Condition (L2) says that every $\mathcal{L}$-object should be a
limit of a $\mathcal{K}$-sequence, which may easily not be the case – some
$\mathcal{L}$-objects may be simply “too far” from $\mathcal{K}$. A natural
thing to do is to define $σ\mathcal{K}$ to be the category of all
$\mathcal{L}$-limits of $\mathcal{K}$-sequences. This is clear with respect to
the objects of $σ\mathcal{K}$, but not so with respect to the morphisms.
Taking $σ\mathcal{K}$ to be a full subcategory of $\mathcal{L}$ may not be
appropriate. Another issue is that the notion of a limit is not absolute.

Let us consider a category $\mathcal{L}^{\prime}$ such that
$\mathcal{K}⊆\mathcal{L}^{\prime}⊆\mathcal{L}$. The ideal situation is that
$\mathcal{L}^{\prime}$-limits of $\mathcal{K}$-sequences are exactly their
$\mathcal{L}$-limits. This really consists of three conditions:

  1. (i)

every $\mathcal{L}$-limit cone for a $\mathcal{K}$-sequence lies in
$\mathcal{L}^{\prime}$,

  2. (ii)

every $\mathcal{L}$-limit cone for a $\mathcal{K}$-sequence $f_{*}$ that lies
in $\mathcal{L}^{\prime}$ is also an $\mathcal{L}^{\prime}$-limit cone for
$f_{*}$,

  3. (iii)

every $\mathcal{L}^{\prime}$-limit cone for a $\mathcal{K}$-sequence $f_{*}$
is also an $\mathcal{L}$-limit cone for $f_{*}$.

The condition (i) is easy to achieve – just add all the $\mathcal{L}$-limit
cones to $\mathcal{L}^{\prime}$. We say that $\mathcal{L}^{\prime}$ is _$σ$
-absolute_ in $\mathcal{L}$ for $\mathcal{K}$ if it satisfies (ii). This is
the key condition and we shall come back to it in a moment. The condition
(iii) may not be achievable – a $\mathcal{K}$-sequence with a limit in
$\mathcal{L}^{\prime}$ may simply not have a limit in $\mathcal{L}$ (which is
fixed). But if $\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (L1) and we
have (i) and (ii), then we have also (iii) – an $\mathcal{L}$-limit cone
$f_{*,∞}$ for a $\mathcal{K}$-sequence $f_{*}$ exists by (L1), lies in
$\mathcal{L}^{\prime}$ by (i), and is an $\mathcal{L}^{\prime}$-limit of
$f_{*}$ by (ii), so the fixed $\mathcal{L}^{\prime}$-limit cone for $f_{*}$ is
$\mathcal{L}^{\prime}$-isomorphic (and so $\mathcal{L}$-isomorphic) to
$f_{*,∞}$.

To clearly describe the construction of $σ\mathcal{K}$ let us introduce
several closure-like operators $\operatorname{cl}^{*}$ on families of maps
$\mathcal{F}⊆\mathcal{L}$. The operators satisfy
$\mathcal{A}⊆\operatorname{cl}^{*}(\mathcal{A})⊆\operatorname{cl}^{*}(\mathcal{B})$
for all families $\mathcal{A}⊆\mathcal{B}$.

  * •

$\operatorname{cl}^{\text{cat}}_{\mathcal{L}}(\mathcal{F})$ is the subcategory
of $\mathcal{L}$ generated by $\mathcal{F}$, i.e. we close the family
$\mathcal{F}$ under identities and composition. Some of the other closures do
not necessarily produce a subcategory, but we definitely want $σ\mathcal{K}$
to be a subcategory. This is why we explicitly consider this closure.

  * •

$\operatorname{cl}^{\text{loc}}_{\mathcal{L}}(\mathcal{F})$ or simply
$\overline{\mathcal{F}}$ denotes the _local closure_ in an MU-category, i.e.
$\overline{\mathcal{F}}∩\mathcal{L}(X,Y)$ is the closure of
$\mathcal{F}∩\mathcal{L}(X,Y)$ in the $∞$-metric space $\mathcal{L}(X,Y)$ for
every $\mathcal{L}$-objects $X,Y$. The family $\mathcal{F}$ is called _locally
closed_ in $\mathcal{L}$ if $\overline{\mathcal{F}}=\mathcal{F}$. In order to
satisfy (L1), $σ\mathcal{K}$ has to be locally complete and so necessarily
locally closed in $\mathcal{L}$. Since composition is continuous in any MU-
category, we have
$\operatorname{cl}^{\text{cat}}_{\mathcal{L}}(\operatorname{cl}^{\text{loc}}_{\mathcal{L}}(\mathcal{F}))⊆\operatorname{cl}^{\text{loc}}_{\mathcal{L}}(\operatorname{cl}^{\text{cat}}_{\mathcal{L}}(\mathcal{F}))$,
and so if $\mathcal{F}$ is a subcategory, so is its local closure.

  * •

$\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{F})$
consists of $\mathcal{F}$ together with all $\mathcal{L}$-maps $f_{n,∞}$,
$n∈ω$, where $f_{*,∞}$ is an $\mathcal{L}$-limit of a
$(\mathcal{K}∩\mathcal{F})$-sequence $f_{*}$ (meaning
$f_{n^{\prime},n^{\prime\prime}}∈\mathcal{K}∩\mathcal{F}$ for every
$n^{\prime}≤n^{\prime\prime}∈ω$). That means, $\mathcal{L}^{\prime}$ satisfies
(i) if and only if
$\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{L}^{\prime})⊆\mathcal{L}^{\prime}$
if and only if
$\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{K})⊆\mathcal{L}^{\prime}$.
The last part is true since for every family $\mathcal{F}⊇\mathcal{K}$ we have
$\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{F})=\mathcal{F}∪\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{K})$.

  * •

$\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{F})$
consists of $\mathcal{F}$ together with all $\mathcal{L}$-maps $h$ such that
there is a $(\mathcal{K}∩\mathcal{F})$-sequence $f_{*}$ and an
$\mathcal{L}$-limit $f_{*,∞}$ of $f_{*}$ lying in $\mathcal{F}$ such that
$f_{n,∞}∘h∈\mathcal{F}$ for every $n∈ω$. This closure addresses the condition
(ii) – it may happen that an $\mathcal{L}$-limit cone $\langle
X_{∞},f_{*,∞}\rangle$ for a $\mathcal{K}$-sequence $f_{*}$ lies in
$\mathcal{L}^{\prime}$, but is not an $\mathcal{L}^{\prime}$-limit, which is
witnessed by an $\mathcal{L}^{\prime}$-cone $\langle Y,γ_{*}\rangle$ for
$f_{*}$ such that the unique factorizing $\mathcal{L}$-map $γ_{∞}\colon Y\to
X_{∞}$ is not in $\mathcal{L}^{\prime}$. The closure
$\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{F})$
collects all limit factorizing maps relevant to $\mathcal{F}$. So
$\mathcal{L}^{\prime}$ is $σ$-absolute in $\mathcal{L}$ for $\mathcal{K}$ if
and only if
$\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{L}^{\prime})⊆\mathcal{L}^{\prime}$.
We may also say that $\mathcal{L}^{\prime}$ is _$σ$ -absolute_ in
$\mathcal{L}^{\prime}$ if
$\operatorname{cl}^{\text{abs}}_{\mathcal{L}^{\prime},\mathcal{L}}(\mathcal{L}^{\prime})⊆\mathcal{L}^{\prime}$.
Note that if $\mathcal{L}^{\prime}⊆\mathcal{L}$ is a full subcategory, then
$\mathcal{L}^{\prime}$ is $σ$-absolute in $\mathcal{L}$.

It seems natural to define $σ\mathcal{K}$ simply as
$\mathcal{A}_{0}:=\operatorname{cl}^{\text{loc}}_{\mathcal{L}}(\operatorname{cl}^{\text{cat}}_{\mathcal{L}}(\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{K})))$,
i.e. the smallest locally closed subcategory of $\mathcal{L}$ containing
$\mathcal{K}$ and all $\mathcal{L}$-limits of $\mathcal{K}$-sequences. But
$\mathcal{A}_{0}$ may not be $σ$-absolute in $\mathcal{L}$ for $\mathcal{K}$,
i.e. even though we have added all $\mathcal{L}$-limits of
$\mathcal{K}$-sequences, they may not be limits from the point of view of
$\mathcal{A}_{0}$. We should really define $σ\mathcal{K}$ as the smallest
locally closed and $σ$-absolute for $\mathcal{K}$ subcategory of $\mathcal{L}$
containing $\mathcal{K}$ and all $\mathcal{L}$-limits of
$\mathcal{K}$-sequences. Is this well-defined? If the category $\mathcal{L}$
is small, then yes – we can simply take the intersection of all such
subcategories, $\mathcal{L}$ being one of them. But if $\mathcal{L}$ is a
large category, we should be more careful.

Note that it is not enough in general to consider just
$\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{A}_{0})$.
The closure $\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}$ may not
be idempotent since adding the limit factorizing maps for the relevant cones
may introduce new relevant cones. Also,
$\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{A}_{0})$
may not be locally closed or even a subcategory. We perform the following
closing-off construction. For every ordinal $α$ let us put
$\mathcal{A}_{α+1}:=\operatorname{cl}^{\text{loc}}_{\mathcal{L}}(\operatorname{cl}^{\text{cat}}_{\mathcal{L}}(\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{A}_{α})))$
and for every limit ordinal $α$ let $\mathcal{A}_{α}:=⋃_{β<α}\mathcal{A}_{β}$.
Note that all families $\mathcal{A}_{α}$ are subcategories of $\mathcal{L}$
with the same class of objects, and that $\mathcal{A}_{1}$ contains all
$\mathcal{L}$-isomorphisms between its objects. We show that the increasing
sequence stabilize after at most $ω_{1}$ many steps, and so
$σ\mathcal{K}:=\mathcal{A}_{ω_{1}}$ is the smallest category between
$\mathcal{K}$ and $\mathcal{L}$ closed under all four closures
$\operatorname{cl}^{*}$. The reason is that for every
$f∈\operatorname{cl}^{*}(\mathcal{A}_{ω_{1}})$ we have
$f∈\operatorname{cl}^{*}(\mathcal{F})$ for a countable family
$\mathcal{F}⊆\mathcal{A}_{ω_{1}}$. Hence, $\mathcal{F}⊆\mathcal{A}_{α}$ for
some $α<ω_{1}$, and so $f∈\mathcal{A}_{α+1}⊆\mathcal{A}_{ω_{1}}$.

Note that formally there may be a problem defining a transfinite sequence of
proper classes, but we may instead define a single class relation $R(f,α)$ by
“$f∈\mathcal{A}_{0}$” for $α=0$, by “$∃β<α,R(f,β)$” for $α$ limit, and by
“there exists a countable family $\mathcal{F}$ such that
$f∈\operatorname{cl}^{\text{loc}}_{\mathcal{L}}(\operatorname{cl}^{\text{cat}}_{\mathcal{L}}(\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{F})))$
and $∀g∈\mathcal{F},R(g,β)$” if $α=β+1$.

This concludes the definition of the _$σ$ -closure_ $σ\mathcal{K}$. If the
supercategory $\mathcal{L}$ is not clear from the context, we write
$σ_{\mathcal{L}}\mathcal{K}$. Note that since $σ\mathcal{K}$ is $σ$-absolute
in $\mathcal{L}$ for $\mathcal{K}$ and closed with respect to the closures
$\operatorname{cl}^{*}$, the corresponding closures $\operatorname{cl}^{*}$ in
$σ\mathcal{K}$ are equal to the original ones, and we have
$σ_{σ\mathcal{K}}\mathcal{K}=σ\mathcal{K}$. Also note that
$σ\mathcal{K}$-isomorphisms are exactly $\mathcal{L}$-isomorphisms between
$σ\mathcal{K}$-objects.

######  Corollary 4.20.

If $\mathcal{K}⊆\mathcal{L}$ are MU-categories satisfying (L1), then the
limits of $\mathcal{K}$-sequences are the same from the point of view of
$\mathcal{L}$ and $σ\mathcal{K}$, i.e. we have the conditions (i), (ii), (iii)
from the previous construction for $\mathcal{L}^{\prime}=σ\mathcal{K}$.
Moreover, we have
$σ_{\mathcal{L}^{\prime}}\mathcal{K}^{\prime}=σ_{\mathcal{L}}\mathcal{K}^{\prime}$
for every $\mathcal{K}^{\prime}⊆\mathcal{K}$.

###### Proof.

The $σ$-closure was constructed so that it satisfies (i) and (ii). Since we
also assume (L1), we have also (iii) as shown in the previous construction.
The last part follows from the fact that the conditions (i), (ii), (iii) are
clearly true also for every $\mathcal{K}^{\prime}⊆\mathcal{K}$. ∎

######  Observation 4.21 (Absolute complete case).

Let $\mathcal{L}$ be an MU-category such that every sequence has a limit. We
say that a subcategory $\mathcal{L}^{\prime}⊆\mathcal{L}$ is _$σ$ -closed_ if
$σ\mathcal{L}^{\prime}=\mathcal{L}^{\prime}$. For such $σ$-closed subcategory
$\mathcal{L}^{\prime}$, limits of $\mathcal{L}^{\prime}$-sequences are the
same in $\mathcal{L}^{\prime}$ and $\mathcal{L}$. It follows that for every
$\mathcal{K}⊆\mathcal{L}^{\prime}$ we have
$σ_{\mathcal{L}^{\prime}}\mathcal{K}=σ_{\mathcal{L}}\mathcal{K}⊆\mathcal{L}^{\prime}$
(a special case of Corollary 4.20), i.e. it does not matter in which ambient
category we take the $σ$-closure. Moreover, a subcategory
$\mathcal{L}^{\prime\prime}⊆\mathcal{L}^{\prime}$ is $σ$-closed in
$\mathcal{L}^{\prime}$ if and only if it is $σ$-closed in $\mathcal{L}$. Also
note that $σ$-closed subcategories are stable under intersections, and that a
full subcategory $\mathcal{L}^{\prime}⊆\mathcal{L}$ is $σ$-closed if and only
if it is closed under limits of sequences.

Let us mention some examples. $\mathsf{MCpt}$ is $σ$-closed in $\mathsf{Top}$
as a full subcategory closed under limits of sequences. Strictly speaking,
$\mathsf{Top}$ is not an MU-category, but $\mathsf{MCpt}$ would be locally
closed with respect to any MU-structure since it is full. The category of all
metrizable continua and all continuous maps $\mathsf{MCont}$ is $σ$-closed in
$\mathsf{MCpt}$, again as a full subcategory closed under limits of sequences.
$\mathsf{MCpt_{s}}$ and $\mathsf{MCont_{s}}:=\mathsf{MCont}∩\mathsf{MCpt_{s}}$
are also $σ$-closed in $\mathsf{MCpt}$. If $\langle X_{∞},f_{*,∞}\rangle$ is a
$\mathsf{MCpt}$-limit of a $\mathsf{MCpt_{s}}$-sequence, then $X_{∞}$ is non-
empty, the maps $f_{n,∞}$ are surjective, and if for a continuous map $h\colon
Y\to X_{∞}$ from a metrizable compactum $Y$ the maps $f_{n,∞}∘h$ are
surjective, then $h$ is surjective as well. This is all well-known. Let us
comment on the last point. The sets $(f_{n,∞})^{-1}[U]$ for $n∈ω$ and
$U⊆X_{n}$ open form an open base of $X_{∞}$, and so the map $h$ has dense
image. We have already mentioned in Example 3.16 that $\mathsf{MCpt_{s}}$ is
locally complete.

The following proposition shows that in presence of (F1) the transfinite
construction of $σ\mathcal{K}$ stops right at the beginning. We also obtain an
alternative description of $σ\mathcal{K}$ in this situation. Note that this
description corresponds with the abstract definition of $\mathcal{C}^{ω}$
given by Panagiotopoulos and Solecki [39] where $\mathcal{C}$ is the category
of all finite connected graphs and connected epimorphisms. The description is
also related to the notion of _admissible extension_ of a category of finite
topological first-order structures by Irwin [19, Definition 2.7].

Moreover, the proposition shows that it is enough to consider (F1) only for
limit cone maps. This weaker condition defined below essentially allows us to
transfer a property of a limit of a $\mathcal{K}$-sequence to any other
$\mathcal{K}$-sequence with the same limit object.

######  Definition 4.22.

We say that a pair of MU-categories $\mathcal{K}⊆\mathcal{L}$ satisfies _(F1)
for limits_ if for every two $\mathcal{K}$-sequences $\langle
X_{*},f_{*}\rangle$, $\langle Y_{*},g_{*}\rangle$ with $\mathcal{L}$-limits
$\langle X_{∞},f_{*,∞}\rangle$, $\langle Y_{∞},g_{*,∞}\rangle$ such that
$X_{∞}=Y_{∞}$ and every $n∈ω$ and $ε>0$ there is $m∈ω$ and a $\mathcal{K}$-map
$h\colon Y_{n}\leftarrow X_{m}$ such that $h∘f_{m,∞}≈_{ε}g_{n,∞}$.

######  Proposition 4.23.

Let $\mathcal{K}⊆\mathcal{L}$ be MU-categories, and let
$\mathcal{L}(\mathcal{K},\mathcal{K})$ and
$\mathcal{L}(σ\mathcal{K},\mathcal{K})$ denote the families of all
$\mathcal{L}$-maps between $\mathcal{K}$-objects and from
$σ\mathcal{K}$-objects to $\mathcal{K}$-objects, respectively.

  * •

Let $\mathcal{B}_{0}$ be the family of all maps
$h∈\mathcal{L}(σ\mathcal{K},\mathcal{K})$ for which (F1) holds, i.e. such for
every $\mathcal{K}$-sequence $f_{*}$ with an $\mathcal{L}$-limit
$\langle\operatorname{dom}(h),f_{*,∞}\rangle$ and every $ε>0$ there is
$g∈\mathcal{K}$ such that $h≈_{ε}g∘f_{n,∞}$.

  * •

Let $\mathcal{B}_{1}$ be the family of all $\mathcal{L}$-maps $h$ between
$σ\mathcal{K}$-objects such that $\mathcal{B}_{0}∘h⊆\mathcal{B}_{0}$.

If $\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F1) for limits, then we
have the following.

  1. (i)

$σ\mathcal{K}∩\mathcal{L}(\mathcal{K},\mathcal{K})=\overline{\mathcal{K}}$,
i.e. $\mathcal{K}$ is locally dense in $σ\mathcal{K}$.

  2. (ii)

$σ\mathcal{K}∩\mathcal{L}(σ\mathcal{K},\mathcal{K})=\mathcal{A}_{0}∩\mathcal{L}(σ\mathcal{K},\mathcal{K})=\mathcal{B}_{0}$.

  3. (iii)

$σ\mathcal{K}=\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{A}_{0})=\mathcal{B}_{1}$,
in particular the construction of $σ\mathcal{K}$ stops after the first step.

Also note that it follows back from
$σ\mathcal{K}∩\mathcal{L}(σ\mathcal{K},\mathcal{K})⊆\mathcal{B}_{0}$ that
$\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F1) for limits.

###### Proof.

  1. (1)

$\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{K})⊆\mathcal{B}_{0}$.
For the limit cone maps $f_{*,∞}$, this is exactly what (F1) for limits says.
It remains to observe that every $\mathcal{K}$-map may be viewed as $f_{0,∞}$
for the canonical $\mathcal{L}$-limit $f_{*,∞}$ of an eventually constant
identity $\mathcal{K}$-sequence $f_{*}$.

  2. (2)

$\overline{\mathcal{B}_{0}}=\mathcal{B}_{0}$. This is clear from the
definition.

  3. (3)

$\mathcal{B}_{0}∩\mathcal{L}(\mathcal{K},\mathcal{K})=\overline{\mathcal{K}}$.
The inclusion
$\mathcal{B}_{0}∩\mathcal{L}(\mathcal{K},\mathcal{K})⊆\overline{\mathcal{K}}$
holds since every $\mathcal{B}_{0}$-map between $\mathcal{K}$-objects
$ε$-factorizes through the constant identity sequence for every $ε>0$. We have
$\mathcal{K}⊆\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{K})⊆\mathcal{B}_{0}$
by (i), and so
$\overline{\mathcal{K}}⊆\overline{\mathcal{B}_{0}}=\mathcal{B}_{0}$ by (ii).
The inclusion $\overline{\mathcal{K}}⊆\mathcal{L}(\mathcal{K},\mathcal{K})$ is
clear.

