# On the proalgebraic fundamental group of topological spaces and amalgamated
products of affine group schemes

Christopher Deninger  and  Michael Wibmer  Christopher Deninger, Mathematical
Institute, University of Münster, Einsteinstr. 62, 48149 Münster, Germany,
https://www.uni-muenster.de/Arithm/deninger/index.html c.deninger@uni-
muenster.de Michael Wibmer, Institute of Analysis and Number Therory, Graz
University of Technology, Kopernikusgasse 24, 8010 Graz, Austria,
https://sites.google.com/view/wibmer<EMAIL_ADDRESS>Dedicated to
Massimo Bertolini on the occation of his 60th birthday

###### Abstract.

The proalgebraic fundamental group of a connected topological space $X$,
recently introduced by the first author, is an affine group scheme whose
representations classify local systems of finite-dimensional vector spaces on
$X$.

In this article, we further develop the theory of the proalgebraic fundamental
group, in particular, we establish homotopy invariance and a Seifert-van
Kampen theorem. To facilitate the latter, we study amalgamated free product of
affine group schemes. We also compute the proalgebraic fundamental group of
the arithmetically relevant Kucharcyzk-Scholze spaces and compare it to the
motivic Galois group.

###### 2020 Mathematics Subject Classification:

14L15, 11G09, 18M25, 14C15

Christopher Deninger: Funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy EXC
2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure and the CRC
878 Groups, Geometry & Actions
Michael Wibmer: Supported by the Lise Meitner grant M 2582-N32 of the Austrian
Science Fund FWF

In [KS18], Kucharczyk and Scholze introduced the étale fundamental group
$\pi^{\operatorname{et}}_{1}(X,x)$ of a pointed connected topological space
$(X,x)$, using Grothendieck’s formalism of Galois categories. It is a
profinite group which classifies the finite coverings of $X$. For path-
connected, locally path-connected and semi-locally simply connected spaces
$X$, the group $\pi^{\operatorname{et}}_{1}(X,x)$ is the profinite completion
of the ordinary fundamental group $\pi_{1}(X,x)$. For a field $F$ containing
the maximal cyclotomic extension $\mathbb{Q}(\mu_{\infty})$ of $\mathbb{Q}$,
Kucharczyk and Scholze defined conneted topological spaces
$\mathfrak{X}_{F}(\mathbb{C})$ and $X_{F}$ whose étale fundamental groups can
be identified with the absolute Galois group of $F$. These spaces are in
general not (locally) path-connected and are rather wild compared to
$CW$-complexes for example; hence the need for the introduction of
$\pi^{\operatorname{et}}_{1}(X,x)$. One could hope to realize not only the
Galois group of a field $F$, but even the motivic Galois group of $F$, which
for motives over $F$ with coefficients in a field $k$ is a $k$-group i.e. an
affine group scheme over $k$. The maximal pro-étale quotient of the motivic
Galois group is the ordinary Galois group viewed as a $k$-group. For these
reasons, in [Den] a fundamental $k$-group $\pi_{k}(X,x)$ for connected pointed
topological spaces $(X,x)$ was introduced and studied. It is defined using the
Tannakian formalism [DM82], as the $k$-group whose finite dimensional
representations classify the local systems $\mathcal{E}$ of finite-dimensional
$k$-vector spaces on $X$ with respect to the fibre functor
$\mathcal{E}\mapsto\mathcal{E}_{x}$. It is shown in [Den] that the maximal
pro-étale quotient of $\pi_{k}(X,x)$ is $\pi^{\operatorname{et}}_{1}(X,x)$ as
a $k$-group. For well behaved topological spaces as above, $\pi_{k}(X,x)$ is
the proalgebraic completion of $\pi_{1}(X,x)$. Moreover, using ideas of Nori
and a theorem of Deligne [Del] on isomorphisms of fibre functors, a
replacement for the ordinary universal covering space was constructed in [Den]
if $k$ is algebraically closed. Only the beginnings of the theory of the
proalgebraic fundamental group $\pi_{k}(X,x)$ were established in [Den]. The
most glaring omissions were discussions of homotopy invariance and of a
Seifert-van Kampen theorem. Moreover the proof in [Den] that $\pi_{k}(X,x)$ is
always reduced was not based on a general Tannakian criterion but on a precise
study of the special situation. Finally, the proalgebraic fundamental groups
of the Kucharczyk-Scholze spaces $\mathfrak{X}_{F}(\mathbb{C})$ and $X_{F}$
were not calculated, although this is a necessary first step to find
topological spaces with $\pi_{k}$ related to motivic Galois groups. In the
present paper, we clarify all these items.

For the Seifert-van Kampen theorem, we construct and study amalgamated free
products of $k$/̄groups. It is known [Por15] that the category of commutative
Hopf algebras over $k$ is complete. In particular, it has fibre products and
so amalgamated free products of $k$-groups exist. However, this existence
result offers no insight into the structure and the algebraic properties of
amalgamated free products of $k$-groups. Here we offer two constructions of
amalgamated free products of $k$/̄groups. The first construction is based on
the formalism of Tannakian categories applied to fibre products of categories.
It has the advantage that it leads to a proof of the Seifert-van Kampen
theorem for the proalgebraic fundamental group rather directly. The second
construction is based on an explicit, albeit quite involved construction with
Hopf algebras. It has the advantage that it allows us to deduce properties of
the amalgamated free product from properties of the factors. For example, we
show that the amalgamated free product of two geometrically reduced $k$-groups
is geometrically reduced.

We conclude the introduction with an overview of the article. In section 1 we
construct the amalgamated free product of $k$-groups using the formalism of
Tannakian categories applied to fibre products of categories. In section 2 we
present an explicit construction for the Hopf algebras of free amalgamated
products of $k$-groups and use this to derive several properties of
amalgamated free products. The proof in section 3 of the Seifert-van Kampen
theorem for the proalgebraic fundamental group follows immediately from our
categorial construction of the amalgamated free product of $k$-groups. In
section 4 we prove Tannakian criteria for an affine group scheme over a
perfect field $k$ of positive characteristic to be (geometrically) reduced and
even stronger to be perfect. Although these criteria are not stated
explicitely in [Cou20], all the necessary ideas can be found there. As an
application we show that the proalgebraic fundamental group $\pi_{k}(X,x)$ of
a connected pointed topological space $(X,x)$ is always perfect. This improves
the result of [Den] asserting that $\pi_{k}(X,x)$ is reduced with an easier
and more conceptual proof. In the last section 5 we first prove homotopy
invariance of $\pi_{k}(X,x)$ for all fields $k$. The proof requires some care
since we make no assumptions on the topologial space $X$ apart from being
connected. We then recall the construction of the spaces
$\mathfrak{X}_{F}(\mathbb{C})$ and $X_{F}$ for fields
$F\supset\mathbb{Q}(\mu_{\infty})$ in [KS18] and compute $\pi_{k}$ along the
same lines as $\pi^{\operatorname{et}}_{1}$ in [KS18]. Whereas
$\pi^{\operatorname{et}}_{1}$ vanishes if $F$ is algebraically closed this is
not true for $\pi_{k}$. We find that $\pi_{k}(X_{\overline{F}},\overline{x})$
is the identity component $\pi_{k}(X_{F},x)^{0}$ which turns out to be
commutative. The pro-unipotent part of $\pi_{k}(X_{F},x)^{0}$ is related to
the pro-unipotent part of the motivic Galois group for motives over $F$ with
coefficients in $k\supset\mathbb{Q}$. For the pro-reductive quotients there
does not seem to be a relation. This shows again that as already noted in the
introduction to [KS18] the spaces are not yet the optimal ones (in [KS18] it
was noted that the Steinberg relations do not hold in the rational cohomology
of $X_{F}$).