  4. (4)

$\mathcal{B}_{0}∘\mathcal{B}_{0}⊆\mathcal{B}_{0}$. We have
$\mathcal{B}_{0}∘\mathcal{B}_{0}=(\mathcal{B}_{0}∩\mathcal{L}(\mathcal{K},\mathcal{K}))∘\mathcal{B}_{0}=\overline{\mathcal{K}}∘\mathcal{B}_{0}⊆\overline{\mathcal{K}∘\mathcal{B}_{0}}⊆\overline{\mathcal{B}_{0}}=\mathcal{B}_{0}$.
We use (iii), the continuity of composition, the clear fact that
$\mathcal{K}∘\mathcal{B}_{0}⊆\mathcal{B}_{0}$, and (ii).

  5. (5)

$\mathcal{B}_{0}=\mathcal{A}_{0}∩\mathcal{L}(σ\mathcal{K},\mathcal{K})$. The
inclusion $\mathcal{B}_{0}⊆\mathcal{L}(σ\mathcal{K},\mathcal{K})$ is trivial.
To prove $\mathcal{B}_{0}⊆\mathcal{A}_{0}$, let $h\colon X\to Y$ be a
$\mathcal{B}_{0}$-map. There is a $\mathcal{K}$-sequence $\langle
X_{*},f_{*}\rangle$ with an $\mathcal{L}$-limit $\langle X,f_{*,∞}\rangle$.
Since $h∈\mathcal{B}_{0}$, we have
$h∈\overline{⋃_{n∈ω}\mathcal{K}(X_{n},Y)∘f_{n,∞}}⊆\overline{\operatorname{cl}^{\text{cat}}_{\mathcal{L}}(\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{K}))}=\mathcal{A}_{0}$.
On the other hand, by (i), (ii), and (iv), the only thing $\mathcal{B}_{0}$
lacks from $\mathcal{A}_{0}$ are the identities on $σ_{0}\mathcal{K}$-objects
that are not $\mathcal{K}$-objects, added by
$\operatorname{cl}^{\text{cat}}_{\mathcal{L}}$.

  6. (6)

$\mathcal{B}_{0}=\mathcal{B}_{1}∩\mathcal{L}(σ\mathcal{K},\mathcal{K})$. We
have $\mathcal{B}_{0}⊆\mathcal{B}_{1}$ since
$\mathcal{B}_{0}∘\mathcal{B}_{0}⊆\mathcal{B}_{0}$ by (iv). The inclusion
$\mathcal{B}_{0}⊆\mathcal{L}(σ\mathcal{K},\mathcal{K})$ is trivial. To show
that $\mathcal{B}_{1}∩\mathcal{L}(σ\mathcal{K},\mathcal{K})⊆\mathcal{B}_{0}$
let $h\colon X\to Y$ be a $\mathcal{B}_{1}$-map to a $\mathcal{K}$-object. We
have $\operatorname{id}_{Y}∈\mathcal{K}⊆\mathcal{B}_{0}$ by (i), and so
$h={\operatorname{id}_{Y}}∘h∈\mathcal{B}_{0}∘h⊆\mathcal{B}_{0}$.

  7. (7)

$\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{B}_{1})⊆\mathcal{B}_{1}$.
Let
$h∈\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{B}_{1})$
and let $\mathcal{F}:=\\{f∈\mathcal{B}_{0}:f∘h∈\mathcal{B}_{0}$ if
composable$\\}$. We want to show that $\mathcal{F}=\mathcal{B}_{0}$. Let
$f_{*}$ be a $\mathcal{K}$-sequence with an $\mathcal{L}$-limit $f_{*,∞}$ such
that $f_{*,∞}∘h⊆\mathcal{B}_{1}$. We have
$f_{*,∞}∘h⊆\mathcal{B}_{1}∩\mathcal{L}(σ\mathcal{K},\mathcal{K})=\mathcal{B}_{0}$
by (vi), and so $f_{*,∞}⊆\mathcal{F}$. We show that for any other
$\mathcal{K}$-sequence $f^{\prime}_{*}$ with an $\mathcal{L}$-limit
$\langle\operatorname{cod}(h),f^{\prime}_{*,∞}\rangle$ we have
$f^{\prime}_{*,∞}∘h⊆\mathcal{B}_{0}$, and so
$\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{K})⊆\mathcal{F}$.
By (F1) for limits we have $f^{\prime}_{*,∞}⊆\overline{\mathcal{K}∘f_{*,∞}}$
and so
$f^{\prime}_{*,∞}∘h⊆\overline{\mathcal{K}∘f_{*,∞}∘h}⊆\overline{\mathcal{K}∘\mathcal{B}_{0}}⊆\mathcal{B}_{0}$.
For the last inclusion see (iv). Besides
$\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{K})⊆\mathcal{F}$
we have also $\overline{\mathcal{F}}⊆\mathcal{F}$ since
$\overline{\mathcal{F}}∘h⊆\overline{\mathcal{F}∘h}⊆\overline{\mathcal{B}_{0}}=\mathcal{B}_{0}$,
and we have $\mathcal{F}∘\mathcal{F}⊆\mathcal{F}$ since
$(\mathcal{F}∘\mathcal{F})∘h⊆\mathcal{F}∘\mathcal{B}_{0}⊆\mathcal{B}_{0}∘\mathcal{B}_{0}⊆\mathcal{B}_{0}$.
Finally,
$\mathcal{F}⊇\mathcal{A}_{0}∩\mathcal{L}(σ\mathcal{K},\mathcal{K})=\mathcal{B}_{0}$
by (v).

  8. (8)

$σ\mathcal{K}⊆\mathcal{B}_{1}$. We show that $\mathcal{B}_{1}$ contains
$\mathcal{K}$ and is closed under all four closures $\operatorname{cl}^{*}$.
The claim will follow since $σ\mathcal{K}$ is the smallest such family. By (i)
and (vi) we have
$\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{K})⊆\mathcal{B}_{0}⊆\mathcal{B}_{1}$.
Hence, $\mathcal{K}⊆\mathcal{B}_{1}$, and also
$\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{B}_{1})⊆\mathcal{B}_{1}$
since
$\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{B}_{1})=\mathcal{B}_{1}∪\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{K})$.
$\mathcal{B}_{1}$ is a category since clearly
$\operatorname{id}_{X}∈\mathcal{B}_{1}$ for every $σ\mathcal{K}$-object $X$,
and $\mathcal{B}_{1}∘\mathcal{B}_{1}⊆\mathcal{B}_{1}$ since
$\mathcal{B}_{0}∘(\mathcal{B}_{1}∘\mathcal{B}_{1})=(\mathcal{B}_{0}∘\mathcal{B}_{1})∘\mathcal{B}_{1}⊆\mathcal{B}_{0}∘\mathcal{B}_{1}⊆\mathcal{B}_{0}$.
Also, $\overline{\mathcal{B}_{1}}⊆\mathcal{B}_{1}$ since
$\mathcal{B}_{0}∘\overline{\mathcal{B}_{1}}⊆\overline{\mathcal{B}_{0}∘\mathcal{B}_{1}}⊆\overline{\mathcal{B}_{0}}=\mathcal{B}_{0}$.
Finally, $\mathcal{B}_{1}$ is closed under
$\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}$ by (vii).

  9. (9)

$σ\mathcal{K}=\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{A}_{0})=\mathcal{B}_{1}$.
We clearly have
$\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{A}_{0})⊆σ\mathcal{K}$,
and by (viii) also $σ\mathcal{K}⊆\mathcal{B}_{1}$. To show that
$\mathcal{B}_{1}⊆\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{A}_{0})$
let $h\colon X\to Y$ be a $\mathcal{B}_{1}$-map. There is a
$\mathcal{K}$-sequence $g_{*}$ with an $\mathcal{L}$-limit $\langle
Y,g_{*,∞}\rangle$. We have
$g_{*,∞}∘h⊆\operatorname{cl}^{\text{lim}}_{\mathcal{K},\mathcal{L}}(\mathcal{K})∘h⊆\mathcal{B}_{0}∘h⊆\mathcal{B}_{0}⊆\mathcal{A}_{0}$
by (i) and (v), and hence
$h∈\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{A}_{0})$.

  10. (10)

$σ\mathcal{K}∩\mathcal{L}(σ\mathcal{K},\mathcal{K})=\mathcal{B}_{0}$ since
both sides are equal to
$\mathcal{B}_{1}∩\mathcal{L}(σ\mathcal{K},\mathcal{K})$ by (ix) and (vi).

  11. (11)

$σ\mathcal{K}∩\mathcal{L}(\mathcal{K},\mathcal{K})=\overline{\mathcal{K}}$
since both sides are equal to
$\mathcal{B}_{0}∩\mathcal{L}(\mathcal{K},\mathcal{K})$ by (x) and (iii). ∎

######  Corollary 4.24.

Let $\mathcal{K}⊆\mathcal{L}$ be MU-categories.

  1. (i)

$\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F1) if and only if
$\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F1) for limits and
$σ\mathcal{K}⊆\mathcal{L}$ is full.

  2. (ii)

If $\langle\mathcal{K},σ\mathcal{K}\rangle$ satisfies (F1), then
$\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F1) for limits. The other
implication holds as well if $\langle\mathcal{K},\mathcal{L}\rangle$ satisfies
(L1).

###### Proof.

Suppose that $\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F1). Then it
clearly satisfies (F1) for limits. We also show that
$\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{A}_{0})$ is
a full subcategory of $\mathcal{L}$, and so $σ\mathcal{K}$ is full as well.
Let $h\colon X_{∞}\to Y_{∞}$ be an $\mathcal{L}$-map between
$σ\mathcal{K}$-objects. By the definition of $σ\mathcal{K}$ there are
$\mathcal{K}$-sequences $f_{*}$ and $g_{*}$ with $\mathcal{L}$-limits $\langle
X_{∞},f_{*,∞}\rangle$ and $\langle Y_{∞},g_{*,∞}\rangle$. By (F1), for every
$n∈ω$ and $ε>0$ there is a $\mathcal{K}$-map $h^{\prime}\colon Y_{n}\leftarrow
X_{n^{\prime}}$ for some $n^{\prime}$ such that
$h^{\prime}∘f_{n^{\prime},∞}≈_{ε}g_{n,∞}∘h$. Hence, the cone $g_{*,∞}∘h$ lies
in $\mathcal{A}_{0}$, and $h$ as the corresponding limit factorizing map lies
in
$\operatorname{cl}^{\text{abs}}_{\mathcal{K},\mathcal{L}}(\mathcal{A}_{0})$.
On the other hand, if $\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F1)
for limits and $σ\mathcal{K}⊆\mathcal{L}$ is full, then
$\mathcal{L}(σ\mathcal{K},\mathcal{K})=σ\mathcal{K}∩\mathcal{L}(σ\mathcal{K},\mathcal{K})=\mathcal{B}_{0}$
by Proposition 4.23. Hence, $\langle\mathcal{K},\mathcal{L}\rangle$ satisfies
(F1).

If $\langle\mathcal{K},σ\mathcal{K}\rangle$ satisfies (F1), then
$\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F1) for limits since every
$\mathcal{L}$-limit cone $f_{*,∞}$ of a $\mathcal{K}$-sequence $f_{*}$ lies in
$σ\mathcal{K}$ and is a $σ\mathcal{K}$-limit of $f_{*}$, and so we have the
$ε$-factorizations. On the other hand, if
$\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (L1) and (F1) for limits,
then by Corollary 4.20 every $σ\mathcal{K}$-limit $\langle
X_{∞},f_{*,∞}\rangle$ of a $\mathcal{K}$-sequence is also an
$\mathcal{L}$-limit, and by Proposition 4.23 we have
$σ\mathcal{K}∩\mathcal{L}(σ\mathcal{K},\mathcal{K})=\mathcal{B}_{0}$, and so
every $σ\mathcal{K}$-map $h\colon X_{∞}\to Y$ to a $\mathcal{K}$-object admits
$ε$-factorizations by the definition of $\mathcal{B}_{0}$. ∎

######  Proposition 4.25.

Let $\mathcal{K}⊆\mathcal{L}$ be MU-categories such that
$\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (L1).

  1. (i)

$\langle\mathcal{K},σ\mathcal{K}\rangle$ satisfies (L1) and (L2).

  2. (ii)

$\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F1) if and only if
$\langle\mathcal{K},σ\mathcal{K}\rangle$ satisfies (F1) and
$σ\mathcal{K}⊆\mathcal{L}$ is full.

  3. (iii)

$\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F2) if and only if
$\langle\mathcal{K},σ\mathcal{K}\rangle$ satisfies (F2).

  4. (iv)

If $\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (C), then
$\langle\mathcal{K},σ\mathcal{K}\rangle$ satisfies (C).

Summarizing, if $\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (L1), (F1),
(F2), and (C), then $\langle\mathcal{K},σ\mathcal{K}\rangle$ is a free
completion, and $σ\mathcal{K}⊆\mathcal{L}$ is full. Moreover,
$\langle\mathcal{K},\mathcal{L}\rangle$ is a free completion if and only if
$\langle\mathcal{K},σ\mathcal{K}\rangle$ is a free completion and
$σ\mathcal{K}=\mathcal{L}$.

###### Proof.

By Corollary 4.20, the limits of $\mathcal{K}$-sequences are the same from the
point of view of $\mathcal{L}$ and $σ\mathcal{K}$. We use this in the proof of
all claims. Claim (i) also uses the fact that $σ\mathcal{K}$ is locally closed
in $\mathcal{L}$ and that $σ\mathcal{K}$-objects are exactly
$\mathcal{L}$-limit objects of $\mathcal{K}$-sequences. Claim (ii) follows
from Corollary 4.24. Claim (iii) is clear. The difference in (C) between
$\mathcal{L}$ and $σ\mathcal{K}$ is only in the family of the maps
$h,h^{\prime}\colon X_{∞}\leftarrow Y$ to be MU-continuously separated, so (C)
for $\mathcal{L}$ implies (C) for $σ\mathcal{K}$, which is claim (iv). The
last part mostly follows from the claims. If
$\langle\mathcal{K},\mathcal{L}\rangle$ is a free completion, then
$σ\mathcal{K}⊆\mathcal{L}$ is full by (ii) and also wide by (L2). ∎

######  Corollary 4.26.

If $\langle\mathcal{K},\mathcal{L}\rangle$ is a free completion of MU-
categories $\mathcal{K}⊆\mathcal{L}$, then for every full subcategory
$\mathcal{F}⊆\mathcal{K}$ we have that
$\langle\mathcal{F},σ\mathcal{F}\rangle$ is a free completion and
$σ\mathcal{F}⊆\mathcal{L}$ is a full subcategory.

###### Proof.

$\langle\mathcal{F},\mathcal{L}\rangle$ satisfies (L1), (F1), (F2), and (C) by
Observation 4.18. The rest follows from Proposition 4.25. ∎

Next, we describe some concrete situations such that
$\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F2) or (C).

######  Definition 4.27.

By a _metric epimorphism_ in an MU-category $\mathcal{K}$ we mean a
$\mathcal{K}$-map $f\colon X\to Y$ such that for every
$Z∈\operatorname{Ob}(\mathcal{K})$ and every $g,h∈\mathcal{K}(Y,Z)$ we have
$d(g∘f,h∘f)=d(g,h)$. Clearly, a metric epimorphism in $\mathcal{K}$ is also a
metric epimorphism in any MU-subcategory containing it, and every metric
epimorphism is an epimorphism. On the other hand, if $\mathcal{K}$ is discrete
with the $0$-$1$ metric, then metric epimorphisms are precisely epimorphisms.
The dual notion of _metric monomorphism_ was considered in [27, page 7].

######  Example 4.28.

In $\mathsf{Met_{u}}$, metric epimorphisms are exactly epimorphisms, i.e.
uniformly continuous maps with dense image. Similarly, in $\mathsf{MCpt}$,
metric epimorphisms are exactly epimorphisms, i.e. continuous surjections.
Hence, the MU-category $\mathsf{MCpt_{s}}$ consists of metric epimorphisms.

######  Observation 4.29.

Let $\mathcal{K}⊆\mathcal{L}$ be MU-categories.
$\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (F2) (even with $δ=ε$) in
the following two cases.

  1. (i)

$\mathcal{L}$ consists of metric epimorphisms.

  2. (ii)

$\mathcal{L}=\mathsf{MCpt}$.

###### Proof.

Let $\langle X_{*},f_{*}\rangle$ be a sequence in $\mathcal{K}$ with a limit
$\langle X_{∞},f_{*,∞}\rangle$ in $\mathcal{L}$, let $g,g^{\prime}\colon
Y\leftarrow X_{n}$ be $\mathcal{K}$-maps, and let $ε>0$ such that
$g∘f_{n,∞}≈_{ε}g^{\prime}∘f_{n,∞}$. In case (i) we can just cancel out
$f_{n,∞}$ and obtain $g≈_{ε}g^{\prime}$. In case (ii) let us put
$A:=\\{x_{*}∈∏_{k∈ω}X_{k}:d(g(x_{n}),g^{\prime}(x_{n}))≥ε\\}$. Note that $A$
is a closed subset of the product. We also put
$F_{m}:=\\{x_{*}∈∏_{k∈ω}X_{k}:x_{i}=f_{i,j}(x_{j})$ for every $i≤j≤m\\}$ for
$m∈ω$, and $F_{∞}:=⋂_{m≥n}F_{m}$. Without loss of generality, $F_{∞}$ with the
restricted projections is the chosen limit of $\langle X_{*},f_{*}\rangle$.
For every $n≤n^{\prime}≤∞$ we have
$g∘f_{n,n^{\prime}}≈_{ε}g^{\prime}∘f_{n,n^{\prime}}$ if and only if
$A∩F_{n^{\prime}}=∅$. Hence, the claim follows from compactness: if
$A∩F_{∞}=∅$, then $A∩F_{n^{\prime}}=∅$ for some $n^{\prime}≥n$. ∎

######  Definition 4.30.

Let $\mathcal{L}$ be an MU-category. We say that an $\mathcal{L}$-map $f\colon
X\to Y$ is _$ε$ -monic_ for $ε>0$ if for every two $\mathcal{L}$-maps
$h,h^{\prime}$ such that $f∘h=f∘h^{\prime}$ we have $h≈_{ε}h^{\prime}$.
Similarly, the map $f$ is _$\langle ε,δ\rangle$ -monic_ for $ε,δ>0$ if for
every two $\mathcal{L}$-maps $h,h^{\prime}$ such that $f∘h≈_{δ}f∘h^{\prime}$
we have $h≈_{ε}h^{\prime}$. Note that the condition (C) for a pair
$\langle\mathcal{K},\mathcal{L}\rangle$ says that for every
$\mathcal{K}$-sequence $f_{*}$ with an $\mathcal{L}$-limit $\langle
X_{∞},f_{*,∞}\rangle$ and $ε>0$ there is $n∈ω$ and $δ>0$ such that $f_{n,∞}$
is $\langle ε,δ\rangle$-monic. Clearly, an $ε$-monic or $\langle
ε,δ\rangle$-monic map in $\mathcal{L}$ is such also in every MU-subcategory
$\mathcal{L}^{\prime}⊆\mathcal{L}$ containing it.

######  Remark 4.31.

Recall that a continuous surjection $f\colon X\to Y$ between metric compacta
is an _$ε$ -map_ (see [38, 2.11] or [35]) if all fibers $f^{-1}(y)$, $y∈Y$,
have diameter $<ε$ (sometimes $≤ε$ is used). These are exactly $ε$-monic
$\mathsf{MCpt_{s}}$-maps since by the compactness both conditions are
equivalent to $f(x)=f(x^{\prime})\implies d(x,x^{\prime})<ε$. Also, $f$ is
$\langle ε,δ\rangle$-monic if and only if $d(f(x),f(x^{\prime}))<δ\implies
d(x,x^{\prime})<ε$.

######  Observation 4.32.

For every sequence $\langle X_{*},f_{*}\rangle$ and its limit $\langle
X_{∞},f_{*,∞}\rangle$ in $\mathsf{MCpt}$ and every $ε>0$ there is $n∈ω$ such
that the map $f_{n,∞}$ is $ε$-monic. For every $ε$-monic $\mathsf{MCpt}$-map
$f\colon X\to Y$ there is $δ>0$ such that $f$ is $\langle ε,δ\rangle$-monic.
It follows that $\langle\mathcal{K},\mathsf{MCpt}\rangle$ satisfies (C) for
every $\mathcal{K}⊆\mathsf{MCpt}$.