The authors are grateful to Andy Magid and Hans Porst for helpful comments.
The second author is grateful for the invitation to the Cluster of Excellency,
Mathematics Münster, where this work originated.

Conventions. Throughout this article $k$ is a field. All schemes are assumed
to be over $k$ unless the contrary is indicated. All rings and algebras are
assumed to be unital and commutative. In particular, all Hopf algebras are
assumed to be commutative. Unadorned tensor products of $k$/̄vector spaces are
understood to be over $k$. For simplicity, we use the term “$k$-group” for
“affine group scheme over $k$”. For a $k$-group $G$ we denote its Hopf algebra
of global sections by $k[G]$. A morphism $\phi\colon G\to H$ of $k$-groups is
a quotient map if it is faithfully flat (equivalently, the dual map
$\phi^{*}\colon k[H]\to k[G]$ is injective).

##  1\. Fibre products of Tannakian categories

In this section we discuss fibre products of neutralized Tannakian categories.
These fibre products can be used to establish the existence of amalgamated
free products of $k$-groups (Corollary 1.12) and they play a crucial role in
the proof of our Seifert-van Kampen theorem for the proalgebraic fundamental
group (Theorem 3.2).

###  1.1. Fibre products of categories

While the product category of two categories is a well-known construction, the
fibre product of two categories may be less familiar. We recall the definition
from [Sta23, Tag 003O, Example 4.31.3] or [Bas68, Chapter VII, §3].

Let $F\colon\mathcal{A}\to\mathcal{C}$ and $G\colon\mathcal{B}\to\mathcal{C}$
be functors. The _fibre product_ $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$
is the category defined as follows: Objects are triples $(A,B,c)$, where $A,B$
are objects of $\mathcal{A}$, $\mathcal{B}$ and $c\colon F(A)\to G(B)$ is an
isomorphism. A morphism $(A,B,c)\to(A^{\prime},B^{\prime},c^{\prime})$ in
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ is a pair $(a,b)$ of morphisms
$a\colon A\to A^{\prime}$, $b\colon B\to B^{\prime}$ such that

$\textstyle{F(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F(a)}$$\scriptstyle{c}$$\textstyle{F(A^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{\prime}}$$\textstyle{G(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G(b)}$$\textstyle{G(B^{\prime})}$

commutes. The composition of two morphisms
$(a,b)\colon(A,B,c)\to(A^{\prime},B^{\prime},c^{\prime})$ and
$(a^{\prime},b^{\prime})\colon(A^{\prime},B^{\prime},c^{\prime})\to(A^{\prime\prime},B^{\prime\prime},c^{\prime\prime})$
in $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ is
$(a^{\prime}a,b^{\prime}b)$. If $\mathcal{C}$ is the trivial category with one
object and one morphism (the identity), then there are unique functors $F$ and
$G$, and
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}=\mathcal{A}\times\mathcal{B}$ is
the product of $\mathcal{A}$ and $\mathcal{B}$.

######  Example 1.1.

For an (abstract) group $G$, let $\operatorname{Rep}(G)$ denote the category
of finite dimensional $k$-linear representations of $G$. Let $\phi_{1}\colon
H\to G_{1}$ and $\phi_{2}\colon H\to G_{2}$ be morphisms of groups. We then
have restriction functors
$\operatorname{Res}_{\phi_{1}}\colon\operatorname{Rep}(G_{1})\to\operatorname{Rep}(H)$
and
$\operatorname{Res}_{\phi_{2}}\colon\operatorname{Rep}(G_{2})\to\operatorname{Rep}(H)$,
which we can use to form the fibre product
$\operatorname{Rep}(G_{1})\times_{\operatorname{Rep}(H)}\operatorname{Rep}(G_{2})$.
The free product $G_{1}*_{H}G_{2}$ of $G_{1}$ and $G_{2}$ with amalgamation
over $H$, comes with a commutative diagram

$\textstyle{G_{1}*_{H}G_{2}}$$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\textstyle{G_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{2}}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\scriptstyle{\phi_{2}}$

Define a functor

$F\colon\operatorname{Rep}(G_{1}*_{H}G_{2})\to\operatorname{Rep}(G_{1})\times_{\operatorname{Rep}(H)}\operatorname{Rep}(G_{2}),\
V\rightsquigarrow(\operatorname{Res}_{i_{1}}(V),\operatorname{Res}_{i_{2}}(V),\operatorname{id}_{V})$

by $F(f)=(\operatorname{Res_{i_{1}}(f)},\operatorname{Res_{i_{2}}(f)})$ for a
morphism $f\colon V\to W$ in $\operatorname{Rep}(G_{1}*_{H}G_{2})$. A
$k$-linear map $V\to W$ between $G_{1}*_{H}G_{2}$-representation is a
$G_{1}*_{H}G_{2}$-morphism if and only if it is a $G_{1}$/̄morphism and a
$G_{2}$-moprhism. Thus $F$ is fully faithful. To see that $F$ is essentially
surjective, consider an object $(V_{1},V_{2},c)$ of
$\operatorname{Rep}(G_{1})\times_{\operatorname{Rep}(H)}\operatorname{Rep}(G_{2})$.
We can use the $k$-linear isomorphism $c\colon V_{1}\to V_{2}$ to define a
$G_{2}$-representation on $V_{1}$. Explicitly, we have
$g_{2}(v_{1})=c^{-1}(g_{2}(c(v_{1})))$ for $v_{1}\in V_{1}$ and $g_{2}\in
G_{2}$. Because $c$ is an $H$-morphism, an $h\in H$ acts in this new
representation on $V_{1}$ in the same fashion as it would act through the
original $G_{1}$ representation on $V_{1}$. That is, the morphisms
$G_{1}\to\operatorname{GL}(V_{1})$ and $G_{2}\to\operatorname{GL}(V_{1})$ have
the same restriction to $H$. The universal property of $G_{1}*_{H}G_{2}$
yields a morphism $G_{1}*_{H}G_{2}\to\operatorname{GL}(V_{1})$, i.e., $V_{1}$
is a $G_{1}*_{H}G_{2}$-representation with $\operatorname{Res_{i_{1}}}(V_{1})$
the original $G_{1}$-representation on $V_{1}$ and
$\operatorname{Res_{i_{2}}}(V_{1})\simeq V_{2}$ as $G_{2}$-representation via
$c$. So
$F(V_{1})=(V_{1},\operatorname{Res_{i_{2}}}(V_{1}),\operatorname{id}_{V_{1}})$
is isomorphic to $(V_{1},V_{2},c)$ via the isomorphism
$(\operatorname{id}_{V_{1}},c)$ in
$\operatorname{Rep}(G_{1})\times_{\operatorname{Rep}(H)}\operatorname{Rep}(G_{2})$.
Thus $F$ is an equivalence of categories. Note that for the trivial group $H$,
the category $\operatorname{Rep}H$ is trivial and the free product $G_{1}\ast
G_{2}$ corresponds to
$\operatorname{Rep}(G_{1})\times\operatorname{Rep}(G_{2})$.

######  Example 1.2.

For a topological space $X$, let $\operatorname{Sh}(X)$ denote the category of
sheaves of abelian groups on $X$. Let $U_{1},U_{2}$ be open subset of $X$ such
that $X=U_{1}\cup U_{2}$. We have restriction functors
$\operatorname{Sh}(U_{1})\to\operatorname{Sh(U_{1}\cap U_{2})},\
\mathcal{F}\rightsquigarrow\mathcal{F}|_{U_{1}\cap U_{2}}$ and
$\operatorname{Sh}(U_{2})\to\operatorname{Sh(U_{1}\cap U_{2})},\
\mathcal{F}\rightsquigarrow\mathcal{F}|_{U_{1}\cap U_{2}}$, which we can use
to form the fibre product
$\operatorname{Sh}(U_{1})\times_{\operatorname{Sh}(U_{1}\cap
U_{2})}\operatorname{Sh}(U_{2})$. Basic sheaf theory ([Sta23, Tag 00AK])
implies that

$\operatorname{Sh(X)}\to\operatorname{Sh}(U_{1})\times_{\operatorname{Sh}(U_{1}\cap
U_{2})}\operatorname{Sh}(U_{2}),\
\mathcal{F}\rightsquigarrow(\mathcal{F}|_{U_{1}},\mathcal{F}|_{U_{2}},\operatorname{id}_{\mathcal{F}|_{U_{1}\cap
U_{2}}})$

is an equivalence of categories. Note that since $\mathcal{F}(\emptyset)=0$
for any sheaf of abelian groups, $\operatorname{Sh}(\emptyset)$ is the trivial
category. So in case $U_{1}\cap U_{2}$ is empty, the above construction yields
the product category.

We return to the discussion of the general case. We have a functor
$P_{1}\colon\mathcal{A}\times_{\mathcal{C}}\mathcal{B}\to\mathcal{A}$ given by
$P_{1}(A,B,c)=A$ and $P_{1}(a,b)=a$. Similarly,
$P_{2}\colon\mathcal{A}\times_{\mathcal{C}}\mathcal{B}\to\mathcal{B}$ is given
by $P_{2}(A,B,c)=B$ and $P_{2}(a,b)=b$. The diagram

$\textstyle{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{1}}$$\scriptstyle{P_{2}}$$\textstyle{\mathcal{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{\mathcal{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{\mathcal{C}}$

$2$-commutes in the sense that the functors $FP_{1}$ and $GP_{2}$ are
isomorphic via an isomorphism $\psi\colon FP_{1}\simeq GP_{2}$. In fact, for
any object $(A,B,c)$ of $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ we have
an isomorphism

$\psi_{(A,B,c)}\colon(FP_{1})(A,B,c)=F(A)\xrightarrow{c}G(B)=(GP_{2})(A,B,c).$

To state the ($2$-categorical) universal property satisfied by the fibre
product $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$, we formaly introduce
$2$-commutative diagrams as follows. A _$2$ -commutative diagram_ (with
respect to $F\colon\mathcal{A}\to\mathcal{C}$ and
$G\colon\mathcal{B}\to\mathcal{C}$) is a quadruple
$(\mathcal{D},Q_{1},Q_{2},\alpha)$, where
$Q_{1}\colon\mathcal{D}\to\mathcal{A}$ and
$Q_{1}\colon\mathcal{D}\to\mathcal{B}$ are functors and $\alpha\colon
FQ_{1}\simeq GQ_{2}$ is an isomorphism of functors. A morphism
$(\mathcal{D},Q_{1},Q_{2},\alpha)\to(\mathcal{D}^{\prime},Q^{\prime}_{1},Q^{\prime}_{2},\alpha^{\prime})$
of $2$-commutative diagrams is a triple $(H,\beta_{1},\beta_{2})$, with
$H\colon\mathcal{D}\to\mathcal{D}^{\prime}$ a functor and $\beta_{1}\colon
Q_{1}\to Q_{1}^{\prime}H$ and $\beta_{2}\colon Q_{2}\to Q_{2}^{\prime}H$
isomorphisms of functors, such that

$\textstyle{FQ_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{F}\star\beta_{1}}$$\scriptstyle{\alpha}$$\textstyle{FQ_{1}^{\prime}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\prime}\star\operatorname{id}_{H}}$$\textstyle{GQ_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{G}\star\beta_{2}}$$\textstyle{GQ_{2}^{\prime}H}$

commutes. An isomorphism between two morphisms $(H,\beta_{1},\beta_{2})$,
$(H^{\prime},\beta_{1}^{\prime},\beta_{2}^{\prime})$ (with the same source and
target) is an isomorphism $\theta\colon H\to H^{\prime}$ such that

$\textstyle{Q_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\beta_{1}^{\prime}}$$\textstyle{Q_{1}^{\prime}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{Q_{1}^{\prime}}\star\theta}$$\textstyle{Q_{1}^{\prime}H^{\prime}}$
and
$\textstyle{Q_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{2}}$$\scriptstyle{\beta_{2}^{\prime}}$$\textstyle{Q_{2}^{\prime}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{Q_{2}^{\prime}}\star\theta}$$\textstyle{Q_{2}^{\prime}H^{\prime}}$

commute. The fibre product of categories satisfies the following
($2$-categorical) universal property.

######  Lemma 1.3.

Let $F\colon\mathcal{A}\to\mathcal{C}$ and $G\colon\mathcal{B}\to\mathcal{C}$
be functors. For every $2$-commutative diagram
$(\mathcal{D},Q_{1},Q_{2},\alpha)$ there exists a morphism
$(H,\beta_{1},\beta_{2})\colon(\mathcal{D},Q_{1},Q_{2},\alpha)\to(\mathcal{A}\times_{\mathcal{C}}\mathcal{B},P_{1},P_{2},\psi)$
of $2$/̄commutative diagrams. If
$(H^{\prime},\beta_{1}^{\prime},\beta_{2}^{\prime})$ is another such morphism,
then there exists a unique isomorphism $\theta$ between
$(H,\beta_{1},\beta_{2})$ and
$(H^{\prime},\beta_{1}^{\prime},\beta_{2}^{\prime})$.

###### Proof.

This is [Sta23, Tag 02X9]. For what follows, it will be helpful to have
explicit formulas for $(H,\beta_{1},\beta_{2})$ and $\theta$. On objects let
$H$ be given by $H(D)=(Q_{1}(D),Q_{2}(D),\alpha_{D})$ and for a morphism
$f\colon D\to D^{\prime}$ in $\mathcal{D}$ set
$H(f)=(Q_{1}(f),Q_{2}(f))\colon(Q_{1}(D),Q_{2}(D),\alpha_{D})\to(Q_{1}(D^{\prime}),Q_{2}(D^{\prime}),\alpha_{D^{\prime}})$.
As $P_{1}H=Q_{1}$, $\beta_{1}\colon Q_{1}\simeq P_{1}H$ can be chosen to be
the identity. Similarly, $\beta_{2}\colon Q_{2}\simeq P_{2}H$ is the identity.

For another morphism
$(H^{\prime},\beta^{\prime}_{1},\beta^{\prime}_{2})\colon(\mathcal{D},Q_{1},Q_{2},\alpha)\to(\mathcal{A}\times_{\mathcal{C}}\mathcal{B},P_{1},P_{2},\psi)$
of $2$/̄commutative diagrams, the commutativity of

$\textstyle{FQ_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{F}\star\beta^{\prime}_{1}}$$\scriptstyle{\alpha}$$\textstyle{FP_{1}H^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi\star\operatorname{id}_{H^{\prime}}}$$\textstyle{GQ_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{G}\star\beta^{\prime}_{2}}$$\textstyle{GP_{2}H^{\prime}}$

shows that for any object $D$ of $\mathcal{D}$ we have a morphism

$\theta_{D}=(\beta^{\prime}_{1,D},\beta^{\prime}_{2,D})\colon
H(D)=(Q_{1}(D),Q_{2}(D),\alpha_{D})\longrightarrow
H^{\prime}(D)=(P_{1}(H^{\prime}(D)),P_{2}(H^{\prime}(D)),\alpha^{\prime}_{D})$

in $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$. ∎

Before considering fibre products of Tannakian categories, we need to a take a
brief look at fibre products of abelian categories.