###### Proof.

For the first part we take for every $x∈X_{∞}$ an open set $x∈U_{x}⊆X_{∞}$ of
diameter $<ε$, and $n_{x}∈ω$ and an open set $V_{x}⊆X_{n_{x}}$ such that
$x∈f_{n_{x},∞}^{-1}[V_{x}]⊆U_{x}$. By the compactness there is a finite set
$F⊆X_{∞}$ such that $⋃_{x∈F}f_{n_{x},∞}^{-1}[V_{x}]=X_{∞}$, and so
$\mathcal{V}:=\\{f_{n_{x},n}^{-1}[V_{x}]:x∈F\\}$ where
$n:=\max\\{n_{x}:x∈F\\}$ is a cover of $f_{n,∞}[X_{∞}]⊆X_{n}$ such that for
every $V∈\mathcal{V}$, $f_{n,∞}^{-1}[V]$ has diameter $<ε$. This proves that
$f_{n,∞}$ is $ε$-monic. For the second part it is enough to put
$δ:=\min\\{d(f(x),f(x^{\prime})):d(x,x^{\prime})≥ε\\}>0$. ∎

By combining the previous observations with Proposition 4.25 be obtain the
following corollary.

######  Corollary 4.33.

$\langle\mathcal{K},σ\mathcal{K}\rangle$ satisfies (L1), (L2), (F2), and (C)
for every $\mathcal{K}⊆\mathsf{MCpt}$.

Next, we prove that (F1) holds for full categories of connected polyhedra.
Recall that an _(abstract) simplicial complex_ is a pair $\langle
V,\mathcal{S}\rangle$ where $V$ is a set and $\mathcal{S}$ is a hereditary
family of non-empty finite subsets of $V$ covering it. A _simplicial map_
$\langle V,\mathcal{S}\rangle\to\langle
V^{\prime},\mathcal{S}^{\prime}\rangle$ is a map $f\colon V\to V^{\prime}$
such that $f[S]∈\mathcal{S}^{\prime}$ for every $S∈\mathcal{S}$. The
_geometric realization_ $\lvert\langle V,\mathcal{S}\rangle\rvert$ of a
simplicial complex is the metric space $\\{x∈\mathbb{I}^{V}:∑x=1$ and
$\operatorname{supp}(x)∈\mathcal{S}\\}$ where
$\operatorname{supp}(x):=\\{v∈V:x(v)>0\\}$ and the metric is inherited from
the supremum metric on $\mathbb{I}^{V}$. The topology is equivalently
inherited from the product topology on $\mathbb{I}^{V}$. The topology is
compact if and only if $V$ is finite. Each point $x∈\lvert\langle
V,\mathcal{S}\rangle\rvert$ can also be uniquely written as
$∑_{v∈V}x(v)\,\lvert v\rvert_{V}$ where $\lvert v\rvert_{V}∈\mathbb{I}^{V}$ is
the characteristic function of $\\{v\\}$. The _geometric realization_ of a
simplicial map $f\colon\langle V,\mathcal{S}\rangle\to\langle
V^{\prime},\mathcal{S}^{\prime}\rangle$ is the continuous map $\lvert
f\rvert\colon\lvert\langle V,\mathcal{S}\rangle\rvert\to\lvert\langle
V^{\prime},\mathcal{S}^{\prime}\rangle\rvert$ defined by $∑_{v∈V}x(v)\,\lvert
v\rvert_{V}\mapsto∑_{v∈V}x(v)\,\lvert f(v)\rvert_{V^{\prime}}$, i.e.
$\bigl{(}\lvert f\rvert(x)\bigr{)}(v^{\prime})=∑_{v∈f^{-1}(v^{\prime})}x(v)$.
The geometric realization is a functor from simplicial complexes and
simplicial maps to metric spaces and continuous maps. On the full subcategory
$\\{\mathbb{I}_{n}:n∈ω\\}$ the functor is isomorphic to the geometric
realization functor defined in Section 2, so the definitions are consistent.

Recall that a _polyhedron_ is a (necessarily metrizable compact) space
homeomorphic to the geometric realization of a finite simplicial complex. Let
$\mathsf{CPol_{s}}$ denote the category of all non-empty connected polyhedra
and all continuous surjections. Let $\mathcal{P}$ be a family of non-empty
connected polyhedra (identified also with the corresponding full subcategory
of $\mathsf{CPol_{s}}$). Recall that a metrizable continuum $X$ is called
_$\mathcal{P}$ -like_ if for every $ε>0$ there is an $ε$-map from $X$ onto a
$\mathcal{P}$-object. This is equivalent (see [35]) to being a limit of a
$\mathcal{P}$-sequence, so $\mathcal{P}$-like continua are exactly
$σ\mathcal{P}$-objects. One implication follows from Observation 4.32, the
other one from the following proposition. Since we have all the needed
machinery at hand, we prove the second implication in Corollary 4.35. Also
note that the notion of arc-like continuum and the category $σ\mathcal{I}$ fit
into this scheme.

######  Proposition 4.34 ([35, Lemma 4]).

For every continuous surjection $f\colon X\to P$ from a metrizable continuum
onto a connected polyhedron and for every $ε>0$ there is $δ>0$ such that for
every connected polyhedron $P^{\prime}$ and every $δ$-monic continuous
surjection $f^{\prime}\colon X\to P^{\prime}$ there is a continuous surjection
$g\colon P^{\prime}\to P$ such that $f≈_{ε}g∘f^{\prime}$.

######  Corollary 4.35 ([35, Theorem 1]).

Let $\mathcal{P}⊆\mathsf{CPol_{s}}$ be a full subcategory and let $X$ be a
metrizable compactum. If for every $ε>0$ there is an $ε$-map from $X$ onto a
$\mathcal{P}$-object, then $X$ is a $σ\mathcal{P}$-object.

###### Proof.

We inductively build a sequence $\langle δ_{n}\rangle_{n∈ω}$ of strictly
positive numbers converging to zero, a sequence of $δ_{n}$-maps $φ_{n}\colon
X\to P_{n}$, a $\mathcal{P}$-sequence $\langle P_{*},g_{*}\rangle$, and an
epsilon sequence $\langle ε_{n}\rangle_{n∈ω}$ for $g_{*}$ such that every
$φ_{n}$ is $\langle δ_{n},4ε_{n}\rangle$-monic and
$φ_{n}≈_{ε_{n}}g_{n}∘φ_{n+1}$ for $n∈ω$. This is possible: $φ_{n}$ depending
only on $δ_{n}$ exists by the assumption; $ε_{n}$ depending only on $δ_{n}$
and $\langle ε_{k},g_{k}\rangle_{k<n}$ exists by Observation 4.32;
$δ_{n+1}≤1/n$ depending only on $φ_{n}$ and $ε_{n}$ is obtained by Proposition
4.34, and so a suitable map $g_{n}$ depending on $φ_{n},ε_{n},φ_{n+1}$ exists
as well.

Let $\langle P_{∞},g_{*,∞}\rangle$ be a limit of $g_{*}$ in
$\mathsf{MCpt_{s}}$. By Proposition 3.19 (iii) applied to the identity
sequence on $X$, $φ_{*}$, and $g_{*}$ in $\mathsf{MCpt_{s}}$ there is a
continuous surjection $φ_{∞}\colon X\to P_{∞}$ such that
$φ_{n}≈_{2ε_{n}}g_{n,∞}∘φ_{∞}$ for every $n∈ω$. To show that $φ_{∞}$ is a
homeomorphism it is enough to prove that it is one-to-one. Let $Y$ be any
metrizable compactum and let $h,h^{\prime}\colon Y\to X$ be any continuous
maps such that $φ_{∞}∘h=φ_{∞}∘h^{\prime}$. For every $n∈ω$ we have
$φ_{n}∘h≈_{2ε_{n}}g_{n,∞}∘φ_{∞}∘h=g_{n,∞}∘φ_{∞}∘h^{\prime}≈_{2ε_{n}}φ_{n}∘h^{\prime}$.
Since $φ_{n}$ is $\langle δ_{n},4ε_{n}\rangle$-monic, we have
$h≈_{δ_{n}}h^{\prime}$. Since $δ_{n}≤1/n$ and $n$ was arbitrary, we have
$h=h^{\prime}$, and so $φ_{∞}$ is one-to-one. ∎

Recall that $\mathsf{MCont_{s}}$ denotes the full MU-subcategory of
$\mathsf{MCpt_{s}}$ of all metrizable continua.

######  Theorem 4.36.

$\langle\mathsf{CPol_{s}},\mathsf{MCont_{s}}\rangle$ is a free completion.
Hence, for every full subcategory $\mathcal{P}⊆\mathsf{CPol_{s}}$ we have that
$σ\mathcal{P}⊆\mathsf{MCont_{s}}$ is a full subcategory and
$\langle\mathcal{P},σ\mathcal{P}\rangle$ is a free completion.

###### Proof.

Since $\mathsf{MCont_{s}}$ is $σ$-closed in $\mathsf{MCpt}$, we have
$σ\mathsf{CPol_{s}}⊆\mathsf{MCont_{s}}$, and
$\langle\mathsf{CPol_{s}},σ\mathsf{CPol_{s}}\rangle$ satisfies (L1), (L2),
(F2), and (C) by Corollary 4.33. We show that
$\langle\mathsf{CPol_{s}},\mathsf{MCont_{s}}\rangle$ satisfies (F1), and it
will follow by Proposition 4.25 (ii) that
$σ\mathsf{CPol_{s}}⊆\mathsf{MCont_{s}}$ is full. Let $f_{*}$ be a
$\mathsf{CPol_{s}}$-sequence with limit $\langle X_{∞},f_{*,∞}\rangle$, let
$h\colon Y\leftarrow X_{∞}$ be a continuous surjection onto a connected
polyhedron, and let $ε>0$. Take $δ>0$ obtained from Proposition 4.34. By
Observation 4.32 there is $n∈ω$ such that $f_{n,∞}$ is $δ$-monic, and so there
is a continuous surjection $g\colon Y\leftarrow X_{n}$ such that
$h≈_{ε}g∘f_{n,∞}$.

To obtain that $\langle\mathsf{CPol_{s}},\mathsf{MCont_{s}}\rangle$ is a free
completion, it remains to show $σ\mathsf{CPol_{s}}=\mathsf{MCont_{s}}$, namely
$\operatorname{Ob}(\mathsf{MCont_{s}})⊆\operatorname{Ob}(σ\mathsf{CPol_{s}})$,
i.e. every non-empty metrizable continuum $X$ is a limit of a
$\mathsf{CPol_{s}}$-sequence. By Corollary 4.35 it is enough to show that $X$
admits an $ε$-map onto a polyhedron for every $ε>0$, which is known [35,
Example 1]. See also [38, Theorem 2.13].

The rest follows from Corollary 4.26. ∎

######  Observation 4.37.

To wrap up the story of the pseudo-arc, observe that $σ\mathcal{I}$ is a full
subcategory of $\mathsf{MCont_{s}}$, and so the general definition is
consistent with the particular definition of $σ\mathcal{I}$ from Section 2.
Also observe that $\langle\mathcal{I},σ\mathcal{I}\rangle$ is a free
completion, and so we can use the characterization of the Fraïssé limit
(Theorem 4.15). We already know that $\mathbb{P}$ is a $σ\mathcal{I}$-limit of
a dominating sequence in $\mathcal{I}$ (Remark 3.28). To show that
$\mathbb{P}$ is a Fraïssé limit of $\mathcal{I}$ in $σ\mathcal{I}$ it remains
to observe that since $\mathcal{I}$ consists of a single object, every
dominating sequence in $\mathcal{I}$ is projective: if $f_{*}$ is a dominating
sequence in $\mathcal{I}$, then for every $\mathcal{I}$-maps $f,g$ and $ε>0$
there is $n∈ω$ and $\mathcal{I}$-maps $f^{\prime},g^{\prime}$ such that
$f∘f^{\prime}≈_{ε/2}f_{0,n}≈_{ε/2}g∘g^{\prime}$, and so we have the
amalgamation property, which is enough by Proposition 4.12.

In fact, it is well-known that suitable dense subcategories
$\mathcal{J}⊆\mathcal{I}$ have even the strict amalgamation property (i.e. the
amalgamation property with respect to the discrete MU-structure), from which
the amalgamation property of $\mathcal{I}$ clearly follows. Results of this
type are known as “mountain climbing theorems” or “mountain climbers’
theorems”. Homma [15] proved the result for $\mathcal{J}$ consisting of all
nowhere constant maps, Sikorski and Zarankiewicz [43] for $\mathcal{J}$
consisting of all piecewise monotone maps. (The $\mathcal{J}$-maps $f$
considered were limited by the condition $f(0)=0$ and $f(1)=1$, but for every
$f∈\mathcal{J}$ one can take a piecewise linear map $f^{\prime}$ such that
$f∘f^{\prime}∈\mathcal{J}$ and fixes the end-points.) On the other hand, whole
$\mathcal{I}$ does not have the strict amalgamation property (see [15]). It
follows that there is no strictly dominating sequence in $\mathcal{I}$.

The amalgamation property of $\mathcal{I}$ also follows directly from Theorem
2.24. Let $f,g$ be $\mathcal{I}$-maps and let $ε>0$. We take a suitable $δ>0$
for $g$ and $ε$ obtained by Theorem 2.24. There is $δ^{\prime}>0$ such that
$f$ is $\langle δ,δ^{\prime}\rangle$-continuous. Let $f^{\prime}$ be any
$δ^{\prime}$-crooked $\mathcal{I}$-map. Then $f∘f^{\prime}$ is $δ$-crooked and
by the choice of $δ$ there is an $\mathcal{I}$-map $g^{\prime}$ such that
$f∘f^{\prime}≈_{ε}g∘g^{\prime}$.

Combining everything together, we obtain the following result.

######  Theorem 4.38.

The pseudo-arc $\mathbb{P}$ is characterized (up to homeomorphism) by any of
the following conditions.

  1. (i)

$\mathbb{P}$ is a hereditarily indecomposable arc-like continuum.

  2. (ii)

$\mathbb{P}$ is a Fraïssé limit of $\mathcal{I}$ in $σ\mathcal{I}$ (which
already consists of several equivalent conditions).

  3. (iii)

$\mathbb{P}$ is a generic object in $σ\mathcal{I}$.

  4. (iv)

$\mathbb{P}$ is a generic object in $\langle\mathcal{I},σ\mathcal{I}\rangle$.

  5. (v)

$\mathbb{P}$ is a generic object over any dominating subcategory of
$\mathsf{Peano_{s}}$.

###  4.3 Fraïssé categories and ages

We have turned the problem of obtaining a Fraïssé object in a locally complete
MU-category $\mathcal{L}$ into obtaining a Fraïssé limit of a free completion
$\langle\mathcal{K},\mathcal{L}\rangle$, which reduces to obtaining a Fraïssé
sequence in $\mathcal{K}$. Now we give a characterization of existence of a
Fraïssé sequence. This is more or less a translation of the known results [28,
Corollary 3.8], [27, Theorem 3.3] to the context of MU-categories.

######  Definition 4.39.

A non-empty MU-category $\mathcal{K}$ is _Fraïssé_ if

  * •

$\mathcal{K}$ is _directed_ , i.e. for every two $\mathcal{K}$-objects $X$ and
$Y$ there are $\mathcal{K}$-maps $f\colon X\leftarrow Z$ and $g\colon
Y\leftarrow Z$ for a $\mathcal{K}$-object $Z$,

  * •

$\mathcal{K}$ has the amalgamation property,

  * •

$\mathcal{K}$ has a countable dominating subcategory.

Of course, the theorem is that an MU-category has a Fraïssé sequence if and
only if it is Fraïssé. We start with some preparations.

We have defined dominating subcategories and sequences (Definition 3.22). Let
us for convenience define a more general notion of a dominating functor so
that a subcategory is dominating if and only if the corresponding inclusion
functor is dominating, and a sequence is dominating if and only if it is
dominating when viewed as a functor from $\langle ω,≤\rangle$.

######  Definition 4.40.

A functor $F\colon\mathcal{K}\to\mathcal{L}$ between MU-categories (not
necessarily MU-continuous) is called _dominating_ if it is both

  * •

_cofinal_ , i.e. for every $\mathcal{L}$-object $X$, there is a
$\mathcal{K}$-object $Y$ and an $\mathcal{L}$-map $f\colon X\leftarrow F(Y)$,

  * •

_absorbing_ , i.e. for every $\mathcal{K}$-object $X$, every $ε>0$, and every
$\mathcal{L}$-map $f\colon F(X)\leftarrow Y$ there is an $\mathcal{L}$-map
$g\colon Y\leftarrow F(Z)$ and a $\mathcal{K}$-map $h\colon X\leftarrow Z$
such that $f∘g≈_{ε}F(h)$.

######  Lemma 4.41.

Let $F\colon\mathcal{K}\to\mathcal{L}$ and $G\colon\mathcal{L}\to\mathcal{M}$
be functors between MU-categories.

  1. (i)

If $F$ and $G$ are cofinal, then $G∘F$ is cofinal.

  2. (ii)

If $F$ and $G$ are absorbing and $G$ is MU-continuous, then $G∘F$ is
absorbing.

###### Proof.

(i) Let $X$ be an $\mathcal{M}$-object. Since $G$ is cofinal, there is an
$\mathcal{L}$-object $Y$ and an $\mathcal{M}$-map $f\colon X\leftarrow G(Y)$.
Since $F$ is cofinal, there is a $\mathcal{K}$-object $Z$ and an
$\mathcal{L}$-map $g\colon Y\leftarrow F(Z)$. Hence, $f∘G(g)\colon X\leftarrow
G(F(Z))$.

(ii) Let $X$ be a $\mathcal{K}$-object, let $ε>0$, and let $f\colon
G(F(X))\leftarrow Y$ be an $\mathcal{M}$-map. Since $G$ is absorbing, there is
an $\mathcal{L}$-map $f^{\prime}\colon F(X)\leftarrow Y^{\prime}$ and an
$\mathcal{M}$-map $g\colon Y\leftarrow G(Y^{\prime})$ such that
$G(f^{\prime})≈_{ε/2}f∘g$. Let $δ>0$ be a witness of MU-continuity of $G$ at
$F(X)$ for $ε/2$. Since $F$ is absorbing, there is a $\mathcal{K}$-map
$f^{\prime\prime}\colon X\leftarrow Y^{\prime\prime}$ and an $\mathcal{L}$-map
$g^{\prime}\colon Y^{\prime}\leftarrow F(Y^{\prime\prime})$ such that
$F(f^{\prime\prime})≈_{δ}f^{\prime}∘g^{\prime}$. Altogether, we have
$G(F(f^{\prime\prime}))≈_{ε/2}G(f^{\prime})∘G(g^{\prime})≈_{ε/2}f∘(g∘G(g^{\prime}))$.
∎

######  Corollary 4.42.

Dominating MU-functors are stable under composition.

######  Corollary 4.43.

If $\langle X_{*},f_{*}\rangle$ is a dominating sequence in $\mathcal{D}$, and
$\mathcal{D}$ is a dominating subcategory of $\mathcal{K}$, then $\langle
X_{*},f_{*}\rangle$ is dominating in $\mathcal{K}$.

######  Lemma 4.44.

Every countable dominating subcategory $\mathcal{D}_{0}$ of a Fraïssé MU-
category $\mathcal{K}$ can be extended to a countable dominating subcategory
$\mathcal{D}$ with
$\operatorname{Ob}(\mathcal{D})=\operatorname{Ob}(\mathcal{D}_{0})$ that is
Fraïssé.