######  Lemma 1.4.

If $F\colon\mathcal{A}\to\mathcal{C}$ and $G\colon\mathcal{B}\to\mathcal{C}$
are exact additive functors between abelian categories, then
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ is an abelian category.

###### Proof.

Let us begin by defining the addition on the hom-sets of
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$. Let
$(a,b),(a^{\prime},b^{\prime})\colon(A,B,c)\to(A^{\prime},B^{\prime},c^{\prime})$
be two morphisms in $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$. Adding the
identities $G(b)c=c^{\prime}F(a)$ and $G(b^{\prime})c=c^{\prime}F(a^{\prime})$
and using the additivity of $F$ and $G$ yields
$G(b+b^{\prime})c=c^{\prime}F(a+a^{\prime})$. So we can define
$(a,b)+(a^{\prime},b^{\prime})=(a+a^{\prime},b+b^{\prime})$. This defines the
structure of an abelian group on the hom-sets such that composition is
bilinear, i.e., $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ is a pre-additive
category.

A zero object of $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ is given by
$(0_{\mathcal{A}},0_{\mathcal{B}},0)$, where $0_{\mathcal{A}}$ and
$0_{\mathcal{B}}$ are zero objects of $\mathcal{A}$ and $\mathcal{B}$
respectively and $0\colon F(0_{\mathcal{A}})\to G(0_{\mathcal{B}})$ is the
zero morphism. The direct sum of $(A,B,c)$ and
$(A^{\prime},B^{\prime},c^{\prime})$ is $(A\oplus A^{\prime},B\oplus
B^{\prime},c\oplus c^{\prime})$. So
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ is an additive category.

Let $(a,b)\colon(A,B,c)\to(A^{\prime},B^{\prime},c^{\prime})$ be a morphism in
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$. Since $F$ and $G$ preserve
kernels, there exists a unique isomorphism $c_{\ker}\colon F(\ker(a))\to
G(\ker(b))$ such that

$\textstyle{F(\ker(a))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{\ker}}$$\textstyle{F(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F(a)}$$\scriptstyle{c}$$\textstyle{F(A^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{\prime}}$$\textstyle{G(\ker(b))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G(b)}$$\textstyle{G(B^{\prime})}$

commutes. The resulting morphism $(\ker(a),\ker(b),c_{\ker})\to(A,B,c)$ in
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ is then a kernel of $(a,b)$. A
similar construction shows that cokernels exist in
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$. Since kernels and cokernels are
constructed componentwise, it follows that for any morphism $(a,b)$ in
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ the morphism
$\operatorname{Coim}(a,b)\to\operatorname{Im}(a,b)$ is an isomorphism. Thus
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ is abelian. ∎

###  1.2. Tannakian categories

Our next goal is to show that the fibre product of Tannakian categories is
naturally a Tannakian category. To this end, we first recall the basic facts
concerning Tannakiann categories. For more background see [DM82] and [Del90].

A _tensor category_ is a tuple $(\mathcal{A},\otimes,\phi,\psi)$, where
$\mathcal{A}$ is a category,
$\otimes\colon\mathcal{A}\times\mathcal{A}\to\mathcal{A},\
(X,Y)\rightsquigarrow X\otimes Y$ is a functor and $\phi$, $\psi$ are
isomophisms of functors expressing the associativity and commutativity of
$\otimes$. More precisely, the _associativity constraint_ $\phi$ has
components $\phi_{X,Y,Z}\colon X\otimes(Y\otimes Z)\to(X\otimes Y)\otimes Z$
and the _commutativity constraint_ $\psi$ has components $\psi_{X,Y}\colon
X\otimes Y\to Y\otimes X$. The functorial isomorphisms $\phi$ and $\psi$ are
required to satisfy three _coherence conditions_ (see [DM82, Section 1] for
details). It is also required that there exists an _identity object_
$(\mathds{1},u)$. This means that $\mathcal{A}\to\mathcal{A},\
X\rightsquigarrow\mathds{1}\otimes X$ is an equivalence of categories and
$u\colon\mathds{1}\to\mathds{1}\otimes\mathds{1}$ is an isomorphism.

Then there exists a unique isomorphism of functors $l$ with component
$l_{X}\colon X\to\mathds{1}\otimes X$ such that certain properties are
satisfied ([DM82, Prop. 1.3 (a)]). Similarly, there is functorial isomorphism
$r$ with components $r_{X}\colon X\to X\otimes\mathds{1}$.

We will usually omit $\phi$ and $\psi$ from the notation and refer to
$(\mathcal{A},\otimes)$, or even $\mathcal{A}$, as a tensor category. A
_tensor functor_
$(\mathcal{A},\otimes)\to(\mathcal{A}^{\prime},\otimes^{\prime})$ is a pair
$(F,\alpha)$, where $F\colon\mathcal{A}\to\mathcal{A}^{\prime}$ is a functor
and $\alpha$ is an isomorphism of functors with components $\alpha_{X,Y}\colon
F(X)\otimes F(Y)\to F(X\otimes Y)$ such that three natural properties
involving respectively the associativity constraints, the commutativity
constraints and the identity objects, are satisfied ([DM82, Def. 1.8]). We
will often omit the functorial isomorphism $\alpha$ from the notation and
simply refer to $F$ as a tensor functor.

A tensor category $(\mathcal{A},\otimes)$ is _rigid_ , if for every object $X$
of $\mathcal{A}$ the exists an object $X^{\vee}$ (a _dual_ of $X$) of
$\mathcal{A}$ together with morphisms $\operatorname{ev}\colon X\otimes
X^{\vee}\to\mathds{1}$ and $\delta\colon\mathds{1}\to X^{\vee}\otimes X$ such
that

$X\xrightarrow{r}X\otimes\mathds{1}\xrightarrow{\operatorname{id}\otimes\delta}X\otimes(X^{\vee}\otimes
X)\xrightarrow{\phi}(X\otimes X^{\vee})\otimes
X\xrightarrow{\operatorname{ev}\otimes\operatorname{id}}\mathds{1}\otimes
X\xrightarrow{l^{-1}}X$

and

$X^{\vee}\xrightarrow{l}\mathds{1}\otimes
X^{\vee}\xrightarrow{\delta\otimes\operatorname{id}}(X^{\vee}\otimes X)\otimes
X^{\vee}\xrightarrow{\phi^{-1}}X^{\vee}\otimes(X\otimes
X^{\vee})\xrightarrow{\operatorname{id}\otimes\operatorname{ev}}X^{\vee}\otimes\mathds{1}\xrightarrow{r^{-1}}X^{\vee}$

are the identity. The above definition of rigid is equivalent to the one used
in [DM82] by [Del90, Section 2]. Given $X,X^{\vee}$ and $\operatorname{ev}$
such that there exists a $\delta$ verifying the above conditions, then
$\delta$ is uniquely determined. Moreover, the pair
$(X^{\vee},\operatorname{ev})$ is unique up to a unique isomorphism. For a
morphism $f\colon X\to Y$ the _transpose_ ${}^{t}f\colon Y^{\vee}\to X^{\vee}$
of $f$ is defined as the composition

${{}^{t}}f\colon Y^{\vee}\simeq\mathds{1}\otimes
Y^{\vee}\xrightarrow{\delta\otimes\operatorname{id}}X^{\vee}\otimes X\otimes
Y^{\vee}\xrightarrow{\operatorname{id}\otimes
f\otimes\operatorname{id}}X^{\vee}\otimes Y\otimes
Y^{\vee}\xrightarrow{\operatorname{id}\otimes\operatorname{ev}}X^{\vee}\otimes\mathds{1}\simeq
X^{\vee}.$

For $f$ invertible one sets $f^{\vee}=({{}^{t}f})^{-1}\colon X^{\vee}\to
Y^{\vee}$.

######  Lemma 1.5.

Let $F\colon\mathcal{A}\to\mathcal{C}$ and $G\colon\mathcal{B}\to\mathcal{C}$
be tensor functors between tensor categories. Then
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ naturally acquires the structure
of a tensor category such that the projections
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}\to\mathcal{A}$ and
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}\to\mathcal{B}$ are tensor
functors. Moreover, if $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ are
rigid, then $\mathcal{A}\otimes_{\mathcal{C}}\mathcal{B}$ is rigid.

###### Proof.

Cf. the discussion on page 360 of [Bas68]. We define the tensor product
$\otimes\colon(\mathcal{A}\times_{\mathcal{C}}\mathcal{B})\times(\mathcal{A}\times_{\mathcal{C}}\mathcal{B})\to\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$
on objects by

$(A,B,c)\otimes(A^{\prime},B^{\prime},c^{\prime})=(A\otimes
A^{\prime},B\otimes B^{\prime},c\otimes c^{\prime})$

and on morphisms by $(a,b)\otimes(a^{\prime},b^{\prime})=(a\otimes
a^{\prime},b\otimes b^{\prime})$. The associativity and commutativity
constraints on $\mathcal{A}$ and $\mathcal{B}$ induce associativity and
commutativity constraints on $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$. The
identity object of $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ is
$(\mathds{1}_{\mathcal{A}},\mathds{1}_{\mathcal{B}},c)$, with $c\colon
F(\mathds{1}_{\mathcal{A}})\to G(\mathds{1}_{\mathcal{B}})$ the unique
isomorphism derived from the fact that $F(\mathds{1}_{\mathcal{A}})$ and
$G(\mathds{1}_{\mathcal{B}})$ are identity objects of $\mathcal{C}$.

If $(A,B,c)$ is an object of $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ and
$A^{\vee}$ and $B^{\vee}$ are duals of $A$ and $B$ respectively, then
$(A^{\vee},B^{\vee},c^{\vee})$ is a dual of $(A,B,c)$. Implicit in this
notation are the identifications $F(A^{\vee})\simeq F(A)^{\vee}$ and
$G(B^{\vee})\simeq G(B)^{\vee}$ derived from the fact that tensor functors
preserve duals. ∎

A rigid tensor category $(\mathcal{A},\otimes)$ is _abelian_ if the underlying
category $\mathcal{A}$ is abelian. In this case $\otimes$ is automatically bi-
additive ([DM82, Prop. 1.16]).

A basic example of a tensor category is the category $\operatorname{Mod}_{R}$
of $R$-modules over a commutative ring $R$, with the usual tensor product and
the usual associativity and commutativity constraints. The most basic example
of a rigid abelian tensor category is the category $\operatorname{Vec}_{k}$ of
finite dimensional $k$-vector spaces.

A _neutral Tannakian category_ over $k$ is a rigid abelian tensor category
$(\mathcal{A},\otimes)$ together with an isomorphism
$k\simeq\operatorname{End}(\mathds{1})$ such that there exists an exact
$k$-linear tensor functor $\omega\colon\mathcal{A}\to\operatorname{Vec}_{k}$.
Any such $\omega$ is called a _(neutral) fibre functor_. We note that by
[Del90, Cor. 2.10] a fibre functor is automatically faithful.

For tensor functors $F,G\colon(\mathcal{A},\otimes)\to(\mathcal{B},\otimes)$,
a morphism $\alpha\colon F\to G$ of functors, is a _morphism of tensor
functors_ if

  1. (i)

$\alpha_{X\otimes Y}=\alpha_{X}\otimes\alpha_{Y}$ for any two objects $X,Y$ of
$\mathcal{A}$ under the identifications $F(X)\otimes F(Y)\simeq F(X\otimes Y)$
and $G(X)\otimes G(Y)\simeq G(X\otimes Y)$ and

  2. (ii)

the diagram

$\textstyle{\mathds{1}_{\mathcal{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(\mathds{1}_{\mathcal{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{\mathds{1}_{\mathcal{A}}}}$$\textstyle{G(\mathds{1_{\mathcal{A}}})}$

commutes.

For a neutral Tannakian category $\mathcal{A}$ with fibre functor
$\omega\colon\mathcal{A}\to\operatorname{Vec}_{k}$, one defines a functor
$\underline{\operatorname{Aut}}^{\otimes}(\omega)$ from the category of
$k$-algebras to the category of groups by
$\underline{\operatorname{Aut}}^{\otimes}(\omega)(R)=\operatorname{Aut}^{\otimes}(\omega_{R})$,
the group of tensor automorphisms of the tensor functor $\omega_{R}$, defined
as the composition of $\omega$ with the tensor functor
$\operatorname{Vec}_{k}\to\operatorname{Mod}_{R},\ V\rightsquigarrow
V\otimes_{k}R$ for any $k$-algebra $R$.

The standard example of a neutral Tannakian category is the category
$\operatorname{Rep}(G)$ of finite dimensional $k$-linear representations of a
$k$-group $G$ with the usual tensor product of representations and the
forgetful functor
$\omega_{G}\colon\operatorname{Rep}(G)\to\operatorname{Vec}_{k}$ as fibre
functor. The main Tannaka reconstruction theorem shows that this is
essentially the only example:

######  Theorem 1.6 ([DM82, Theorem 2.11]).

Let $(\mathcal{A},\otimes)$ be a neutral Tannakian category with fibre functor
$\omega\colon\mathcal{A}\to\operatorname{Vec}_{k}$. Then
$G=\underline{\operatorname{Aut}}^{\otimes}(\omega)$ is a $k$-group and
$\omega$ induces an equivalence of tensor categories
$\mathcal{A}\to\operatorname{Rep}(G)$.