###### Proof.

For every $\mathcal{D}_{0}$-objects $X$ and $Y$ we pick some
$\mathcal{K}$-maps $v_{X,Y,0}\colon X\leftarrow V_{X,Y}$ and $v_{X,Y,1}\colon
Y\leftarrow V_{X,Y}$ witnessing that $\mathcal{K}$ is directed. We can manage
to have $V_{X,Y}∈\operatorname{Ob}(\mathcal{D}_{0})$ since $\mathcal{D}_{0}$
is cofinal. Similarly, for every $\mathcal{D}_{0}$-maps $f\colon Z\leftarrow
X$ and $g\colon Z\leftarrow Y$ and for $n∈ω$ we pick some $\mathcal{K}$-maps
$w_{f,g,n,0}\colon X\leftarrow W_{f,g,n}$ and $w_{f,g,n,1}\colon Y\leftarrow
W_{f,g,n}$ such that $f∘w_{f,g,n,0}≈_{1/n}g∘w_{f,g,n,1}$, witnessing that
$\mathcal{K}$ has the amalgamation property. Again, we can have
$W_{f,g,n}∈\operatorname{Ob}(\mathcal{D}_{0})$. We let
$\mathcal{D}_{1}⊆\mathcal{K}$ be the subcategory generated by
$\mathcal{D}_{0}∪\\{v_{X,Y,0},v_{X,Y,1}:X,Y∈\operatorname{Ob}(\mathcal{D}_{0})\\}∪\\{w_{f,g,n,0},w_{f,g,n,1}:f,g∈\mathcal{D}_{0}$
with $\operatorname{cod}(f)=\operatorname{cod}(g)$, $n∈ω\\}$. Clearly,
$\mathcal{D}_{1}$ is still countable and dominating in $\mathcal{K}$ since
$\operatorname{Ob}(\mathcal{D}_{1})=\operatorname{Ob}(\mathcal{D}_{0})$.
Moreover, it is directed, and it has the amalgamation property with respect to
spans from $\mathcal{D}_{0}$. However, we have potentially added new spans, so
we need to repeat the second part of the procedure to obtain a chain
$\mathcal{D}_{0}⊆\mathcal{D}_{1}⊆\mathcal{D}_{2}⊆\cdots$. Finally, we put
$\mathcal{D}:=⋃_{n∈ω}\mathcal{D}_{n}$, which is still countable and dominating
in $\mathcal{K}$, and is directed and has the amalgamation property.
Altogether, $\mathcal{D}$ is Fraïssé. ∎

######  Lemma 4.45.

Let $\mathcal{K}$ be a countable MU-category. We may view the set
$\mathcal{K}$ of all morphisms as a countable discrete space and consider the
Polish space $\mathcal{K}^{ω}$. The subset $\mathcal{S}⊆\mathcal{K}^{ω}$
consisting of all sequences in $\mathcal{K}$ (i.e. elements of
$\mathcal{K}^{ω}$ consisting of composable maps) is closed and so a Polish
space on its own. If $\mathcal{K}$ is Fraïssé, then the set of all Fraïssé
sequences is dense $G_{δ}$ in $\mathcal{S}$.

###### Proof.

$\mathcal{S}$ is indeed closed since for every
$f_{*}∈\mathcal{K}^{ω}\setminus\mathcal{S}$ there is $n∈ω$ such that
$\operatorname{dom}(f_{n})≠\operatorname{cod}(f_{n+1})$, and so
$g_{*}∉\mathcal{S}$ for every $g_{*}∈\mathcal{K}^{ω}$ with $g_{n}=f_{n}$ and
$g_{n+1}=f_{n+1}$.

For every $\mathcal{K}$-object $X$, let $\mathcal{D}_{X}⊆\mathcal{S}$ be the
set of all $\mathcal{K}$-sequences $\langle X_{*},f_{*}\rangle$ such that
there is $n∈ω$ and a $\mathcal{K}$-map $f\colon X\leftarrow X_{n}$. The set
$\mathcal{D}_{X}$ is open since the membership in $\mathcal{D}_{X}$ is again
witnessed by a single coordinate. The set $\mathcal{D}_{X}$ is dense since for
every finite composable sequence $\langle f_{k}\colon X_{k}\leftarrow
X_{k+1}\rangle_{k<n}$ there are $\mathcal{K}$-maps $f_{n}\colon
X_{n}\leftarrow X_{n+1}$ and $f\colon X\leftarrow X_{n+1}$ by the fact that
$\mathcal{K}$ is directed.

For every $n∈ω$, every $\mathcal{K}$-map $f\colon X\leftarrow Y$ and $ε>0$ let
$\mathcal{E}_{n,f,ε}⊆\mathcal{S}$ be the set of all $\mathcal{K}$-sequences
$\langle X_{*},f_{*}\rangle$ such that $X_{n}≠X$ or there is $n^{\prime}≥n$
and a $\mathcal{K}$-map $g\colon Y\leftarrow X_{n^{\prime}}$ such that
$f∘g≈_{ε}f_{n,n^{\prime}}$. The set $\mathcal{E}_{n,f,ε}$ is open since if
$f_{*}∈\mathcal{E}_{n,f,ε}$, this is witnessed by a finite initial segments
$\langle f_{k}\rangle_{k<n^{\prime}}$ for some $n^{\prime}≥n$, and so every
$\mathcal{K}$-sequence $g_{*}$ with $\langle
g_{k}\rangle_{k<n^{\prime}}=\langle f_{k}\rangle_{k<n^{\prime}}$ is in
$\mathcal{E}_{n,f,ε}$. The set $\mathcal{E}_{n,f,ε}$ is dense since for every
finite composable sequence $\langle f_{k}\colon X_{k}\leftarrow
X_{k+1}\rangle_{k<m}$ with $m≥n$ and with $X_{n}=X$ there are
$\mathcal{K}$-maps $f_{m}\colon X_{m}\leftarrow X_{m+1}$ and $g\colon
Y\leftarrow X_{m+1}$ such that $f_{n,m}∘f_{m}≈_{ε}f∘g$ by the amalgamation
property.

Finally, the set of all dominating sequences is clearly
$⋂\\{\mathcal{D}_{X}:X∈\operatorname{Ob}(\mathcal{K})\\}∩⋂\\{\mathcal{E}_{n,f,1/k}:n∈ω,\,f∈\mathcal{K},\,k∈ω\\}$,
which is dense $G_{δ}$ by Baire category theorem and by the previous claims.
Fraïssé sequences are exactly dominating sequences by Proposition 4.12. ∎

######  Theorem 4.46.

An MU-category $\mathcal{K}$ has a Fraïssé sequence if and only if it is a
Fraïssé MU-category.

###### Proof.

Suppose $\langle X_{*},f_{*}\rangle$ is a Fraïssé sequence in $\mathcal{K}$.
Clearly $\mathcal{K}≠∅$. $\mathcal{K}$ is directed since by the cofinality,
for every $\mathcal{K}$-objects $X,Y$ there are $\mathcal{K}$-maps $f\colon
X\leftarrow X_{m}$ and $g\colon Y\leftarrow X_{n}$ for some $m,n∈ω$, so
$X_{\max(m,n)}$ works. $\mathcal{K}$ has the amalgamation property by
Proposition 4.12. The countable subcategory generated by $\\{f_{n}:n∈ω\\}$
(which may contain more compositions than $f_{n,m}$ since we may have
$X_{n}=X_{m}$ for some $n,m∈ω$) is dominating since the sequence is
dominating. We conclude that $\mathcal{K}$ is Fraïssé.

Suppose that $\mathcal{K}$ is Fraïssé. By Lemma 4.44 there is a countable
dominating subcategory $\mathcal{D}⊆\mathcal{K}$ that is Fraïssé. By Lemma
4.45 and the fact that $\mathcal{K}$ and so $\mathcal{D}$ is non-empty, there
is a Fraïssé sequence $f_{*}$ in $\mathcal{D}$. By Corollary 4.43, $f_{*}$ is
dominating in $\mathcal{K}$. By Proposition 4.12, $f_{*}$ is Fraïssé in
$\mathcal{K}$. ∎

Let us formulate two observations helpful for showing that a given MU-category
is Fraïssé.

######  Observation 4.47.

We may call an MU-category $\mathcal{K}$ _locally separable_ if every hom-set
$\mathcal{K}(X,Y)$ is separable. It is well-known that $\mathsf{MCpt}$ (and so
every $\mathcal{K}⊆\mathsf{MCpt}$) is locally separable (see e.g. [21,
(4.19)]).

If a locally separable MU-category $\mathcal{K}$ has a countable cofinal
subcategory $\mathcal{C}$ (equivalently, a countable _cofinal family of
objects_), then $\mathcal{K}$ has a countable dominating subcategory
$\mathcal{D}$: it is enough to let $\mathcal{D}⊆\mathcal{K}$ be the
subcategory generated by
$⋃\\{\mathcal{D}_{X,Y}:X,Y∈\operatorname{Ob}(\mathcal{C})\\}$ where every
$\mathcal{D}_{X,Y}$ is a countable dense subset of $\mathcal{K}(X,Y)$. It
follows that every locally separable MU-category with a countable directed
cofinal subcategory (in particular with a cofinal object, e.g.
$∅≠\mathcal{K}⊆\mathsf{Peano_{s}}$ full) is Fraïssé if and only if it has the
amalgamation property.

######  Observation 4.48.

Let $\mathcal{K}$ be an MU-category and $\mathcal{D}⊆\mathcal{K}$ a wide
dominating subcategory. If $\mathcal{D}$ has the amalgamation property, then
$\mathcal{K}$ has the amalgamation property as well.

###### Proof.

For every $\mathcal{K}$-maps $f_{0},f_{1}$ with a common codomain and $ε>0$
there are $\mathcal{K}$-maps $f^{\prime}_{i},g_{i}$, $i<2$, such that
$f_{i}∘f^{\prime}_{i}≈_{ε/3}g_{i}∈\mathcal{D}$, and there are
$\mathcal{D}$-maps $g^{\prime}_{i}$ such that
$g_{0}∘g^{\prime}_{0}≈_{ε/3}g_{1}∘g^{\prime}_{1}$. Altogether, we have
$f_{0}∘(f^{\prime}_{0}∘g^{\prime}_{0})≈_{ε}f_{1}∘(f^{\prime}_{1}∘g^{\prime}_{1})$.
∎

We have connected the notions of a Fraïssé object, a Fraïssé sequence, and a
Fraïssé category. These concepts may sometimes overlap – a Fraïssé category
$\mathcal{K}$ of small objects may already have a Fraïssé object,
corresponding to a constant identity Fraïssé sequence (see Proposition 4.13).
But even if it is not the case, $\mathcal{K}$ has a Fraïssé limit in a free
completion $\mathcal{L}$, where the limit is a Fraïssé object, and so
$\mathcal{L}$ itself is a Fraïssé category. To summarize, we formulate the
following observation.

######  Observation 4.49.

Let $\langle\mathcal{K},\mathcal{L}\rangle$ be a free completion of MU-
categories.

  1. (i)

If $\mathcal{K}$ is locally complete with a Fraïssé object $U$, then
$\mathcal{K}$ is a Fraïssé category, and $U$ is also a Fraïssé object in
$\mathcal{L}$.

  2. (ii)

If $\mathcal{K}$ is a Fraïssé category, then $\mathcal{L}$ is a locally
complete Fraïssé category with a Fraïssé object.

In the classical Fraïssé theory of first-order structures, ages of countable
structures are often discussed, while the ambient pair
$\langle\mathcal{K},\mathcal{L}\rangle$ forming a free completion is implicit.
We shall formulate the relative version of the characterization of Fraïssé
limits in our setting, so the correspondence to the classical Fraïssé theory
is more direct. Later, we shall view the classical theory as a special case of
our setting.

######  Definition 4.50.

Let $\mathcal{K}$ be an MU-category. A subcategory $\mathcal{F}⊆\mathcal{K}$
is called _hereditary_ if for every $f∈\mathcal{K}$ we have $f∈\mathcal{F}$ if
$\operatorname{dom}(f)∈\operatorname{Ob}(\mathcal{F})$, i.e. $\mathcal{F}$ is
a “downwards closed” full subcategory, where “downwards” means from the domain
to the codomain, as is natural in the projective setting. Similarly,
$\mathcal{F}$ is called a _component_ if it is both “downwards” and “upwards”
closed full subcategory, i.e. $f∈\mathcal{F}$ whenever
$\operatorname{dom}(f)∈\operatorname{Ob}(\mathcal{F})$ or
$\operatorname{cod}(f)∈\operatorname{Ob}(\mathcal{F})$ for $f∈\mathcal{K}$
(this is related to the notion of connected category – see Observation 4.3).

Let $\mathcal{K}⊆\mathcal{L}$ be MU-categories. For every $\mathcal{L}$-object
$U$ let $\operatorname{Age}(U)$ (or more precisely
$\operatorname{Age}_{\mathcal{K},\mathcal{L}}(U)$) denote the full subcategory
of $\mathcal{K}$ corresponding to the class of objects
$\\{X∈\operatorname{Ob}(\mathcal{K}):\mathcal{L}(U,X)≠∅\\}$. Clearly, this is
a hereditary subcategory. In this context we will identify classes of
$\mathcal{K}$-objects with the corresponding full subcategories.

######  Observation 4.51.

Let $\langle\mathcal{K},\mathcal{L}\rangle$ be a free completion of MU-
categories. If we start with an $\mathcal{L}$-object $U$, we have that
$\langle\operatorname{Age}(U),\operatorname{σAge}(U)\rangle$ is a free
completion and that $\operatorname{σAge}(U)⊆\mathcal{L}$ is full by Corollary
4.26. By (L2) of $\langle\mathcal{K},\mathcal{L}\rangle$, $U$ is a
$\operatorname{σAge}(U)$-object. Altogether, the characterization of a Fraïssé
limit (Theorem 4.15) applies.

On the other hand, if we start with a Fraïssé class $\mathcal{F}⊆\mathcal{K}$,
we have that $\langle\mathcal{F},σ\mathcal{F}\rangle$ is a free completion
with a Fraïssé limit $U$. However, $\mathcal{F}⊆\operatorname{Age}(U)$ and
$σ\mathcal{F}⊆\operatorname{σAge}(U)$ may be strict inclusions, and $U$ may
not be a Fraïssé limit of its age. The inclusions are equalities if and only
if $\mathcal{F}$ is hereditary, and $U$ is a Fraïssé limit of its age if and
only if the age has the amalgamation property. But in any case, $U$ is generic
in $\langle\operatorname{Age}(U),\operatorname{σAge}(U)\rangle$ since
$\mathcal{F}⊆\operatorname{Age}(U)$ is dominating (see Proposition 3.24), and
so $U$ is the only candidate for a Fraïssé limit.

######  Theorem 4.52.

Let $\langle\mathcal{K},\mathcal{L}\rangle$ be a free completion of MU-
categories and let $U$ be an $\mathcal{L}$-object.

  1. (i)

$U$ is homogeneous in $\langle\mathcal{K},\mathcal{L}\rangle$ if and only if
$U$ is a Fraïssé limit of $\operatorname{Age}(U)$ in $\operatorname{σAge}(U)$.

  2. (ii)

$U$ is projective in $\langle\mathcal{K},\mathcal{L}\rangle$ if and only if
$U$ is a Fraïssé limit of $\operatorname{Age}(U)$ in $\operatorname{σAge}(U)$
and $\operatorname{Age}(U)$ is a component of $\mathcal{K}$.

###### Proof.

We use the first part of the previous observation. By the definition of
$\operatorname{Age}(U)$ and by the fact that
$\operatorname{σAge}(U)⊆\mathcal{L}$ is full, we have that $U$ is homogeneous
in $\langle\mathcal{K},\mathcal{L}\rangle$ if and only if it is homogeneous in
$\langle\operatorname{Age}(U),\operatorname{σAge}(U)\rangle$. Clearly, $U$ is
always cofinal in
$\langle\operatorname{Age}(U),\operatorname{σAge}(U)\rangle$, hence the claim
(i) follows. Similarly, $U$ is projective in
$\langle\mathcal{K},\mathcal{L}\rangle$ if and only if it is projective in
$\langle\operatorname{Age}(U),\operatorname{σAge}(U)\rangle$ and
$\operatorname{Age}(U)$ is a component of $\mathcal{K}$. Hence, (ii) follows.
∎

######  Corollary 4.53.

A projective object in a free completion
$\langle\mathcal{K},\mathcal{L}\rangle$ of MU-categories is homogeneous.

######  Observation 4.54.

For a free completion $\langle\mathcal{K},\mathcal{L}\rangle$ we may preorder
homogeneous objects in $\langle\mathcal{K},\mathcal{L}\rangle$ by putting
$U≤V$ if $\operatorname{Age}(U)⊆\operatorname{Age}(V)$. This is a well-defined
preorder, and two homogeneous objects are $≤$-equivalent if and only if they
have the same ages, and so if and only if they are $\mathcal{L}$-isomorphic,
as Fraïssé limits.

###  4.4 Applications

Let us demonstrate the theory built so far on several examples.

######  Example 4.55 (Cantor space).

Let $\mathsf{Fin_{s}}$ be the category of non-empty finite sets and surjective
maps. These are equivalently non-empty finite discrete spaces with continuous
surjections, and so $\mathsf{Fin_{s}}$ may be viewed as an MU-subcategory of
$\mathsf{MCpt_{s}}$. We will observe that $σ\mathsf{Fin_{s}}$ is the category
of all non-empty zero-dimensional metrizable compacta and all continuous
surjections. Clearly every limit $\langle X_{∞},f_{*,∞}\rangle$ of a
$\mathsf{Fin_{s}}$-sequence $\langle X_{*},f_{*}\rangle$ is a non-empty zero-
dimensional metrizable compactum.
$\langle\mathsf{Fin_{s}},\mathsf{MCpt_{s}}\rangle$ satisfies (F1): continuous
surjections $g\colon X_{∞}\to Y$ onto finite spaces correspond to finite
clopen partitions of $X_{∞}$. As in Observation 4.32, there is $n∈ω$ such that
the partition of $X_{∞}$ induced by $f_{n,∞}$ refines the one induced by $g$,
and so there is a surjection $h\colon X_{n}\to Y$ such that $h∘f_{n,∞}=g$. It
follows that $\langle\mathsf{Fin_{s}},σ\mathsf{Fin_{s}}\rangle$ is a free
completion and that $σ\mathsf{Fin_{s}}⊆\mathsf{MCpt_{s}}$ is full (Proposition
4.25, Corollary 4.33). Since clearly for every zero-dimensional metrizable
compactum $X$ and $ε>0$ there is an $ε$-map onto a finite space, $X$ is a
$σ\mathsf{Fin_{s}}$-object as in Corollary 4.35.

It is easy to see that $\mathsf{Fin_{s}}$ is a Fraïssé category, and so there
is a Fraïssé limit $U$ of $\langle\mathsf{Fin_{s}},σ\mathsf{Fin_{s}}\rangle$.
The limit $U$ is characterized by projectivity in
$\langle\mathsf{Fin_{s}},σ\mathsf{Fin_{s}}\rangle$ since cofinality follows
(Remark 4.16), i.e. by the condition that for every continuous surjection
$f\colon U\to F$ onto a finite space and every $\mathsf{Fin_{s}}$-map $g\colon
F^{\prime}\to F$ there is a continuous surjection $h\colon U\to F^{\prime}$
such that $f=g∘h$. Equivalently, for every (necessarily finite) clopen
partition of $U$ and every finite refinement pattern there is a corresponding
refining clopen partition of $U$. Since every $\mathsf{Fin_{s}}$-map $g\colon
F^{\prime}\to F$ is a composition of “simple” maps, namely finite surjections
splitting just one point into two, $U$ is projective in
$\langle\mathsf{Fin_{s}},σ\mathsf{Fin_{s}}\rangle$ if and only if it has no
isolated points. The well-known theorem that there is a unique non-empty zero-
dimensional metrizable compact space without isolated points as well as the
fact that the Fraïssé limit is the Cantor space $2^{ω}$ follow. As a
byproduct, we obtain the known facts that $2^{ω}$ maps onto every non-empty
zero-dimensional metrizable compactum, that every continuous surjection
$2^{ω}\to 2^{ω}$ is a near-homeomorphism, as well as that for every continuous
surjections $f,g\colon 2^{ω}\to Z$ onto a zero-dimensional compactum and every
$ε>0$ there is a homeomorphism $h\colon 2^{ω}\to 2^{ω}$ such that $f≈_{ε}g∘h$.
It is also easy to see that a $\mathsf{Fin_{s}}$-sequence $\langle
X_{*},f_{*}\rangle$ is Fraïssé if and only if it “splits every point”, i.e.
for every $n$ and $x∈X_{n}$ there is $n^{\prime}≥n$ such that
$f_{n,n^{\prime}}^{-1}(x)$ is non-degenerate. The winning strategy for Odd in
$\operatorname{\mathsf{BM}}(\mathsf{Fin_{s}})$ for the Cantor space obviously
consists of splitting every point, e.g. by playing the projection
$X_{n}\leftarrow X_{n}×2$.

The example above was discrete in the sense that we had even strict
commutativity of diagrams into small objects. Let us summarize how the theory
simplifies in the discrete setting in general.

######  Remark 4.56 (Discrete Fraïssé theory).

Let $\mathcal{K}⊆\mathcal{L}$ be categories viewed also as discrete MU-
categories (Definition 3.17). All the properties involving commutativity of
diagrams up to arbitrarily small $ε>0$ simplify to strict commutativity. For
example, an $\mathcal{L}$-object $U$ is homogeneous in
$\langle\mathcal{K},\mathcal{L}\rangle$ if and only if for every
$\mathcal{L}$-maps $f,g\colon U\to X$ to a $\mathcal{K}$-object there is an
automorphism $h\colon U\to U$ such that $f=g∘h$. Similarly, $\mathcal{K}$ has
the amalgamation property if and only if for every $\mathcal{K}$-maps $f,g$
with common codomain there are $\mathcal{K}$-maps $f^{\prime},g^{\prime}$ such
that $f∘f^{\prime}=g∘g^{\prime}$.