The $k$-group
$G_{\mathcal{A}}=\underline{\operatorname{Aut}}^{\otimes}(\omega)$ is called
the _fundamental group_ of $\mathcal{A}$ or the _Tannakian dual_ of
$\mathcal{A}$. A _neutralized Tannakian category_ is a pair
$(\mathcal{A},\omega_{\mathcal{A}})$ consisting of a neutral Tannakian
category $\mathcal{A}$ and a fibre functor
$\omega_{\mathcal{A}}\colon\mathcal{A}\to\operatorname{Vec}_{k}$. Abusing
notation, a _tensor functor_
$(F,\gamma)\colon(\mathcal{A},\omega_{\mathcal{A}})\to(\mathcal{B},\omega_{\mathcal{B}})$
between neutralized Tannakian categories is a pair $(F,\gamma)$ consisting of
a $k$-linear tensor functor $F\colon\mathcal{A}\to\mathcal{B}$ and an
isomorphism $\gamma\colon\omega_{\mathcal{B}}F\simeq\omega_{\mathcal{A}}$ of
tensor functors. Note that this implies that $F$ is exact since
$\omega_{\mathcal{A}}$ and $\omega_{\mathcal{B}}$ are exact. Such a tensor
functor induces a morphism
$\varphi\colon\underline{\operatorname{Aut}}^{\otimes}(\omega_{\mathcal{B}})\to\underline{\operatorname{Aut}}^{\otimes}(\omega_{\mathcal{A}})$
of $k$-groups as follows: For every $k$-algebra $R$, an element
$g\in\underline{\operatorname{Aut}}^{\otimes}(\omega_{\mathcal{B}})(R)=\operatorname{Aut}^{\otimes}(\omega_{\mathcal{B},R})$
gives rises to a tensor automorphism
$g\star\operatorname{id}_{F}\colon\omega_{\mathcal{B},R}F\simeq\omega_{\mathcal{B},R}F$,
that yields a tensor automorphism of $\omega_{\mathcal{A},R}$ via the tensor
isomorphism $\gamma_{R}=\operatorname{id}_{-\otimes
R}\star\gamma\colon\omega_{\mathcal{B},R}F\simeq\omega_{\mathcal{A},R}$, i.e.,
$\varphi_{R}(g)=\gamma_{R}(g\star\operatorname{id}_{F})\gamma_{R}^{-1}$. Note
that in the context of Theorem 1.6 we have an isomorphism
$(\mathcal{A},\omega)\to(\operatorname{Rep}(G),\omega_{G})$ of neutralized
Tannakian categories.

The composition $(G,\delta)(F,\gamma)$ of two tensor functors
$(F,\gamma)\colon(\mathcal{A},\omega_{\mathcal{A}})\to(\mathcal{B},\omega_{\mathcal{B}})$
and
$(G,\delta)\colon(\mathcal{B},\omega_{\mathcal{B}})\to(\mathcal{C},\omega_{\mathcal{C}})$
is defined as
$(G,\delta)(F,\gamma)=(GF,\omega_{\mathcal{C}}GF\xrightarrow{\delta\star\operatorname{id}_{F}}\omega_{\mathcal{B}}F\xrightarrow{\gamma}\omega_{\mathcal{A}})$.

Let
$(F,\gamma),(F^{\prime},\gamma^{\prime})\colon(\mathcal{A},\omega_{\mathcal{A}})\to(\mathcal{B},\omega_{\mathcal{B}})$
be tensor functors. An isomorphism between $(F,\gamma)$ and
$(F^{\prime},\gamma^{\prime})$ is an isomorphism $\delta\colon F\to
F^{\prime}$ of tensor functors such that

$\textstyle{\omega_{\mathcal{B}}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{\omega_{\mathcal{B}}}\star\delta}$$\scriptstyle{\gamma}$$\textstyle{\omega_{\mathcal{B}}F^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma^{\prime}}$$\textstyle{\omega_{\mathcal{A}}}$

commutes. Note that isomorphic $(F,\gamma)$, $(F^{\prime},\gamma^{\prime})$
induce the same morphism
$\underline{\operatorname{Aut}}^{\otimes}(\omega_{\mathcal{B}})\to\underline{\operatorname{Aut}}^{\otimes}(\omega_{\mathcal{A}})$.

Morally, the category of $k$-groups is anti-equivalent to the category of
neutralized Tannakian categories over $k$. In reality, this is not quite true
since a category of categories should be regarded as a $2$-category. The
following precise statement ([OW15, Cor. 3.2]) is sufficient for our purposes.

######  Lemma 1.7.

Let $(\mathcal{A},\omega_{\mathcal{A}})$ and
$(\mathcal{B},\omega_{\mathcal{B}})$ be neutralized Tannakian categories with
fundamental groups $G_{\mathcal{A}}$ and $G_{\mathcal{B}}$ respectively. Then
$\operatorname{Hom}(G_{\mathcal{B}},G_{\mathcal{A}})$ is in bijection with the
isomorphism classes of tensor functors
$(\mathcal{A},\omega_{\mathcal{A}})\to(\mathcal{B},\omega_{\mathcal{B}})$.

As one may expect, the fibre product of neutralized Tannakian categories
inherits all of the structure:

######  Lemma 1.8.

Let
$(F,\gamma)\colon(\mathcal{A},\omega_{\mathcal{A}})\to(\mathcal{C},\omega_{\mathcal{C}})$
and
$(G,\delta)\colon(\mathcal{B},\omega_{\mathcal{B}})\to(\mathcal{C},\omega_{\mathcal{C}})$
be tensor functors between neutralized Tannakian categories. Then
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ inherits the structure of a
neutralized Tannakian category with fibre functor
$\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}$ given by
$\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}(A,B,c)=\omega_{\mathcal{A}}(A)$.

###### Proof.

We already saw in Lemma 1.5 that $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$
is naturally a rigid tensor category. As $F$ and $G$ are exact and additive,
Lemma 1.4 shows that $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ is abelian.

The map
$k\to\operatorname{End}(\mathds{1}_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}})$
is an isomorphism because an endomorphism of
$\mathds{1}_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}=(\mathds{1}_{\mathcal{A}},\mathds{1}_{\mathcal{B}},c)$
is a pair $(a,b)$ with $a\in\operatorname{End}(\mathds{1}_{\mathcal{A}})\simeq
k$ and $b\in\operatorname{End}(\mathds{1}_{\mathcal{B}})\simeq k$ such that
$a$ and $b$ agree as elements of $k$.

Note that the projection
$P_{1}\colon\mathcal{A}\times_{\mathcal{C}}\mathcal{B}\to\mathcal{A}$ is an
exact $k$-linear tensor functor (the implied isomorphism $P_{1}(-)\otimes
P_{1}(-)\simeq P_{1}(-\otimes-)$ is the identity). Therefore, also the
composition $\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}$ of $\omega$
with $P_{1}$ is an exact $k$-linear tensor functor. ∎

######  Remark 1.9.

There are four possibilities to define the fibre functor
$\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}$ in Lemma 1.8: On an
object $(A,B,c)$ of $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ we can define
it as $\omega_{\mathcal{A}}(A)$, $\omega_{\mathcal{B}}(B)$,
$\omega_{\mathcal{C}}(F(A))$ or $\omega_{\mathcal{C}}(G(B))$. The choice is
irrelevant, since all four functors are isomorphic.

###  1.3. Amalgamated free products

In this section we recall the definition of amalgamated free products and
discuss their relation with fibre products of neutralized Tannakian
categories.

Let $\phi_{1}\colon H\to G_{1}$ and $\phi_{2}\colon H\to G_{2}$ be morphisms
of $k$-groups. In line with the common notation in group theory, we denote a
pushout of $\phi_{1}$ and $\phi_{2}$ in the category of $k$-groups with
$G_{1}*_{H}G_{2}$ and call it a _free product of $G_{1}$ and $G_{2}$ with
amalgamation over $H$_ or simply an _amalgamated free product_. So
$G_{1}*_{H}G_{2}$ comes with two morphisms $i_{1}\colon G_{1}\to
G_{1}*_{H}G_{2}$ and $i_{2}\colon G_{2}\to G_{1}*_{H}G_{2}$ such that

$\textstyle{G_{1}*_{H}G_{2}}$$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\textstyle{G_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{2}}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\scriptstyle{\phi_{2}}$

commutes and satisfies the following universal property: If $j_{1}\colon
G_{1}\to G$ and $j_{2}\colon G_{2}\to G$ are morphisms of $k$-groups such that
$j_{1}\phi_{1}=j_{2}\phi_{2}$, then there exists a unique morphism
$G_{1}*_{H}G_{2}\to G$ such that

$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\scriptstyle{j_{1}}$$\textstyle{G_{1}*_{H}G_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{2}}$$\scriptstyle{j_{2}}$$\textstyle{G}$

commutes. For $H=1$ the trivial group, one obtains the _free product_
$G_{1}*G_{2}$ of $G_{1}$ and $G_{2}$.