The pair $\langle\mathcal{K},\mathcal{L}\rangle$ rarely satisfies the discrete
version of the continuity condition (C). Hence, we say that
$\langle\mathcal{K},\mathcal{L}\rangle$ is a _discrete free completion_ if it
satisfies (L1), (L2), (F1), (F2), which simplify to the following:

  * (L1)

Every $\mathcal{K}$-sequence has a limit in $\mathcal{L}$.

  * (L2)

Every $\mathcal{L}$-object is a limit of a $\mathcal{K}$-sequence.

  * (F1)

For every $\mathcal{K}$-sequence $f_{*}$, every its limit $\langle
X_{∞},f_{*,∞}\rangle$ in $\mathcal{L}$, and every $\mathcal{L}$-map $h\colon
Y\leftarrow X_{∞}$ to a $\mathcal{K}$-object there is $n∈ω$ and a
$\mathcal{K}$-map $g\colon Y\leftarrow X_{n}$ such that $h=g∘f_{n,∞}$.

  * (F2)

For every $\mathcal{K}$-sequence $f_{*}$, every its limit $\langle
X_{∞},f_{*,∞}\rangle$ in $\mathcal{L}$, every $n∈ω$, and every
$\mathcal{K}$-maps $g,g^{\prime}\colon Y\leftarrow X_{n}$ to a
$\mathcal{K}$-object such that $g∘f_{n,∞}=g^{\prime}∘f_{n,∞}$ there is
$n^{\prime}≥n$ such that $g∘f_{n,n^{\prime}}=g^{\prime}∘f_{n,n^{\prime}}$.

Since Theorem 4.15 uses (C) only to prove large homogeneity, we still have the
following characterization of Fraïssé limits in the discrete context. For a
discrete free completion $\langle\mathcal{K},\mathcal{L}\rangle$ and an
$\mathcal{L}$-object $U$, the following conditions are equivalent:

  1. (i)

$U$ is an $\mathcal{L}$-limit of a Fraïssé sequence in $\mathcal{K}$,

  2. (ii)

$U$ is cofinal and projective in $\langle\mathcal{K},\mathcal{L}\rangle$,

  3. (iii)

$U$ is cofinal and homogeneous in $\langle\mathcal{K},\mathcal{L}\rangle$.

Moreover, such object $U$ is unique up to an isomorphism, is cofinal in
$\mathcal{L}$, and every $\mathcal{K}$-sequence with limit $U$ is Fraïssé.

It turns out that we may alter the MU-structure on $\mathcal{L}$ so that the
discrete free completion $\langle\mathcal{K},\mathcal{L}\rangle$ becomes a
free completion of MU-categories. For every $\mathcal{L}$-object $X$ we fix
any $\mathcal{K}$-sequence $\langle X_{*},f_{*}\rangle$ with limit $\langle
X,f_{*,∞}\rangle$ and for every $\mathcal{L}$-maps $g,h\colon X\leftarrow Y$
we put $d(g,h):=\inf\\{1/(n+1):f_{n,∞}∘g=f_{n,∞}∘h\\}∈[0,∞]$, so $g≈_{1/n}h$
if and only if $f_{n,∞}∘g=f_{n,∞}∘h$. This turns $\mathcal{L}$ into a locally
complete MU-category. Up to MU-isomorphism, the resulting MU-category does not
depend on the choice of $\mathcal{K}$-sequences and limit maps.
$\langle\mathcal{K},\mathcal{L}\rangle$ with this induced MU-structure is a
free completion. Every $\mathcal{K}$-object is still discrete in $\mathcal{L}$
with the induced MU-structure, so the commutativity of diagrams to
$\mathcal{K}$-objects simplify as before, and hence the Fraïssé limit is the
same as in the discrete setting, but additionally is homogeneous and
projective in $\mathcal{L}$ (with respect to the induced MU-structure). This
explains how our MU-setup can be viewed as a “conservative extension” of the
discrete setup.

###### Proof.

We show that $d$ is a well-defined complete (ultra)metric on every
$\mathcal{L}(Y,X)$. We have $d(g,h)=0\iff g=h$ by the uniqueness of a limit
factorizing map. The map $d$ satisfies the ultrametric triangle inequality
since if $g≈_{1/n}h$ and $h≈_{1/m}k$ for $n≤m$, then
$f_{n,∞}∘g=f_{n,∞}∘h=f_{n,∞}∘k$, and so $g≈_{1/n}k$. The metric $d$ is
complete since for every Cauchy sequence $\langle g_{k}\rangle_{g∈ω}$ in
$\mathcal{L}(Y,X)$ and $n∈ω$ the sequence $\langle f_{n,∞}∘g_{k}\rangle_{k∈ω}$
is eventually constant with value $γ_{n}$, and $\langle Y,γ_{*}\rangle$ is a
cone for $f_{*}$. The limit factorizing map $γ_{∞}$ is the limit of the
sequence $\langle g_{k}\rangle_{g∈ω}$.

$\mathcal{L}$ becomes an MU-category with the induced structure. Clearly, if
$f_{n,∞}∘g=f_{n,∞}∘h$, then $f_{n,∞}∘g∘k=g_{n,∞}∘h∘k$, so $d(g∘k,h∘k)≤d(g,h)$.
On the other hand, for every $f\colon X\to Z$, a chosen $\mathcal{K}$-sequence
$\langle Z_{*},z_{*}\rangle$ with limit $\langle Z,z_{*,∞}\rangle$, and $m∈ω$
there is $n∈ω$ and a $\mathcal{K}$-map $f^{\prime}\colon Z_{m}\leftarrow
X_{n}$ such that $f^{\prime}∘f_{n,∞}=z_{m,∞}∘f$ by (F1). Hence, if
$f_{n,∞}∘g=f_{n,∞}∘h$, then $z_{m,∞}∘f∘g=z_{m,∞}∘f∘h$, and $f$ is $\langle
1/m,1/n\rangle$-continuous.

The induced MU-structure is independent on the choice of
$\mathcal{K}$-sequences and limit maps up to MU-isomorphism. Let $\langle
X^{\prime}_{*},f^{\prime}_{*}\rangle$ be another $\mathcal{K}$-sequence with
limit $\langle X,f^{\prime}_{*,∞}\rangle$. For every $m∈ω$ there is $n∈ω$ and
a map $u\colon X^{\prime}_{m}\leftarrow X_{n}$ such that $u∘f_{n,∞}=f_{m,∞}$
by (F1), and so for any $\mathcal{L}$-object $Y$ and any $\mathcal{L}$-maps
$g,h\colon X\leftarrow Y$ if $g≈_{1/n}h$, then $g≈^{\prime}_{1/m}h$, where
$≈^{\prime}$ comes from the alternative MU-structure. Hence,
$\operatorname{id}_{\mathcal{L}}$ is MU-continuous, and a symmetric argument
gives that $\operatorname{id}_{\mathcal{L}}$ is an MU-isomorphism.

Every $\mathcal{K}$-object $X$ is still discrete in $\mathcal{L}$. We can
choose constant identity sequence and the constant identity cone for
definition of the metric. Then for every $ε>0$ and $\mathcal{L}$-maps
$g,h\colon X\leftarrow Y$, if $g≈_{ε}h$, then $g=h$. It follows that
$\langle\mathcal{K},\mathcal{L}\rangle$ with the induced structure satisfies
(L1), (L2), (F1), (F2) since the conditions with respect to the discrete MU-
structure (with the exception of local completeness in (L1), which we have
proved separately in the induced MU-structure). Condition (C) is true as well:
the metric $d$ is defined exactly so that (C) becomes true. ∎

######  Remark 4.57 (Classical Fraïssé theory).

Let $L$ be a first-order language, let $\mathcal{L}$ be the opposite category
of all countably generated $L$-structures and all embeddings (we formally take
the opposite category so that the natively injective setting fits our
projective setting), and let $\mathcal{K}$ be the full subcategory of all
finitely generated $L$-structures. $\langle\mathcal{K},\mathcal{L}\rangle$ is
a discrete free completion, and so a free completion with respect to the
induced MU-structure on $\mathcal{L}$. (L1) and (L2) follow from the fact that
countably generated structures are exactly limits of sequences of finitely
generated structures, where the limit is essentially the union of an
increasing countable chain. (F1) is true since every finitely generated
substructure of the union of an increasing chain $⋃_{n∈ω}A_{n}$ is covered by
some $A_{n}$. (F2) is true since all $\mathcal{L}$-maps are opposite
monomorphisms. The fact that the conditions for free completion are always
true in the classical setting seems to be the reason why these conditions are
usually not spelled out explicitly. It follows that for every class
$\mathcal{F}$ of finitely generated $L$-structures (viewed as a full
subcategory of $\mathcal{K}$) we have that
$\langle\mathcal{F},σ\mathcal{F}\rangle$ is a free completion and
$σ\mathcal{F}⊆\mathcal{L}$ is full, and hence the characterization of Fraïssé
limits applies. We also recover the correspondence between homogeneous
countable $L$-structures and hereditary Fraïssé classes (Theorem 4.52).

The large homogeneity coming from the induced MU-structure is not interesting
in the classical case: a countably generated structure $U$ is homogeneous in
$σ\mathcal{F}$ if for every structure $X$ in $σ\mathcal{F}$, every embeddings
$f,g\colon X\to U$, any choice of increasing chain of finitely generated
structures $⋃_{n∈ω}X_{n}=X$, and every $n∈ω$ there is an automorphism $h\colon
U\to U$ such that $h∘g=f$ when restricted to $X_{n}$, which is the same as
homogeneity in $\langle\mathcal{F},σ\mathcal{F}\rangle$. On the other hand,
the topology on every $\mathcal{L}(X,Y)$ coming from the induced MU-structure
is the topology of pointwise convergence, which is the correct topology when
studying the automorphism group of the Fraïssé limit, e.g. in the context of
the KPT correspondence [22]. Moreover, the left uniformity of the completely
metrizable topological group $\operatorname{Aut}(U)$ is induced by the metric
coming from the induced MU-structure, while the two-sided uniformity is
complete. For more details on abstractly inducing the topology on
$\operatorname{Aut}(U)$ in the context of the KPT correspondence see [2,
Construction 2.6].

######  Remark 4.58 (Projective Fraïssé theory).

Let us precisely describe how the projective Fraïssé theory of Irwin and
Solecki [18] fits into our framework. Irwin and Solecki consider certain
epimorphisms of _topological $L$-structures_. For a first-order language $L$,
a topological $L$-structure is an $L$-structure endowed with a zero-
dimensional compact metrizable topology making the operations continuous and
the relations closed (as subsets of the corresponding products). By a
_quotient map_ of topological $L$-structures we mean a map that is both
quotient of the corresponding $L$-structures and of the corresponding compact
topological spaces, i.e. a continuous surjective $L$-homomorphism such that
every relation holding in the codomain has a witness in the domain.
Topological $L$-structures and quotient maps form an ambient category
$\mathcal{L}$.

Let $\mathcal{K}⊆\mathcal{L}$ denote the full subcategory of all finite
$L$-structures. An $\mathcal{L}$-limit of a $\mathcal{K}$-sequence is the
limit in $\mathsf{MCpt_{s}}$ with relations and operations defined
coordinatewise. $\langle\mathcal{K},\mathcal{L}\rangle$ satisfies (L1), (F1),
(F2). We have that $σ\mathcal{K}⊆\mathcal{L}$ is full and that
$\langle\mathcal{K},σ\mathcal{K}\rangle$ is a discrete free completion and so
a free completion with the induced MU-structure. (If $L$ is a relational
language, then $σ\mathcal{K}=\mathcal{L}$, but this is not true in general. By
[18, Lemma 2.5], a topological $L$-structure $X$ is a $σ\mathcal{K}$-object if
and only if every clopen partition of $X$ can be refined by partition induced
by a congruence, but this is not the case e.g. for $2^{ω}$ with the shift map
for $L$ consisting of one unary operation.) Since all $\mathcal{L}$-objects
are metrizable compacta and all $\mathcal{L}$-maps are continuous, we may view
$\mathcal{L}$ as an MU-category also as in Example 3.16. However, on
$σ\mathcal{K}$ this “native” MU-structure is the same as the induced one. Note
that Example 4.55 is a special case corresponding to the empty language.

A class $\mathcal{F}$ of topological $L$-structures can be identified with a
full subcategory $\mathcal{F}⊆\mathcal{L}$. Irwin and Solecki define a Fraïssé
limit of $\mathcal{F}$ as (using our language) an $\mathcal{L}$-object $U$
that is cofinal and homogeneous in $\langle\mathcal{F},\mathcal{L}\rangle$
with respect to the discrete MU-structure and that for every $ε>0$ admits an
$ε$-monic $\mathcal{L}$-map onto a $\mathcal{K}$-object. However, if
$\mathcal{F}⊆\mathcal{K}$, i.e. if all $\mathcal{F}$-objects are finite
structures (and so $\mathcal{F}$ is a discrete MU-category), the last
condition becomes equivalent to being a $σ\mathcal{F}$-object, and discrete
homogeneity becomes equivalent to approximate homogeneity. In this case,
$\langle\mathcal{F},σ\mathcal{F}\rangle$ is a free completion with
$σ\mathcal{F}⊆\mathcal{L}$ being full, and the two definitions of a Fraïssé
limit by Irwin–Solecki and ours coincide.

######  Example 4.59.

We consider the language with one binary relation. Let $\mathcal{I}_{Δ}$ be
the category of all finite linear graphs $\mathbb{I}_{n}$, $n∈ω$, with
quotient maps. By the general remarks above,
$\langle\mathcal{I}_{Δ},σ\mathcal{I}_{Δ}\rangle$ is a free completion. It is
easy to see that $\mathcal{I}_{Δ}$ is directed and countable. Irwin and
Solecki [18] proved that $\mathcal{I}_{Δ}$ has the amalgamation property (for
a simpler proof see [31, Remark 3.11]). So by the general theory we have the
Fraïssé limit $ℙ_{Δ}$ of $\mathcal{I}_{Δ}$ in $σ\mathcal{I}_{Δ}$. Irwin and
Solecki showed that $ℙ_{Δ}$ is the Cantor space with a closed equivalence
relation (necessarily with classes of size one and two) and that the
corresponding topological quotient is the pseudo-arc $ℙ$. This way they
characterized $ℙ$ as the unique homogeneous object in $σ\mathcal{I}$. By our
theory we have the large homogeneity of $ℙ_{Δ}$ in $σ\mathcal{I}_{Δ}$, i.e.
for every two quotient maps $f,g\colon ℙ_{Δ}\to X$ onto a
$σ\mathcal{I}_{Δ}$-object and every $ε>0$ there is an automorphism $h\colon
ℙ_{Δ}\to ℙ_{Δ}$ with $f≈_{ε}g∘h$. We may consider the full subcategory
$\mathcal{P}σ\mathcal{I}⊆σ\mathcal{I}_{Δ}$ of _arc-like pre-spaces_ , i.e. of
those $σ\mathcal{I}_{Δ}$-structures whose closed relation is an equivalence.
The operation of taking the topological quotient induces an essentially
surjective functor $\mathcal{P}σ\mathcal{I}\to σ\mathcal{I}$ by [18, Lemma
4.3, 4.5, and 4.7]. Note that $ℙ_{Δ}$ is a Fraïssé object also in the category
of arc-like pre-spaces $\mathcal{P}σ\mathcal{I}$, which is disjoint from
$\mathcal{I}_{Δ}$.

We have built our approximate Fraïssé theory on MU-categories along with the
running example of the pseudo-arc and the arc-like continua, which culminated
in Theorem 4.38. Because of a result by Russo [42], this is essentially the
only case of a class of $\mathcal{P}$-like continua for a family $\mathcal{P}$
of connected polyhedra with a Fraïssé limit. However, there are Fraïssé limits
of categories of circle-like continua with restricted classes of morphisms.
This is the content of the next section. We summarize the situation of full
polyhedral categories in the following theorem. Let $\SS$ denote a circle and
$*$ a one-point space.

######  Theorem 4.60.

Let $\mathcal{P}$ be a non-empty family of non-empty connected polyhedra,
identified with the corresponding full subcategory of $\mathsf{CPol_{s}}$, so
$σ\mathcal{P}$ is the category of all $\mathcal{P}$-like continua (we consider
families $\mathcal{P}$ up to homeomorphic copies).

  1. (i)

$σ\mathcal{P}$ has a Fraïssé object if and only if $\mathcal{P}$ has the
amalgamation property, which happens if and only if
$\mathcal{P}⊆\\{*,\mathbb{I}\\}$. The Fraïssé object is the pseudo-arc
$\mathbb{P}$, unless $\mathcal{P}=\\{*\\}$, in which case the Fraïssé object
is $*$.

  2. (ii)

$σ\mathcal{P}$ has a generic object if and only if $σ\mathcal{P}$ has a
cofinal object, which happens if and only if
$\mathcal{P}⊆\\{*,\mathbb{I},\SS\\}$. The generic object is the _universal
pseudo-solenoid $\mathbb{P}_{Π}$_ (see Section 5.2), unless
$\mathcal{P}⊆\\{*,\mathbb{I}\\}$ and we are in case (i).

  3. (iii)

There is always a generic object in $\langle\mathcal{P},σ\mathcal{P}\rangle$
and it is the pseudo-arc, unless $\mathcal{P}=\\{*\\}$.

###### Proof.

We have that $\langle\mathcal{P},σ\mathcal{P}\rangle$ is a free completion by
Theorem 4.36, so Theorem 4.15 applies. By Theorem 4.46, we have a Fraïssé
limit if and only if $\mathcal{P}$ is a Fraïssé MU-category, which reduces to
having the amalgamation property by Observation 4.47.

Claim (iii) and the positive part of (i) follow from Theorem 4.38. By a result
of Russo [42], there is no cofinal object in $σ\mathcal{P}$ unless
$\mathcal{P}⊆\\{*,\mathbb{I},\SS\\}$, hence we have the negative part of (ii).
From this and from the fact that $\\{\SS\\}$ does not have the amalgamation
property ([41, Example 1], see also Proposition 5.3 below) we have the
negative part of (i). The positive part of (ii) in the case
$\mathcal{P}⊆\\{*,\mathcal{I}\\}$ follows from (i). If $\SS∈\mathcal{P}$, then
the universal pseudo-solenoid $\mathbb{P}_{Π}$ is cofinal in $σ\mathcal{P}$ by
Corollary 5.29. Since also every continuous surjection
$\mathbb{P}_{Π}\to\mathbb{P}_{Π}$ is a near-homeomorphism (again by 5.29),
$\mathbb{P}_{Π}$ is generic in $σ\mathcal{P}$ by Observation 4.9. ∎

##  5 Circle-like continua and pseudo-solenoids

The last section is devoted to the main application of our theory in this
article. Let $\SS$ denote the unit circle in the complex plane, and let
$\mathcal{S}$ be the category of all continuous surjections of $\SS$. By
Theorem 4.36, $σ\mathcal{S}$ is the category of all circle-like continua and
all continuous surjections, and $\langle\mathcal{S},σ\mathcal{S}\rangle$ is a
free completion. Hence, by Theorem 4.15 and 4.46 there is a Fraïssé limit of
$\mathcal{S}$ in $σ\mathcal{S}$ if and only if $\mathcal{S}$ is a Fraïssé MU-
category, which reduces to the amalgamation property by Observation 4.47.
However, Rogers [41] have shown that $\mathcal{S}$ does not have the strict
amalgamation property, and the same example shows that $\mathcal{S}$ does not
have the amalgamation property (see Proposition 5.3 below). Anyway, we would
not obtain any new Fraïssé object since the only candidate for the limit is
the pseudo-arc since it is generic over $\mathcal{S}$ by Theorem 3.9. However,
it turns out that the problem with amalgamation concerns _degree zero maps_ ,
and can be fixed by restricting the set of allowed _degrees_.

Let $φ\colon\mathbb{R}\to\SS$ be the canonical surjection $x\mapsto e^{ix}$.
We endow $\SS$ with the metric induced by $φ$, that is, the inner metric of
the curve. Recall that every continuous map $f\colon\SS\to\SS$ admits a
continuous _lifting_ $\tilde{f}\colon ℝ\to ℝ$, i.e. a continuous map
satisfying $φ∘\tilde{f}=f∘φ$. The family of all such liftings is
$\\{\tilde{f}+2πk:k∈ℤ\\}$ where $\tilde{f}$ is any of them.

There is also a unique number $d∈ℤ$ such that
$\tilde{f}(x+2π)=\tilde{f}(x)+2πd$ for every $x∈ℝ$. This number is called the
_degree_ of $f$ and is denoted by $\deg(f)$. Every continuous map of nonzero
degree is surjective. We have $\deg(\operatorname{id}_{\SS})=1$ and
$\deg(f∘g)=\deg(f)⋅\deg(g)$ for every continuous maps $f,g\colon\SS\to\SS$.

######  Lemma 5.1.

Let $f,g\colon\SS\to\SS$ be continuous. If $f≈_{π}g$, then $\deg(f)=\deg(g)$.

###### Proof.

Let $\tilde{f},\tilde{g}\colon ℝ\to ℝ$ be the liftings of $f$ and $g$,
respectively, such that $\tilde{f}(0)≈_{π}\tilde{g}(0)$. Since the metric on
$\SS$ is induced by the projection $φ$, we have $d(φ(x),φ(y))=\lvert
x-y\rvert$ whenever $\lvert x-y\rvert≤π$, i.e. $φ$ is an isometry on any
interval of length $≤π$. It follows that $\tilde{f}≈_{π}\tilde{g}$ since
otherwise there would be $x∈ℝ$ such that
$\lvert\tilde{f}(x)-\tilde{g}(x)\rvert=π=d(f(φ(x)),g(φ(x)))$. We have
$2π\deg(f)=\tilde{f}(x+2π)-\tilde{f}(x)≈_{2π}\tilde{g}(x+2π)-\tilde{g}(x)=2π\deg(g)$
for every $x∈ℝ$, and hence $\deg(f)≈_{1}\deg(g)$. Since the degrees are
integers, we are done. ∎

######  Remark 5.2.

We view the monoid $ℤ$ with the multiplication as a category with one object
and endow it with the discrete metric. The facts above concerning the degree
can be summarized by saying that $\deg\colon\mathcal{S}\to ℤ$ is an MU-
functor.

######  Proposition 5.3 (based on [41, Example 1]).

Let $f\colon\SS\to\SS$ be the map $z\mapsto z^{2}$, i.e. the canonical map of
degree $2$. Let $g\colon\SS\to\SS$ be the map $z\mapsto z^{2}$ for
$\operatorname{Im}(z)≥0$ and $z\mapsto-z^{2}$ for $\operatorname{Im}(z)≤0$,
i.e. the surjection of degree $0$ corresponding to the tent map. There are no
continuous surjections $f^{\prime},g^{\prime}\colon X\to\SS$ from a connected
space $X$ such that $f^{\prime}∘f≈_{π}g^{\prime}∘g$. In particular, the
category $\mathcal{S}$ does not have the amalgamation property.

###### Proof.

Let us endow $\SS^{2}$ with the $\ell_{1}$ metric and let us put
$G:=\\{\langle x,y\rangle∈\SS^{2}:f(x)=g(y)\\}$ and $G_{ε}:=\\{\langle
x,y\rangle∈\SS^{2}:f(x)≈_{ε}g(y)\\}$ for every $ε>0$. Pairs of continuous maps
$f^{\prime},g^{\prime}\colon\SS\to\SS$ are in one-to-one correspondence with
continuous maps $h\colon\SS\to\SS^{2}$. Moreover, $f∘f^{\prime}=g∘g^{\prime}$
or $f∘f^{\prime}≈_{ε}g∘g^{\prime}$ if and only if $\operatorname{rng}(h)⊆G$ or
$\operatorname{rng}(h)⊆G_{ε}$, respectively. Also $f^{\prime}$ or $g^{\prime}$
is surjective if and only if $π_{1}[\operatorname{rng}(h)]=\SS$ or
$π_{2}[\operatorname{rng}(h)]=\SS$, respectively.

In our case, $G$ consists of two similar components separated by distance of
$π$, and the second projection of each component is just a half of the circle.
For our map $f$, it is easy to see that whenever we have $f(x)≈_{ε}y$ there is
$x^{\prime}≈_{ε/2}x$ such that $f(x^{\prime})=y$. It follows that for every
$\langle x,y\rangle∈G_{ε}$ we have $d(\langle x,y\rangle,G)<ε/2$. Hence,
$G_{π}$ is contained in the $π/2$-neighborhood of $G$, which has two
components, and the second projection of each component still does not cover
the whole circle (by one point), so there is no suitable map $h$. ∎

######  Definition 5.4.

Let $Π$ denote the set of all primes. For every $P⊆Π$ let
$\mathcal{S}_{P}⊆\mathcal{S}$ be the MU-subcategory of all $\mathcal{S}$-maps
whose degree uses only primes from $P$, i.e.
$\mathcal{S}_{P}:=\deg^{-1}[D_{P}]$ where $D_{P}$ is the set of all $0≠k∈ℤ$
such that if a prime $p$ divides $k$, then $p∈P$. For example, we have
$D_{∅}=\\{-1,1\\}$, $D_{\\{2\\}}=\\{±2^{n}:n∈ℕ_{0}\\}$, and
$D_{Π}=ℤ\setminus\\{0\\}$. Also, let $σ\mathcal{S}_{P}$ be the $σ$-closure of
$\mathcal{S}_{P}$ in $\mathsf{MCpt}$ or equivalently in $σ\mathcal{S}$ (see
Corollary 4.20).

######  Proposition 5.5.

$\langle\mathcal{S}_{P},σ\mathcal{S}_{P}\rangle$ is a free completion for
every $P⊆Π$.

###### Proof.

By Corollary 4.33, $\langle\mathcal{S}_{P},σ\mathcal{S}_{P}\rangle$ satisfies
(L1), (L2), (F2), and (C). By Corollary 4.24, it is enough to prove that
$\langle\mathcal{S}_{P},σ\mathcal{S}_{P}\rangle$ satisfies (F1) for limits. So
let $\langle X_{*},f_{*}\rangle$ and $\langle Y_{*},g_{*}\rangle$ be sequences
in $\mathcal{S}_{P}$ with limits $\langle X_{∞},f_{*,∞}\rangle$ and $\langle
Y_{∞},g_{*,∞}\rangle$ such that $X_{∞}=Y_{∞}$. Let $n∈ω$ and $0<ε<π/2$. Since
$\langle\mathcal{S},σ\mathcal{S}\rangle$ is a free completion, there is $m∈ω$
and an $\mathcal{S}$-map $f\colon X_{m}\to Y_{n}$ such that
$f∘f_{m,∞}≈_{ε}g_{n,∞}$. We show that $f$ is an $\mathcal{S}_{P}$-map, i.e.
$\deg(f)∈D_{P}$.

There is $δ>0$ such that $f$ is $\langle ε,δ\rangle$-continuous,
$n^{\prime}≥n$, and an $\mathcal{S}$-map $g\colon Y_{n^{\prime}}\to X_{m}$
such that $g∘g_{n^{\prime},∞}≈_{δ}f_{m,∞}$. It follows that
$f∘g∘g_{n^{\prime},∞}≈_{ε}f∘f_{m,∞}≈_{ε}g_{n,∞}$, and so
$f∘g≈_{2ε}g_{n,n^{\prime}}$. Since $2ε<π$, by Lemma 5.1 we have
$\deg(f)⋅\deg(g)=\deg(f∘g)=\deg(g_{n,n^{\prime}})∈D_{P}$. Finally,
$\deg(f),\deg(g)∈D_{P}$ since $D_{P}$ is closed under divisors. ∎

######  Proposition 5.6 (Rogers [41, Uniformization Theorem]).

Let $\mathcal{S}^{\prime}⊆\mathcal{S}$ denote the subcategory of all piecewise
linear maps (i.e. continuous surjections $f\colon\SS\to\SS$ such that the
restriction of the lifting $\tilde{f}\colon[0,2π]\to ℝ$ is piecewise linear).
For every $\mathcal{S}^{\prime}$-maps $f,g$ of non-zero degree there are
$\mathcal{S}^{\prime}$-maps $f^{\prime},g^{\prime}$ such that
$f∘f^{\prime}=g∘g^{\prime}$ and
$\deg(f∘f^{\prime})=\deg(g∘g^{\prime})=\operatorname{lcm}(\deg(f),\deg(g))$,
so the degrees of $f^{\prime}$ and $g^{\prime}$ are non-zero and optimal.

It follows that $\mathcal{S}^{\prime}∩\mathcal{S}_{P}$ has the strict
amalgamation property for every $P⊆Π$. Since piecewise linear surjections are
dense in all continuous surjections, and since close maps have equal degrees,
$\mathcal{S}^{\prime}∩\mathcal{S}_{P}$ is dominating in $\mathcal{S}_{P}$, and
so $\mathcal{S}_{P}$ has the amalgamation property by Observation 4.48. If we
also use Observation 4.47, we obtain the following.

######  Corollary 5.7.

$\mathcal{S}_{P}$ is a Fraïssé MU-category for every $P⊆Π$.

######  Theorem 5.8.

For every $P⊆Π$, there is a Fraïssé limit $ℙ_{P}$ of $\mathcal{S}_{P}$ in
$σ\mathcal{S}_{P}$. The continuum $ℙ_{P}$ is characterized (up to a
homeomorphism) by any of the following conditions.

  1. (i)

$ℙ_{P}$ is a projective object in
$\langle\mathcal{S}_{P},σ\mathcal{S}_{P}\rangle$.

  2. (ii)

$ℙ_{P}$ is a homogeneous object in
$\langle\mathcal{S}_{P},σ\mathcal{S}_{P}\rangle$.

  3. (iii)

$ℙ_{P}$ is a generic object in
$\langle\mathcal{S}_{P},σ\mathcal{S}_{P}\rangle$.

  4. (iv)

$ℙ_{P}$ is a projective object in $σ\mathcal{S}_{P}$.

  5. (v)

$ℙ_{P}$ is a homogeneous object in $σ\mathcal{S}_{P}$.

  6. (vi)

$ℙ_{P}$ is a generic object in $σ\mathcal{S}_{P}$.

###### Proof.

Since $\mathcal{S}_{P}$ is a Fraïssé MU-category and
$\langle\mathcal{S}_{P},σ\mathcal{S}_{P}\rangle$ is a free completion, we have
a Fraïssé limit characterized by any of the conditions in Theorem 4.15. As
with the pseudo-arc, the cofinality is trivial (see Remark 4.16), and so
$ℙ_{P}$ is characterized by any of (i), (ii), (iv), (v). From the general
theory we have uniqueness up to a $σ\mathcal{S}_{P}$-isomorphism, but
$σ\mathcal{S}_{P}$-isomorphisms are exactly homeomorphisms between
$σ\mathcal{S}_{P}$-objects from the construction of $σ$-closure. Also, the
genericity conditions (iii), (vi) are weaker than being a Fraïssé limit
(Proposition 3.25, Observation 4.9), and yet guarantee uniqueness. ∎

Recall that a hereditarily indecomposable circle-like continuum is called a
_pseudo-solenoid_. We will observe that $\mathbb{P}_{P}$ is the unique _$P$
-adic pseudo-solenoid_, but first we need to recall the notion of a _type_ of
a circle-like continuum according to degrees of maps in the corresponding
inverse sequence. Moreover, the Fraïssé limit properties of $\mathbb{P}_{P}$
are stated in terms of $σ\mathcal{S}_{P}$-maps, but which continuous
surjections are $σ\mathcal{S}_{P}$-maps? The statement that
$σ\mathcal{S}_{P}$-maps are exactly those $σ\mathcal{S}$-maps that can be
sufficiently approximated by an $\mathcal{S}_{P}$-map will be made more
precise by considering a certain extension of the notion of the degree of a
map.

###  5.1 Types of circle-like continua and continuous surjections

Let us recall the notion of _supernatural numbers_. By $ℕ$ we denote the set
of all natural numbers including $0$, and by $ℕ_{+}$ we denote the set of all
(strictly) positive natural numbers. A positive supernatural number is a
function $s\colon Π\to ℕ∪\\{∞\\}$ representing the formal product
$∏_{p∈Π}p^{s(p)}$. We denote the set of all positive supernatural numbers
$(ℕ∪\\{∞\\})^{Π}$ by $\overline{ℕ}_{+}$. The set $\overline{ℕ}_{+}$ carries a
natural multiplication defined by $(s⋅s^{\prime})(p)=s(p)+s^{\prime}(p)$ for
every $p∈Π$, and a natural order defined by $s≤s^{\prime}$ if
$s(p)≤s^{\prime}(p)$ for every $p∈Π$. In fact, the order is exactly the
divisibility order induced by the multiplication, i.e. $s≤s^{\prime}$ if and
only if there is some $t$ with $t⋅s=s^{\prime}$. Since there is a natural
infinitary addition on the set $ℕ∪\\{∞\\}$, we may define an infinitary
multiplication on $\overline{ℕ}_{+}$ by $(∏_{i∈I}s_{i})(p)=∑_{i∈I}s_{i}(p)$.
We have the monoid and divisibility order embedding $ℕ_{+}\to\overline{ℕ}_{+}$
mapping $n$ to the supernatural number $s$ corresponding to the prime
decomposition of $n$, i.e. $n=∏_{p∈Π}p^{s(p)}$. Because of this, we view
$ℕ_{+}⊆\overline{ℕ}_{+}$. Together, we have for example
$2^{∞}∈\overline{ℕ}_{+}$, $s=∏_{p∈Π}p^{s(p)}$ for every $s∈\overline{ℕ}_{+}$,
$1∈\overline{ℕ}$ is the divisibility minimum, and
$∏_{p∈Π}p^{∞}∈\overline{ℕ}_{+}$ is the divisibility maximum. Finally, we
define the set of all supernatural numbers
$\overline{ℕ}:=\overline{ℕ}_{+}∪\\{0\\}$, and we extend the multiplication by
$s⋅0=0$ and the order by $s≤0$ for every $s∈\overline{ℕ}$. We also have the
extended embedding $ℕ\to\overline{ℕ}$.

We consider an equivalence $\sim$ on $\overline{ℕ}$ where $0$ is equivalent
only to itself and where for $s,s^{\prime}∈\overline{ℕ}_{+}$ we have $s\sim
s^{\prime}$ if and only if $s^{-1}(∞)=(s^{\prime})^{-1}(∞)$ and
$\\{p∈Π:s(p)≠s^{\prime}(p)\\}$ is finite. We denote the $\sim$-equivalence
class of $s∈\overline{ℕ}$ by $[s]$, but we also identify an equivalence class
with its representative when convenient, e.g. we may view
$1=[1]=ℕ_{+}∈\overline{ℕ}/{\sim}$. It is easy to see that $\sim$ is indeed an
equivalence and that it is moreover a congruence: for every $s\sim s^{\prime}$
and $t\sim t^{\prime}$ we have $s⋅t\sim s^{\prime}⋅t^{\prime}$. Hence, we have
the induced multiplication on $\overline{ℕ}/{\sim}$, and the divisibility
order induced by the multiplication is induced also by the original
divisibility order, i.e. for $S,S^{\prime}∈\overline{ℕ}/{\sim}$ there is some
$T∈\overline{ℕ}/{\sim}$ such that $T⋅S=S^{\prime}$ if and only if there are
some $s∈S$ and $s^{\prime}∈S^{\prime}$ with $s≤s^{\prime}$, in which case
there is $t∈\overline{ℕ}$ with $t⋅s=s^{\prime}$ and moreover $t⋅S⊆S^{\prime}$.
The order is indeed antisymmetric since the equivalence is convex: if
$s≤s^{\prime}$ and $s\sim s^{\prime}$, then also $s\sim t$ for every
$t∈[s,s^{\prime}]$, and if $S≤S^{\prime}≤S$, then there is $s∈S$ and
$t,t^{\prime}∈\overline{ℕ}$ such that $t⋅s∈S^{\prime}$ and $t^{\prime}⋅t⋅s∈S$,
so $s≤t⋅s≤t^{\prime}⋅t⋅s\sim s$ and $s\sim t⋅s$ and $S=S^{\prime}$.

######  Definition 5.9.

For $S,S^{\prime}∈\overline{ℕ}_{+}/{\sim}$ we say that a map $f\colon S\to
S^{\prime}$ is a _positive multiplication_ if for some $t∈\overline{ℕ}_{+}$ we
have $f(s)=t⋅s$ for every $s∈S$. Similarly, for
$S,S^{\prime}∈\overline{ℕ}/{\sim}$ we say that a map $f\colon S∪\\{0\\}\to
S^{\prime}∪\\{0\\}$ is a _multiplication_ if for some $t∈\overline{ℕ}$ we have
$f(s)=t⋅s$ for every $s∈S$. Note that a multiplication $S∪\\{0\\}\to
S^{\prime}∪\\{0\\}$ is either the extension of a positive multiplication $S\to
S^{\prime}$ by $0\mapsto 0$ (in which case we also call it positive), or it is
a constant zero map (i.e. the multiplication by $0$).

######  Observation 5.10.

A multiplication $f\colon S∪\\{0\\}\to S^{\prime}∪\\{0\\}$ mapping $s∈S$ to
$s^{\prime}∈S^{\prime}∪\\{0\\}$ exists if and only if $s≤s^{\prime}$, and it
is unique. We know that $s≤s^{\prime}$ if and only if there is
$t∈\overline{ℕ}$ such that $t⋅s=s^{\prime}$, and every such $t$ induces a
multiplication since $\sim$ is a congruence. If $s=0$ (i.e. $S=\\{0\\}$) and
$s^{\prime}≠0$, then there is no solution to the equation $t⋅s=s^{\prime}$. If
$s≠0$ and $s^{\prime}=0$, then the unique solution is $t=0$. If
$s=s^{\prime}=0$, then every $t∈\overline{ℕ}$ represents $f$. Otherwise, the
multiplication $f$ is positive and $t(p)$ for $p∈Π\setminus s^{-1}(∞)$ is
uniquely determined by $t⋅s=s^{\prime}$, while $t(p)$ for $p∈s^{-1}(∞)$ can be
arbitrary. If $t⋅s=t^{\prime}⋅s$, then $t(p)=t^{\prime}(p)$ for every
$p∈Π\setminus s^{-1}(∞)$, and so $t⋅u=t^{\prime}⋅u$ for every $u∈S$ since
$s^{-1}(∞)=u^{-1}(∞)$.

######  Definition 5.11.

We define the category of _positive types_ $\mathcal{T}_{+}$, where the set of
objects is $\overline{ℕ}_{+}/{\sim}$ and a $\mathcal{T}_{+}$-map $S\to
S^{\prime}$ is a positive multiplication. Composition is the actual
composition and identities are the actual identities. We also define the
category of _types_ $\mathcal{T}$ by freely adding a zero object to
$\mathcal{T}_{+}$, i.e. there is one new object $0$, and between every two
$\mathcal{T}$-objects there is one new map, a zero map, such that any
composition with a zero map is a zero map. Equivalently, we may view
$\mathcal{T}$-objects as $\overline{ℕ}/{\sim}$ and $\mathcal{T}$-maps $S\to
S^{\prime}$ as multiplications $S∪\\{0\\}\to S^{\prime}∪\\{0\\}$.

For every $P⊆Π$ let $P^{∞}$ denote the supernatural number $∏_{p∈P}p^{∞}$. The
types of the form $[P^{∞}]$ together with $[0]$ are exactly the types with a
$≤$-smallest representative. Sometimes we denote types by their
representatives and call $[0]$ the “type $0$”, $[1]=S_{∅}$ the “type $1$”, and
$[Π^{∞}]$ the “type $∞$”. Types are ordered as $\overline{ℕ}/{\sim}$. Clearly,
$1$ is the minimum, $0$ the maximum, and $[Π^{∞}]$ the maximum among positive
types.

Note that for every positive type $S$ and $s∈S$, the $\mathcal{T}_{+}$-maps
from $S$ are in a one-to-one correspondence with positive supernatural numbers
$s^{\prime}≥s$: by Observation 5.10 for every $s^{\prime}≥s$ there is a unique
positive multiplication $[s]\to[s^{\prime}]$ mapping $s$ to $s^{\prime}$. (In
particular, $\mathcal{T}_{+}$-maps from $1$ are in one-to-one correspondence
with $\overline{ℕ}_{+}$: every $s∈\overline{ℕ}_{+}$ induces the multiplication
by $s\colon[1]\to[s]$.) It follows that there is a $\mathcal{T}_{+}$-map $S\to
S^{\prime}$ if and only is $S≤S^{\prime}$. It also follows that $∞$ is the
terminal object in $\mathcal{T}_{+}$ since $Π^{∞}$ is the unique
representative of $[Π^{∞}]$.