It is known ([Por15, Prop. 22 (2)]) that the category of (commutative) Hopf
algebras over a field is complete. In particular, it has fibre products.
Therefore amalgamated free products exist. Beyond this existence result, the
only literature on amalgamated free products of $k$-groups we could find is
[Ant12], which discusses amalgamated free products of finite flat group
schemes over complete discrete valuation rings. We will provide two existence
proofs for amalgamated free products of $k$-groups independent from [Por15].
The first one (Corollary 1.12), based on the fibre product of neutralized
Tannakian categories, leads to a proof of the Seifert-van Kampen Theorem for
the proalgebraic fundamental group rather directly. The second one
(Proposition 2.4), based on an explicit construction with Hopf algebras, is
helpful for determining properties of the amalgamated free product.

We next discuss the universal property of the fibre product constructed in
Lemma 1.8. As before, let
$(F,\gamma)\colon(\mathcal{A},\omega_{\mathcal{A}})\to(\mathcal{C},\omega_{\mathcal{C}})$
and
$(G,\delta)\colon(\mathcal{B},\omega_{\mathcal{B}})\to(\mathcal{C},\omega_{\mathcal{C}})$
be tensor functors between neutralized Tannakian categories. We consider
$\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$ as a neutralized Tannakian
category as described in Lemma 1.8 The projections
$P_{1}\colon\mathcal{A}\times_{\mathcal{C}}\mathcal{B}\to\mathcal{A}$ and
$P_{2}\colon\mathcal{A}\times_{\mathcal{C}}\mathcal{B}\to\mathcal{B}$ are
$k$-linear tensor functors.

In fact, $P_{1}$ and $P_{2}$ can be regarded as tensor functors between
neutralized Tannakian categories as follows. With
$\chi_{1}=\operatorname{id}\colon\omega_{\mathcal{A}}P_{1}\to\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}$
we obtain a tensor functor
$(P_{1},\chi_{1})\colon(\mathcal{A}\times_{\mathcal{C}}\mathcal{B},\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}})\to(\mathcal{A},\omega_{\mathcal{A}})$
between neutralized Tannakian categories. We define an isomorphism
$\chi_{2}\colon\omega_{\mathcal{B}}P_{2}\to\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}$
by

$\chi_{2,(A,B,c)}\colon(\omega_{\mathcal{B}}P_{2})(A,B,c)=\omega_{\mathcal{B}}(B)\xrightarrow{\delta_{B}^{-1}}\omega_{\mathcal{C}}(G(B))\xrightarrow{\omega_{\mathcal{C}}(c)^{-1}}\omega_{\mathcal{C}}(F(A))\xrightarrow{\gamma_{A}}\omega_{\mathcal{A}}(A)=\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}(A,B,c)$

for any object $(A,B,c)$ of $\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$. Then
also
$(P_{2},\chi_{2})\colon(\mathcal{A}\times_{\mathcal{C}}\mathcal{B},\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}})\to(\mathcal{B},\omega_{\mathcal{B}})$
is a tensor functor between neutralized Tannakian categories. The isomorphism
$\psi\colon FP_{1}\to GP_{2}$ is an isomorphism of tensor functors
$(F,\gamma)(P_{1},\chi_{1})\to(G,\delta)(P_{2},\chi_{2})$ since the diagram

$\textstyle{\omega_{\mathcal{C}}FP_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{\omega_{\mathcal{C}}}\star\psi}$$\scriptstyle{\gamma\star\operatorname{id}_{P_{1}}}$$\textstyle{\omega_{\mathcal{C}}GP_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta\star\operatorname{id}_{P_{2}}}$$\textstyle{\omega_{\mathcal{A}}P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{1}}$$\textstyle{\omega_{\mathcal{B}}P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{2}}$$\textstyle{\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}}$

commutes by definition of $\chi_{2}$.

To state the universal property of the fibre product
$(\mathcal{A}\times_{\mathcal{C}}\mathcal{B},\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}})$
we need $2$-commutative diagrams of neutralized Tannakian categories. Let
$(F,\gamma)\colon(\mathcal{A},\omega_{\mathcal{A}})\to(\mathcal{C},\omega_{\mathcal{B}})$
and
$(G,\delta)\colon(\mathcal{B},\omega_{\mathcal{B}})\to(\mathcal{C},\omega_{\mathcal{C}})$
be tensor functors between neutralized Tannakian categories. A $2$-commutative
diagram (with respect to $(F,\gamma)$ and $(G,\delta)$) is a quadruple
$((\mathcal{D},\omega_{\mathcal{D}}),(Q_{1},\epsilon_{1}),(Q_{2},\epsilon_{2}),\alpha)$
consisting of a neutralized Tannakian category
$(\mathcal{D},\omega_{\mathcal{D}})$, tensor functors
$(Q_{1},\epsilon_{1})\colon(\mathcal{D},\omega_{\mathcal{D}})\to(\mathcal{A},\omega_{\mathcal{A}})$
and
$(Q_{2},\epsilon_{2})\colon(\mathcal{D},\omega_{\mathcal{D}})\to(\mathcal{B},\omega_{\mathcal{B}})$
and an isomorphism
$\alpha\colon(F,\gamma)(Q_{1},\epsilon_{1})\to(G,\delta)(Q_{2},\epsilon_{2})$
of tensor functors.

A morphism

$((H,\eta),\beta_{1},\beta_{2})\colon((\mathcal{D},\omega_{\mathcal{D}}),(Q_{1},\epsilon_{1}),(Q_{2},\epsilon_{2}),\alpha)\to((\mathcal{D}^{\prime},\omega_{\mathcal{D}^{\prime}}),(Q^{\prime}_{1},\epsilon^{\prime}_{1}),(Q^{\prime}_{2},\epsilon^{\prime}_{2}),\alpha^{\prime})$

of $2$/̄commutative diagrams consists of a tensor functor
$(H,\eta)\colon(\mathcal{D},\omega_{\mathcal{D}})\to(\mathcal{D}^{\prime},\omega_{\mathcal{D}^{\prime}})$
and isomorphisms
$\beta_{1}\colon(Q_{1},\epsilon_{1})\to(Q^{\prime}_{1},\epsilon_{1}^{\prime})(H,\eta)$
and
$\beta_{2}\colon(Q_{2},\epsilon_{2})\to(Q^{\prime}_{2},\epsilon_{2}^{\prime})(H,\eta)$
of tensor functors such that

$\textstyle{FQ_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{F}\star\beta_{1}}$$\scriptstyle{\alpha}$$\textstyle{FQ_{1}^{\prime}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\prime}\star\operatorname{id}_{H}}$$\textstyle{GQ_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{G}\star\beta_{2}}$$\textstyle{GQ_{2}^{\prime}H}$

commutes. Finally, an isomorphism between two morphisms
$((H,\eta),\beta_{1},\beta_{2})$,
$((H^{\prime},\eta^{\prime}),\beta^{\prime}_{1},\beta^{\prime}_{2})$ of
$2$-commutative diagrams with the same source and target, is an isomorphism
$\theta\colon(H,\eta)\to(H^{\prime},\eta^{\prime})$ of tensor functors such
that

$\textstyle{Q_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\beta_{1}^{\prime}}$$\textstyle{Q_{1}^{\prime}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{Q_{1}^{\prime}}\star\theta}$$\textstyle{Q_{1}^{\prime}H^{\prime}}$
and
$\textstyle{Q_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{2}}$$\scriptstyle{\beta_{2}^{\prime}}$$\textstyle{Q_{2}^{\prime}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{Q_{2}^{\prime}}\star\theta}$$\textstyle{Q_{2}^{\prime}H^{\prime}}$

commute.