######  Notation 5.12.

Let $\langle X_{*},u_{*}\rangle$ be an $\mathcal{S}$-sequence with a
$σ\mathcal{S}$-limit $\langle X_{∞},u_{*,∞}\rangle$. For every $n_{0}∈ω$ we
write
$\lvert\deg(u_{n_{0},∞})\rvert:=∏_{n≥n_{0}}\lvert\deg(u_{n})\rvert∈\overline{ℕ}$.
Similarly, for every $\mathcal{S}$-map $f\colon X_{n_{0}}\to Y$ we write
$\lvert\deg(f∘u_{n_{0},∞})\rvert:=\lvert\deg(f)\rvert⋅\lvert\deg(u_{n_{0},∞})\rvert$.

######  Lemma 5.13.

Let $\langle X_{*},u_{*}\rangle$, $\langle Y_{*},v_{*}\rangle$ be sequences in
$\mathcal{S}$ and let $\langle X_{∞},u_{*,∞}\rangle$, $\langle
Y_{∞},v_{*,∞}\rangle$ be their limits in $σ\mathcal{S}$ such that
$X_{∞}=Y_{∞}$. If $f\colon X_{m_{0}}\to Y_{n_{0}}$ is an $\mathcal{S}$-map
such that $f∘u_{m_{0},∞}≈_{π}v_{n_{0},∞}$, then
$\lvert\deg(f∘u_{m_{0},∞})\rvert=\lvert\deg(v_{n_{0},∞})\rvert$.

###### Proof.

First, for every $m_{1}≥m_{0}$ there is arbitrarily large $n_{1}≥n_{0}$ and an
$\mathcal{S}$-map $g\colon Y_{n_{1}}\to X_{m_{1}}$ such that
$f∘u_{m_{0},m_{1}}∘g≈_{π}v_{n_{0},n_{1}}$, and so
$\lvert\deg(f∘u_{m_{0},m_{1}}∘g)\rvert=\lvert\deg(v_{n_{0},n_{1}})\rvert$ by
Lemma 5.1. To find such map $g$, we take $ε>0$ such that
$f∘u_{m_{0},∞}≈_{π-ε}v_{n_{0},∞}$ and $δ>0$ such that $f∘u_{m_{0},m_{1}}$ is
$\langle ε,δ\rangle$-continuous. By (F1) there is $g\colon Y_{n_{1}}\to
X_{m_{1}}$ with $n_{1}≥n_{0}$ arbitrarily large such that
$g∘v_{n_{1},∞}≈_{δ}u_{m_{1},∞}$, and so
$f∘u_{m_{0},m_{1}}∘g∘v_{n_{1},∞}≈_{ε}f∘u_{m_{0},∞}≈_{π-ε}v_{n_{0},∞}$. We get
$f∘u_{m_{0},m_{1}}∘g≈_{π}v_{n_{0},n_{1}}$ since $v_{n_{1},∞}$ is a surjection.

We have
$\lvert\deg(f∘u_{m_{0},m_{1}})\rvert≤\lvert\deg(f∘u_{m_{0},m_{1}}∘g)\rvert=\lvert\deg(v_{n_{0},n_{1}})\rvert≤\lvert\deg(v_{n_{0},∞})\rvert$
for arbitrary $m_{1}≥m_{0}$, and so
$\lvert\deg(f∘u_{m_{0},∞})\rvert≤\lvert\deg(v_{n_{0},∞})\rvert$.

Since we could have chosen $δ<π$ above, we can apply the already proved half
of the lemma to $g\colon Y_{n_{1}}\to X_{m_{1}}$, i.e. we have
$\lvert\deg(g∘v_{n_{1},∞})\rvert≤\lvert\deg(u_{m_{1},∞})\rvert$. Hence we
obtain the other inequality:
$\lvert\deg(v_{n_{0},∞})\rvert=\lvert\deg(f∘u_{m_{0},m_{1}}∘g∘v_{n_{1},∞})\rvert≤\lvert\deg(f∘u_{m_{0},∞})\rvert$.
∎

######  Construction 5.14.

We define a contravariant _type functor_ $T\colon σ\mathcal{S}\to\mathcal{T}$.
For every circle-like continuum $X$ we define its type
$T(X)∈\overline{ℕ}/{\sim}$ as follows. Pick any $\mathcal{S}$-sequence
$\langle X_{*},u_{*}\rangle$ with limit $\langle X,u_{*,∞}\rangle$. If
$\deg(u_{n})=0$ for infinitely many $n∈ω$, we put $T(X):=0$. Otherwise, we
take any $n_{0}∈ω$ such that $\deg(u_{n})≠0$ for every $n≥n_{0}$ and put
$T(X):=\lvert\deg(u_{n_{0},∞})\rvert$ viewed as an element of
$\overline{ℕ}/{\sim}$. We say that $T(X)$ is the _type_ of $X$ as well as the
_type_ of $\langle X_{*},u_{*}\rangle$.

For every continuous surjection $f\colon X\to Y$ between circle-like continua
we take any $\mathcal{S}$-sequences $\langle X_{*},u_{*}\rangle$, $\langle
Y_{*},v_{*}\rangle$ with limits $\langle X,u_{*,∞}\rangle$, $\langle
Y,v_{*,∞}\rangle$, and any $\mathcal{S}$-map $f_{0}\colon X_{m_{0}}\to
Y_{n_{0}}$ such that $v_{n_{0},∞}∘f≈_{π}f_{0}∘u_{m_{0},∞}$ and such that
$s:=\lvert\deg(v_{n_{0},∞})\rvert∈T(Y)$ and
$s^{\prime}:=\lvert\deg(u_{m_{0},∞})\rvert∈T(X)$ (i.e. each of them is
positive if possible), and we let $T(f)$ be the unique multiplication
$T(Y)∪\\{0\\}\to T(X)∪\\{0\\}$ that maps $s$ to
$\lvert\deg(f_{0})\rvert⋅s^{\prime}$.

###### Proof.

We need to prove that everything is well-defined and that $T$ is indeed a
functor. First, $T(X)$ does not depend on the choice of the sequence $\langle
X_{*},u_{*}\rangle$ and $n_{0}∈ω$. Let $T(X,u_{*})$ denote the type obtained
from the sequence $\langle X_{*},u_{*}\rangle$. Clearly,
$\lvert\deg(u_{n_{0},∞})\rvert\sim\lvert\deg(u_{n_{1},∞})\rvert$ for every
$n_{1}≥n_{0}$ if $\deg(u_{n_{0},n_{1}})≠0$, so $T(X,u_{*})$ is well-defined.
Let $\langle X^{\prime}_{*},u^{\prime}_{*}\rangle$ be another
$\mathcal{S}$-sequence with limit $\langle X,u^{\prime}_{*,∞}\rangle$. We show
that $T(X,u_{*})≤T(X,u^{\prime}_{*})$, and so by a symmetrical argument they
are equal. By (F1), for every $n^{\prime}_{0}∈ω$ there is $n_{0}∈ω$ and an
$\mathcal{S}$-map $g\colon X_{n_{0}}\to X^{\prime}_{n^{\prime}_{0}}$ such that
$g∘u_{n_{0},∞}≈_{π}u^{\prime}_{n^{\prime}_{0},∞}$ and so by Lemma 5.13 we have
$\lvert\deg(g∘u_{n_{0},∞})\rvert=\lvert\deg(u^{\prime}_{n^{\prime}_{0},∞})\rvert$.
If the right-hand side is non-zero for some $n^{\prime}_{0}$, we have
$T(X,u_{*})∋\lvert\deg(u_{n_{0},∞})\rvert≤\lvert\deg(u^{\prime}_{n^{\prime}_{0},∞})\rvert∈T(X,u^{\prime}_{*})$.
Otherwise, $T(X,u_{*})≤0=T(X,u^{\prime}_{*})$.

Let $T(f,u_{*},v_{*},f_{0})$ denote the definition of $T(f)$ for fixed
sequences $u_{*}$, $v_{*}$ and a fixed $\mathcal{S}$-map $f_{0}\colon
X_{m_{0}}\to Y_{n_{0}}$. By Observation 5.10, $T(f,u_{*},v_{*},f_{0})$ is a
well-defined multiplication $T(Y)∪\\{0\\}\to T(X)∪\\{0\\}$ if
$\lvert\deg(v_{n_{0},∞})\rvert≤\lvert\deg(f_{0}∘u_{m_{0},∞})\rvert$. We have
$f_{0}∘u_{m_{0},∞}≈_{π-ε}v_{n_{0},∞}∘f$ for some $ε>0$. By (F1), for every
$n_{1}≥n_{0}$ there is $m_{1}≥m_{0}$ and $f_{1}\colon X_{m_{1}}\to Y_{n_{1}}$
with $f_{1}∘u_{m_{1},∞}≈_{δ}v_{n_{1},∞}∘f$ for $δ>0$ such that
$v_{n_{0},n_{1}}$ is $\langle ε,δ\rangle$-continuous. Hence, we have
$v_{n_{0},n_{1}}∘f_{1}∘u_{m_{1},∞}≈_{ε}v_{n_{0},∞}∘f≈_{π-ε}f_{0}∘u_{m_{0},∞}$,
and so by Lemma 5.13,
$\lvert\deg(v_{n_{0},n_{1}})\rvert≤\lvert\deg(v_{n_{0},n_{1}}∘f_{1}∘u_{m_{1},∞})\rvert=\lvert\deg(f_{0}∘u_{m_{0},∞})\rvert$.
Since $n_{1}$ was arbitrary, we are done.

Next, we show that $T(f,u_{*},v_{*},f_{0})$ does not depend on $f_{0}$, and so
can be denoted by $T(f,u_{*},v_{*})$. Note that is this trivial if $T(X)=0$ or
$T(Y)=0$. Otherwise, let $f_{1}\colon X_{m_{1}}\to Y_{n_{1}}$ be another
suitable $\mathcal{S}$-map. Without loss of generality, $n_{0}≤n_{1}$. We have
$f_{i}∘u_{m_{i},∞}≈_{π-ε_{i}}v_{n_{i},∞}∘f$ for some $ε_{i}>0$ and
$i∈\\{0,1\\}$. Let $δ∈(0,ε_{1})$ be such that $v_{n_{0},n_{1}}$ is $\langle
ε_{0},δ\rangle$-continuous. By (F1) there is $m≥m_{0},m_{1}$ and an
$\mathcal{S}$-map $g\colon X_{m}\to Y_{n_{1}}$ such that
$g∘u_{m,∞}≈_{δ}v_{n_{1},∞}∘f$. Since $δ<ε_{1}$, we have
$g∘u_{m,∞}≈_{π}f_{1}∘u_{m_{1},∞}$. Hence,
$\lvert\deg(g)\rvert=\lvert\deg(f_{1}∘u_{m_{1},m})\rvert$ and
$T(f,u_{*},v_{*},g)=T(f,u_{*},v_{*},f_{1})$. Since $v_{n_{0},n_{1}}$ is
$\langle ε_{0},δ\rangle$-continuous, we have
$f_{0}∘u_{m_{0},∞}≈_{π-ε_{0}}v_{n_{0},∞}∘f≈_{ε_{0}}v_{n_{0},n_{1}}∘g∘u_{m,∞}$.
Hence,
$\lvert\deg(f_{0}∘u_{m_{0},m})\rvert=\lvert\deg(v_{n_{0},n_{1}}∘g)\rvert$ and
$T(f,u_{*},v_{*},f_{0})=T(f,u_{*},v_{*},f_{0}∘u_{m_{0},m})=T(f,u_{*},v_{*},v_{n_{0},n_{1}}∘g)$
since all three multiplications map $s$ to
$\lvert\deg(f_{0})\rvert⋅s^{\prime}$. Also, $T(f,u_{*},v_{*},g)$ maps
$\lvert\deg(v_{n_{1},∞})\rvert$ to $\lvert\deg(g∘u_{m_{1},∞})\rvert$, and so
maps $\lvert\deg(v_{n_{0},∞})\rvert$ to
$\lvert\deg(v_{n_{0},n_{1}}∘g∘u_{m_{1},∞})\rvert$, so
$T(f,u_{*},v_{*},g)=T(f,u_{*},v_{*},v_{n_{0},n_{1}}∘g)$. Together, we have
$T(f,u_{*},v_{*},f_{0})=T(f,u_{*},v_{*},g)=T(f,u_{*},v_{*},f_{1})$.

Now we show that $T(f,u_{*},v_{*})$ contravariantly preserves composition,
i.e. if $g\colon Y\to Z$ is another $σ\mathcal{S}$-map and $\langle
Z_{*},w_{*}\rangle$ is an $\mathcal{S}$-sequence with limit $\langle
Z,w_{*,∞}\rangle$, we have
$T(g∘f,u_{*},w_{*})=T(f,u_{*},v_{*})∘T(g,v_{*},w_{*})$. We pick $k_{0}∈ω$ such
that $t:=\lvert\deg(w_{k_{0},∞})\rvert∈T(Z)$. There is $n_{0}∈ω$ and an
$\mathcal{S}$-map $g_{0}\colon Y_{n_{0}}\to Z_{k_{0}}$ such that
$g_{0}∘v_{n_{0},∞}≈_{π/2}w_{k_{0},∞}∘g$. By possibly replacing $n_{0}$ with
$n^{\prime}_{0}$ and $g_{0}$ with $g_{0}∘v_{n_{0},n^{\prime}_{0}}$ we may
arrange that $t^{\prime}:=\lvert\deg(v_{n_{0},∞})\rvert∈T(Y)$. There is
$m_{0}∈ω$ such that $t^{\prime\prime}:=\lvert\deg(u_{m_{0},∞})\rvert∈T(X)$ and
an $\mathcal{S}$-map $f_{0}\colon X_{m_{0}}\to Y_{n_{0}}$ with
$f_{0}∘u_{m_{0},∞}≈_{δ}v_{n_{0},∞}∘f$ where $δ∈(0,π)$ is such that $g_{0}$ is
$\langle π/2,δ\rangle$-continuous. We have
$g_{0}∘f_{0}∘u_{m_{0},∞}≈_{π/2}g_{0}∘v_{n_{0},∞}∘f≈_{π/2}w_{k_{0},∞}∘g∘f$.
Hence, $T(g∘f,u_{*},w_{*})$ maps $t$ to
$\lvert\deg(g_{0}∘f_{0})\rvert⋅t^{\prime\prime}$. If $T(g,v_{*},w_{*})$ is the
multiplication by $r$ and $T(f,u_{*},v_{*})$ is the multiplication by
$r^{\prime}$, we have
$r^{\prime}⋅r⋅t=r^{\prime}⋅\lvert\deg(g_{0})\rvert⋅t^{\prime}=\lvert\deg(g_{0}∘f_{0})\rvert⋅t^{\prime\prime}$,
so $T(g∘f,u_{*},w_{*})$ is the multiplication by $r^{\prime}⋅r$.

Finally, we observe that
$T(\operatorname{id}_{X},u_{*},u^{\prime}_{*})=\operatorname{id}_{T(X)}$ for
every sequences $u_{*},u^{\prime}_{*}$ with limit $X$. It will follow that
$T(f,u_{*},v_{*})$ does not depend on the choice of $u_{*}$ and $v_{*}$ since
for every alternative sequences $u^{\prime}_{*}$ and $v^{\prime}_{*}$ we would
have
$T(f,u^{\prime}_{*},v^{\prime}_{*})=T(\operatorname{id}_{X},u^{\prime}_{*},u_{*})∘T(f,u_{*},v_{*})∘T(\operatorname{id}_{Y},v_{*},v^{\prime}_{*})=T(f,u_{*},v_{*})$.
It will also follow that $T(\operatorname{id}_{X})=\operatorname{id}_{T(X)}$.
So let $\langle X_{*},u_{*}\rangle$ and $\langle
X^{\prime}_{*},u^{\prime}_{*}\rangle$ be $\mathcal{S}$-sequences with limits
$\langle X,u_{*,∞}\rangle$ and $\langle X,u^{\prime}_{*,∞}\rangle$. There is
an $\mathcal{S}$-map $f_{0}\colon X_{n_{0}}\to X^{\prime}_{n^{\prime}_{0}}$
such that $f_{0}∘u_{n_{0},∞}≈_{π}u^{\prime}_{n^{\prime}_{0},∞}$ and
$\lvert\deg(u_{n_{0},∞})\rvert∈T(X)$ and
$\lvert\deg(u^{\prime}_{n^{\prime}_{0},∞})\rvert∈T(Y)$. Hence,
$T(\operatorname{id}_{X},u_{*},u^{\prime}_{*})$ maps
$\lvert\deg(u^{\prime}_{n^{\prime}_{0},∞})\rvert$ to
$\lvert\deg(f_{0}∘u_{n_{0},∞})\rvert$. But by Lemma 5.13 we have
$\lvert\deg(u^{\prime}_{n^{\prime}_{0},∞})\rvert=\lvert\deg(f_{0}∘u_{n_{0},∞})\rvert$,
and so $T(\operatorname{id}_{X},u_{*},u^{\prime}_{*})=\operatorname{id}_{X}$.
∎

######  Observation 5.15.

Let $I\colon\mathcal{S}\to σ\mathcal{S}$ denote the inclusion functor, and let
$J\colon ℤ\to\mathcal{T}$ be the functor mapping every $k∈ℤ$ to the
multiplication by $\lvert k\rvert\colon[1]\to[1]$. Then we have $T∘I=J∘\deg$,
so $T$ may be viewed as an extension of the degree from $\mathcal{S}$ to
$σ\mathcal{S}$. To see this, for an $\mathcal{S}$-map $f$, we just consider
the constant $\mathcal{S}$-sequence consisting of $\operatorname{id}_{\SS}$,
so we have $T(\SS)=[1]$ and $T(f)(1)=\lvert\deg(f)\rvert$.

Note that it is natural to forget the sign of the degree of an
$\mathcal{S}$-map. To consider the degree of a continuous surjection $f\colon
X\to Y$ between _some_ circles (as opposed to _the_ circle $\SS$), we would
need to choose homeomorphisms $g\colon\SS\to X$ and $h\colon Y\to\SS$ and to
put $\deg(f):=\deg(h∘f∘g)$. However, different choices of homeomorphisms $g,h$
may change the sign of the degree.

######  Observation 5.16.

The category $\mathcal{T}$ is rigid and skeletal, i.e. if $f\colon S\to
S^{\prime}$ is a $\mathcal{T}$-isomorphism, then $S=S^{\prime}$ and
$f=\operatorname{id}$. This follows from Observation 5.10 because for $s∈S$ we
have $s≤f(s)≤f^{-1}(f(s))=s$. It follows that for a homeomorphism $f\colon
X\to Y$ of circle-like continua we have $T(X)=T(Y)$ and
$T(f)=\operatorname{id}$.

######  Remark 5.17.

Classification of solenoids and pseudo-solenoids is often stated in terms of
the first _Čech cohomology_ $H^{1}$. Let us make a clear connection between
$H^{1}$ and our type functor $T$. The functor $H^{1}\colon
σ\mathcal{S}\to\mathsf{Ab}$ can be equivalently described as $[-,\SS]$, i.e.
for every circle-like continuum $X$, $[X,\SS]$ denotes the set of all
continuous surjections $X\to\SS$ up to homotopy, endowed with the group
structure induced by the pointwise multiplication in $\SS⊆\mathbb{C}$, see
e.g. [23, page 42].

Moreover, if $\langle X_{*},u_{*}\rangle$ is an $\mathcal{S}$-sequence with
limit $\langle X,u_{*,∞}\rangle$, applying $[-,\SS]$ preserves the limit. To
see this, every $σ\mathcal{S}$-map $X\to\SS$ approximately factorizes through
$u_{*}$ by (F1), and every sufficiently close maps to $\SS$ are homotopic. On
the other hand, for every $\mathcal{S}$-maps $f,g\colon X_{n}\to\SS$ such that
$f∘u_{n,∞}$ and $g∘u_{n,∞}$ are homotopic, there are maps
$f∘u_{n,∞}=h_{0}≈_{π}h_{1}≈_{π}\cdots≈_{π}h_{k}=g∘u_{n,∞}\colon X\to\SS$,
which can be approximately factorized through $u_{*}$, and so there is $m≥n$
such that $f∘u_{n,m}$ and $g∘u_{n,m}$ are homotopic.

Since the degree induces an isomorphism $[\SS,\SS]\to ℤ$, the abelian groups
$H^{1}(X)$ are limits of sequences of group endomorphisms of the additive
group $ℤ$, and so are isomorphic to subgroups of $ℚ$. Together, we may view
$H^{1}\colon σ\mathcal{S}\to\operatorname{Sub}(ℚ)$, where
$\operatorname{Sub}(ℚ)$ denotes the category of all subgroups of $ℚ$ and group
homomorphisms.

Now for every subgroup $\\{0\\}≠G⊆ℚ$ we may consider $n:=\min(G∩ℤ_{+})$, and
for every $q∈G$ we put $s(q,G):=\sup_{\overline{ℕ}}\\{k∈ℕ_{+}:q/k∈G\\}$. Then
we have $G=\\{nl/k:l∈ℤ,k∈ℕ_{+},k≤s(n,G)\\}$. Pick $q∈G\setminus\\{0\\}$ and
put $Q(G):=[s(q,G)]∈\overline{ℕ}/{\sim}$. For every group homomorphism
$f\colon G\to G^{\prime}$ we have $s(q,G)≤s(f(q),G^{\prime})$, and we may let
$Q(f)\colon Q(G)\to Q(G^{\prime})$ be the supernatural multiplication
$s(q,G)\mapsto s(f(q),G^{\prime})$. In fact, $Q(G)$ and $Q(f)$ do not depend
on $q$, and the construction gives a functor
$Q\colon\operatorname{Sub}(ℚ)\to\mathcal{T}$. It turns out that $T=Q∘H^{1}$,
and $Q$ is the “rigid skeletal modification” of $\operatorname{Sub}(ℚ)$, i.e.
$f\colon G\to G^{\prime}$ is an isomorphism if and only if
$Q(G)=Q(G^{\prime})$ and $Q(f)=\operatorname{id}$.

######  Observation 5.18.

The type functor $T\colon σ\mathcal{S}\to\mathcal{T}$ is MU-continuous when
$\mathcal{T}$ is endowed with the $0$-$1$ discrete MU-structure (and $T$ is
viewed as a covariant functor from $σ\mathcal{S}$ to the opposite category of
$\mathcal{T}$). That means, for every $σ\mathcal{S}$-object $Y$ there is $δ>0$
such that for every $σ\mathcal{S}$-object $X$ and every $σ\mathcal{S}$-maps
$f,g\colon X\to Y$ such that $f≈_{δ}g$ we have $T(f)=T(g)$. Moreover, if
$Y=\SS$, the choice $δ=π$ works.

###### Proof.

This is trivial if $T(Y)=0$. Otherwise, by (L2) there is an
$\mathcal{S}$-sequence $\langle Y_{*},v_{*}\rangle$ with limit $\langle
Y,v_{*,∞}\rangle$ such that $\lvert\deg(v_{0,∞})\rvert∈T(Y)$. We take $δ>0$
such that $v_{0,∞}$ is $\langle π,δ\rangle$-continuous. If $Y=\SS$, we can
have $v_{0,∞}=\operatorname{id}_{\SS}$ and $δ=π$. Now let $f,g\colon X\to Y$
be $σ\mathcal{S}$-maps such that $f≈_{δ}g$. We have
$v_{0,∞}∘f≈_{π-ε}v_{0,∞}∘g$ for some $ε∈(0,π)$. Let $\langle
X_{*},u_{*}\rangle$ be an $\mathcal{S}$-sequence with limit $\langle
X,u_{*,∞}\rangle$. By (F1) there is an $\mathcal{S}$-map $f_{0}\colon X_{n}\to
Y_{0}$ such that $f_{0}∘u_{n,∞}≈_{ε}v_{0,∞}∘f≈_{π-ε}v_{0,∞}∘g$. Hence, $f_{0}$
is a witness for both $T(f)$ and $T(g)$. ∎

######  Remark 5.19.

The formalism of types allows to comfortably express several known results.
For example,

  1. (i)

a circle-like continuum $X$ is also arc-like if and only if $T(X)=0$,

  2. (ii)

a circle-like continuum $X$ can be embedded into the plane if and only if
$T(X)∈\\{0,1\\}$.

The fact that $T(X)=0$ implies arc-like is proved for example in [16, Theorem
2]. We can also view this directly by observing that a continuous surjection
$f\colon\SS\to\SS$ has degree $0$ if and only if it factorizes through
$\mathbb{I}$. This is easily seen by considering a lifting $\tilde{f}\colon
ℝ\to ℝ$, which is periodic if and only if $\deg(f)=0$. Similarly, if $\langle
X_{*},u_{*}\rangle$ is an $\mathcal{S}$-sequence with limit $\langle
X,u_{*,∞}\rangle$, and $\langle Y_{*},v_{*}\rangle$ is an
$\mathcal{I}$-sequence with limit $\langle X,v_{*,∞}\rangle$, by (F1) and (F2)
of $\langle\mathsf{CPol_{s}},\mathsf{MCpt_{s}}\rangle$, for every $m∈ω$ there
are continuous surjections $g\colon Y_{n}\to X_{m}$ and $h\colon
X_{m^{\prime}}\to Y_{n}$ such that $g∘h≈_{π}u_{m,m^{\prime}}$, so
$\deg(u_{m,m^{\prime}})=\deg(f∘g)=0$.

Claim (ii) follows from the results of Bing [7], who however uses the
formalism of refining covers instead of inverse limits. Every arc-like
continuum is embeddable into the plane by [7, Theorem 5], see also [38,
Theorem 12.20]. Every circle-like continuum of type $1$ is embeddable into the
plane by [7, Theorem 4]. It follows from [7, Theorem 3] that a circle-like
continuum of non-zero type in the plane has type $1$. See also [23, 4.1].

Now we can characterize $σ\mathcal{S}_{P}$ in terms of $T$.

######  Proposition 5.20.

Let $P⊆Π$.

  1. (i)

A circle-like continuum $X$ is a $σ\mathcal{S}_{P}$-object if and only if
$T(X)≤[P^{∞}]$.

  2. (ii)

A continuous surjection $f\colon X\to Y$ from a $σ\mathcal{S}$-object to a
$σ\mathcal{S}_{P}$-object is a $σ\mathcal{S}_{P}$-map if and only if $T(f)$ is
the multiplication by $t≤P^{∞}$, i.e. $T(f)$ is allowed to increase only
coordinates from $P$.

Together, $σ\mathcal{S}_{P}=T^{-1}[\mathcal{T}_{P}]$ where
$\mathcal{T}_{P}⊆\mathcal{T}$ is the subcategory consisting of types
$≤[P^{∞}]$ and multiplications by numbers $≤P^{∞}$.

###### Proof.

(i) is clear since $T(X)≤S_{P}$ if and only if $X$ is a limit of some
$\mathcal{S}$-sequence $\langle X_{*},u_{*}\rangle$ such that
$\lvert\deg(u_{n_{0},∞})\rvert≤P^{∞}$ for some $n_{0}∈ω$.

For (ii) let $\langle X_{*},u_{*}\rangle$ and $\langle Y_{*},v_{*}\rangle$ be
$\mathcal{S}$-sequences with $σ\mathcal{S}$-limits $\langle X,u_{*,∞}\rangle$
and $\langle Y,v_{*,∞}\rangle$, and let $n∈ω$. Since $Y$ is an
$σ\mathcal{S}_{P}$-object, we may assume that $v_{*}$ is an
$\mathcal{S}_{P}$-sequence, and so $v_{n,∞}$ is a $σ\mathcal{S}_{P}$-map and
$\lvert\deg(v_{n,∞})\rvert≤P^{∞}$. For every $ε∈(0,π)$ there is an
$\mathcal{S}$-map $f_{0}\colon X_{m}\to Y_{n}$ such that
$f_{0}∘u_{m,∞}≈_{ε}v_{n,∞}∘f$. Let $t∈\overline{ℕ}$ be such $T(f)$ is the
multiplication by $t$. We have
$t⋅\lvert\deg(v_{n_{0},∞})\rvert=\lvert\deg(f_{0}∘u_{m_{0},∞})\rvert$.

Now if $f∈σ\mathcal{S}_{P}$, we may have chosen $u_{*}$ that is an
$\mathcal{S}_{P}$-sequence and $f_{0}$ that is an $\mathcal{S}_{P}$-map, so
$t≤\lvert\deg(f_{0}∘u_{m,∞})\rvert≤P^{∞}$. On the other hand, if $t≤P^{∞}$,
then $\lvert\deg(f_{0}∘u_{m,∞})\rvert=t⋅\lvert\deg(v_{n,∞})\rvert≤P^{∞}$, and
so $f_{0}$ is an $\mathcal{S}_{P}$-map and $u_{*}$ is an
$\mathcal{S}_{P}$-sequence after $m$. Hence,
$v_{n,∞}∘f≈_{ε}f_{0}∘u_{m,∞}∈σ\mathcal{S}_{P}$. Since $ε>0$ was arbitrarily
small, $v_{n,∞}∘f∈σ\mathcal{S}_{P}$ by the definition of $σ$-closure.
Similarly, since $n$ was arbitrary, $f∈σ\mathcal{S}_{P}$. ∎

Let us summarize properties of $σ\mathcal{S}_{P}$-maps in the following
corollary.

######  Corollary 5.21.

Let $P⊆Π$.

  1. (i)

Every $σ\mathcal{S}_{P}(X,Y)⊆σ\mathcal{S}(X,Y)$ is clopen. More precisely, for
every $σ\mathcal{S}$-object $Y$ there is $δ>0$ such that for every
$σ\mathcal{S}$-object $X$, $σ\mathcal{S}_{P}(X,Y)$ is $δ$-separated from its
complement.

  2. (ii)

We have $g∘f∈σ\mathcal{S}_{P}$ if and only if $g∈σ\mathcal{S}_{P}$ and
$f∈σ\mathcal{S}_{P}$ for every $σ\mathcal{S}$-maps $f\colon X\to Y$ and
$g\colon Y\to Z$ between $σ\mathcal{S}$-objects.

  3. (iii)

A $σ\mathcal{S}$-map $f\colon X\to Y$ between $σ\mathcal{S}_{P}$-objects is a
$σ\mathcal{S}_{P}$-map if and only if $g∘f$ is a $σ\mathcal{S}_{P}$-map for
some $σ\mathcal{S}_{P}$-map $g\colon Y\to\SS$.

  4. (iv)

For every $σ\mathcal{S}$-maps $f,g\colon X\to\SS$ such that $f≈_{π}g$ we have
$f∈σ\mathcal{S}_{P}$ if and only if $g∈σ\mathcal{S}_{P}$.

  5. (v)

For every $σ\mathcal{S}$-map $f\colon X\to Y$ between
$σ\mathcal{S}_{P}$-objects, every $\mathcal{S}_{P}$-sequences $\langle
X_{*},u_{*}\rangle$, $\langle Y_{*},v_{*}\rangle$ with limits $\langle
X,u_{*,∞}\rangle$, $\langle Y,v_{*,∞}\rangle$ and every $\mathcal{S}$-map
$f_{0}\colon X_{m_{0}}\to Y_{n_{0}}$ such that
$f_{0}∘u_{m_{0},∞}≈_{π}v_{n_{0},∞}∘f$ we have $f∈σ\mathcal{S}_{P}$ if and only
if $f_{0}∈\mathcal{S}_{P}$.

  6. (vi)

For every $σ\mathcal{S}_{P}$-object $X$ there is $δ>0$ such that every
$δ$-monic $σ\mathcal{S}$-map $f\colon X\to\SS$ is a $σ\mathcal{S}_{P}$-map.

###### Proof.

We will use Proposition 5.20 without noting. Claims (i) and (iv) follow from
the continuity of $T$ (Observation 5.18). Claim (ii) follows since if $T(g)$
and $T(f)$ are the multiplications by $s$ and $t$, we have $s⋅t≤P^{∞}$ if and
only if both $s,t≤P^{∞}$. Claim (iii) follows from (ii). Claim (v) follows
since $T(f)$ maps $\lvert\deg(v_{n_{0},∞})\rvert∈T(Y)$ to
$\lvert\deg(f_{0}∘u_{m_{0},∞})\rvert$ and
$\lvert\deg(u_{m_{0},∞})\rvert≤P^{∞}$. To prove (vi) let $\langle
X_{*},u_{*}\rangle$ be a $\mathcal{S}_{P}$-sequence with a
$σ\mathcal{S}_{P}$-limit $\langle X,u_{*,∞}\rangle$. By Proposition 4.34 there
is $δ>0$ such that for every $δ$-monic $σ\mathcal{S}$-map $f\colon X\to\SS$
there is an $\mathcal{S}$-map $g$ such that $u_{0,∞}≈_{π}g∘f$. Since
$u_{0,∞}∈σ\mathcal{S}_{P}$, we have $g∘f∈σ\mathcal{S}_{P}$ by (iv) and
$f∈σ\mathcal{S}_{P}$ by (ii). ∎

So to show that a $σ\mathcal{S}$-map $f\colon X\to Y$ between
$σ\mathcal{S}_{P}$-objects is a $σ\mathcal{S}_{P}$-map, we may either use (v)
directly, or use (iii) to reduce the problem to the case $Y=\SS$. A
$σ\mathcal{S}$-map $f\colon X\to\SS$ is a $σ\mathcal{S}_{P}$-map if it is
$π$-close to a map of the form $u_{n_{0},∞}$ for a sequence $u_{*}$ with limit
$X$ such that $\lvert\deg(u_{n_{0},∞})\rvert≤P^{∞}$ or $π$-close to a map of
the form $h∘g$ where $g\colon X\to\SS$ is a $δ$-monic $σ\mathcal{S}$-map for
sufficiently small $δ>0$ and $h\colon\SS\to\SS$ is an $\mathcal{S}_{P}$-map.

This description is somewhat similar to the definition of maps of _positive
rank_ in the discrete setting of Irwin [19, Definition 4.5]. See also the
related definition of _positive rank_ of a continuous surjection $f\colon X\to
Y$ between circle-like continua [19, Definition 4.8].

###  5.2 Pseudo-solenoids as Fraïssé limits

For every $\mathcal{T}$-object $S$ there is a unique (up to a homeomorphism)
hereditarily indecomposable circle-like continuum $X$ of type $S$. We shall
call it the _$S$ -adic pseudo-solenoid_. For $P⊆Π$ the $[P^{∞}]$-adic pseudo-
solenoid is called just _$P$ -adic pseudo-solenoid_, and for $P=\\{p\\}$ just
_$p$ -adic pseudo-solenoid_. The $[1]$-adic pseudo-solenoid is also called the
_pseudo-circle_. (But note that Rogers [41] calls every pseudo-solenoid of
non-zero type a pseudo-circle.) The $Π$-adic pseudo-solenoid is also called
the _universal pseudo-solenoid_.

For $S=0$, the $S$-adic pseudo-solenoid is the pseudo-arc since by Remark 5.19
it is also arc-like, and Bing’s theorem [5] and Theorem 4.38 apply. The
pseudo-circle was constructed by Bing [5] and its uniqueness was proved by
Fearnley [12, Theorem 6.3]. Later, Fearnley [13, Theorem 3.2] extended the
previous results by showing that two pseudo-solenoids are homeomorphic if and
only if they have the same type. As Irwin [19, page 9] notes, Fearnley’s
definition of equivalent fundamental sequences needs to be fixed by allowing
to cut their finite initial segments. The same classification based on
sequences of primes up to an equivalence applies to _solenoids_ as well as
proved earlier by McCord [36, page 198].

Note that the existence of the $S$-adic pseudo-solenoid for
$S∈\overline{ℕ}/{\sim}$ follows from Theorem 2.17 since it is enough to build
a crooked sequence of type $S$. There exists an $ε$-crooked $\mathcal{S}$-map
of degree $1$ for every $ε>0$ by a construction similar to Construction 2.22.
By composing with the map $z\mapsto z^{n}$ (or a tent map for $n=0$) we get an
$ε$-crooked $\mathcal{S}$-map of degree $n$. Then we proceed as in
Construction 2.18 to obtain a crooked sequence.

We finally show the following.

######  Proposition 5.22.

The Fraïssé limit $ℙ_{P}$ of $\mathcal{S}_{P}$ in $σ\mathcal{S}_{P}$ is the
$P$-adic pseudo-solenoid for every $P⊆Π$.

###### Proof.

Since $ℙ_{P}$ is a $σ\mathcal{S}_{P}$-object, we have $T(ℙ_{P})≤[P^{∞}]$ by
Proposition 5.20. But since every $\mathcal{S}_{P}$-sequence $\langle
X_{*},u_{*}\rangle$ with limit $\langle ℙ_{P},u_{*,∞}\rangle$ is Fraïssé, for
every $p∈P$ and $n∈ω$ there are $\mathcal{S}_{P}$-maps $f\colon
X_{n}\leftarrow\SS$ and $g\colon\SS\leftarrow X_{m}$ such that $\deg(f)=p$ and
$f∘g≈_{π}u_{n,m}$. Hence, $p$ divides $u_{n,m}$ and $T(ℙ_{P})=[P^{∞}]$.
Similarly, for every $n∈ω$ and $ε>0$ there is an $ε$-crooked
$\mathcal{S}_{P}$-map $f\colon X_{n}\leftarrow\SS$ and an
$\mathcal{S}_{P}$-map $g\colon\SS\leftarrow X_{m}$ such that
$f∘g≈_{ε}u_{n,m}$, and so $u_{n,m}$ is $3ε$-crooked. Hence $\langle
X_{*},u_{*}\rangle$ is a crooked sequence, $ℙ_{P}$ is a hereditarily
indecomposable circle-like continuum of type $[P^{∞}]$, and the classification
applies. ∎

As with the pseudo-arc and Bing’s theorem (Remark 3.28), it is possible to
view Fearnley’s classification of pseudo-solenoids as coming from uniqueness
of a generic object. In place of the crookedness factorization theorem (2.24)
we use the following version of a theorem by Kawamura [20, Theorem 7].

######  Theorem 5.23.

For an $\mathcal{S}_{Π}$-sequence $\langle X_{*},f_{*}\rangle$, the following
conditions are equivalent.

  1. (i)

$\langle X_{*},f_{*}\rangle$ is a crooked sequence.

  2. (ii)

For every $ε>0$ and $\mathcal{S}_{Π}$-map $g\colon X_{n}\leftarrow\SS$ such
that $\deg(g)$ divides $\deg(f_{n,n^{\prime}})$ for some $n^{\prime}≥n$
(equivalently, $\lvert\deg(g)\rvert≤\lvert\deg(f_{n,∞})\rvert$) there is an
$\mathcal{S}_{Π}$-map $h\colon\SS\leftarrow X_{n^{\prime\prime}}$ for some
$n^{\prime\prime}≥n^{\prime}$ such that $g∘h≈_{ε}f_{n,n^{\prime}}$. In other
words, $\langle X_{*},f_{*}\rangle$ is absorbing in $\mathcal{S}_{Π}$, but
only for maps of relatively bounded degree.

###### Proof.

As we have already seen, in order for the sequence to be crooked, it is enough
to absorb all $\mathcal{S}$-maps of degree $1$. The other implication follows
from [20, Theorem 7], where we replace the limit being hereditarily
indecomposable by the sequence being crooked (via Theorem 2.17) and we replace
the condition AEOP (which corresponds to projectivity) by its weaker version
corresponding to the absorption property. ∎

######  Corollary 5.24.

Every two crooked $\mathcal{S}$-sequences $\langle X_{*},f_{*}\rangle$,
$\langle Y_{*},g_{*}\rangle$ of the same type $S∈\overline{ℕ}/{\sim}$ have
homeomorphic limits, and hence there is a unique $S$-adic pseudo-solenoid.

###### Proof.

For $S=0$, the limit is arc-like and hence the pseudo-arc. Otherwise, we can
cut initial parts of the sequences and assume that
$\lvert\deg(f_{0,∞})\rvert=\lvert\deg(g_{0,∞})\rvert≠0$. With the use of the
previous theorem we construct a back and forth sequence as in the proof of
Proposition 3.25. Let $0<ε_{0}≤π$, let $n_{0}=m_{0}=0$, and let $h_{0}\colon
Y_{n_{0}}\leftarrow X_{m_{0}}$ be the identity. Suppose we have defined
$0<ε_{k}≤π$ and $h_{2k}\colon Y_{n_{k}}\leftarrow X_{m_{k}}$ such that
$\lvert\deg(h_{2k}∘f_{m_{k},∞})\rvert=\lvert\deg(g_{n_{k},∞})\rvert$. Since
$\lvert\deg(h_{2k})\rvert≤\lvert\deg(g_{n_{k},∞})\rvert$, by the previous
theorem there is $n_{k+1}>n_{k}$ and an $\mathcal{S}$-map $h_{2k+1}\colon
X_{m_{k}}\leftarrow Y_{n_{k+1}}$ such that
$h_{2k}∘h_{2k+1}≈_{ε_{k}/2}g_{n_{k},n_{k+1}}$. Since $ε_{k}≤π$, we have
$\deg(h_{2k}∘h_{2k+1})=\deg(g_{n_{k},n_{k+1}})$, and so