######  Proposition 1.10.

Let
$(F,\gamma)\colon(\mathcal{A},\omega_{\mathcal{A}})\to(\mathcal{C},\omega_{\mathcal{B}})$
and
$(G,\delta)\colon(\mathcal{B},\omega_{\mathcal{B}})\to(\mathcal{C},\omega_{\mathcal{C}})$
be tensor functors between neutralized Tannakian categories. Then
$((\mathcal{A}\times_{\mathcal{C}}\mathcal{B},\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}),(P_{1},\chi_{1}),(P_{2},\chi_{2}),\psi)$
satisfies the following universal property. For every $2$-commutative diagram
$((\mathcal{D},\omega_{\mathcal{D}}),(Q_{1},\epsilon_{1}),(Q_{2},\epsilon_{2}),\alpha)$
of neutralized Tannakian categories, there exists a morphism

$((H,\eta),\beta_{1},\beta_{2})\colon((\mathcal{D},\omega_{\mathcal{D}}),(Q_{1},\epsilon_{1}),(Q_{2},\epsilon_{2}),\alpha)\to((\mathcal{A}\times_{\mathcal{C}}\mathcal{B},\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}),(P_{1},\chi_{1}),(P_{2},\chi_{2}),\psi)$
(1)

of $2$-commutative diagrams. If
$((H^{\prime},\eta^{\prime}),\beta^{\prime}_{1},\beta^{\prime}_{2})$ is
another such morphism, then there exists an isomorphism between
$((H,\eta),\beta_{1},\beta_{2})$ and
$((H^{\prime},\eta^{\prime}),\beta^{\prime}_{1},\beta^{\prime}_{2})$.

###### Proof.

The functor $H\colon\mathcal{D}\to\mathcal{A}\times_{\mathcal{C}}\mathcal{B}$
defined in Lemma 1.3 can be considered as a tensor functor with functorial
isomorphism $H(-)\otimes H(-)\simeq H(-\otimes-)$ induced by the functorial
ismorphisms $Q_{i}(-)\otimes Q_{i}(-)\simeq Q_{i}(-\otimes-)$ for $i=1,2$.
Setting
$\eta\colon\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}H=\omega_{\mathcal{A}}Q_{1}\xrightarrow{\epsilon_{1}}\omega_{\mathcal{D}}$
we obtain a tensor functor
$(H,\eta)\colon(\mathcal{D},\omega_{\mathcal{D}})\to(\mathcal{A}\times_{\mathcal{C}}\mathcal{B},\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}})$
between neutralized Tannakian categories. As
$(P_{1},\chi_{1})(H,\eta)=(Q_{1},\epsilon_{1})$, the identity
$\beta_{1}\colon(Q_{1},\epsilon_{1})\to(P_{1},\chi_{1})(H,\eta)$ is a morphism
of tensor functors. By definition of $\chi_{2}$, the diagram

$\textstyle{\omega_{\mathcal{C}}FP_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{\omega_{\mathcal{C}}}\star\psi}$$\scriptstyle{\gamma\star\operatorname{id}_{P_{1}}}$$\textstyle{\omega_{\mathcal{C}}GP_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta\star\operatorname{id}_{P_{2}}}$$\textstyle{\omega_{\mathcal{A}}P_{1}}$$\textstyle{\omega_{\mathcal{B}}P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{2}}$

commutes. Therefore, the upper square in

$\textstyle{\omega_{\mathcal{C}}FP_{1}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{\omega_{\mathcal{C}}}\star\psi\star\operatorname{id}_{H}}$$\scriptstyle{\gamma\star\operatorname{id}_{P_{1}H}}$$\textstyle{\omega_{\mathcal{C}}GP_{2}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta\star\operatorname{id}_{P_{2}H}}$$\textstyle{\omega_{\mathcal{A}}P_{1}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{1}}$$\textstyle{\omega_{\mathcal{B}}P_{2}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{2}\star\operatorname{id}_{H}}$$\scriptstyle{\epsilon_{2}}$$\textstyle{\omega_{\mathcal{D}}}$
(2)

commutes. Noting that $P_{1}H=Q_{1}$, $P_{2}H=Q_{2}$ and
$\psi\star\operatorname{id}_{H}=\alpha$, we see that the outer diagram in (2)
commutes because
$\alpha\colon(F,\gamma)(Q_{1},\epsilon_{1})\to(G,\delta)(Q_{2},\epsilon_{2})$
is an isomorphism of tensor functors. Thus also the lower triangle in (2)
commutes. This shows that $(P_{2},\chi_{2})(H,\eta)=(Q_{2},\epsilon_{2})$.
Thus the identity
$\beta_{2}\colon(Q_{2},\epsilon_{2})\to(P_{2},\chi_{2})(H,\eta)$ is an
isomorphism of tensor functors. We have thus constructed the morphism (1).

If $((H^{\prime},\eta^{\prime}),\beta_{1}^{\prime},\beta_{2}^{\prime})$ is
another such morphism, the isomorphism $\theta\colon H\to H^{\prime}$
constructed in Lemma 1.3 is a tensor isomorphism because
$\beta_{1}^{\prime}\colon Q_{1}\to P_{1}H^{\prime}$ and
$\beta_{2}^{\prime}\colon Q_{2}\to P_{2}H^{\prime}$ are tensor isomorphisms.
Moreover, the fact that
$\beta_{1}^{\prime}\colon(Q_{1},\epsilon_{1})\to(P_{1},\chi_{1})(H^{\prime},\eta^{\prime})$
is an isomorphism of tensor functors, implies that

$\textstyle{\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}H=\omega_{\mathcal{A}}Q_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}}\star\theta=\operatorname{id}_{\omega_{\mathcal{A}}}\star\beta_{1}^{\prime}}$$\scriptstyle{\eta=\epsilon_{1}}$$\textstyle{\omega_{\mathcal{A}}P_{1}H^{\prime}=\omega_{\mathcal{A}\times_{\mathcal{C}}\mathcal{B}}H^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\prime}}$$\textstyle{\omega_{\mathcal{D}}}$

commutes. So $\theta\colon(H,\eta)\to(H^{\prime},\eta^{\prime})$ is an
isomorphism of tensor functors. ∎

######  Corollary 1.11.

Let $(\mathcal{A}_{1},\omega_{\mathcal{A}_{1}})$,
$(\mathcal{A}_{2},\omega_{\mathcal{A}_{2}})$ and
$(\mathcal{B},\omega_{\mathcal{B}})$ be neutralized Tannakian categories with
fundamental groups $G_{1},G_{2}$ and $H$. For tensor functors
$(\mathcal{A}_{1},\omega_{\mathcal{A}_{1}})\to(\mathcal{B},\omega_{\mathcal{B}})$
and
$(\mathcal{A}_{2},\omega_{\mathcal{A}_{2}})\to(\mathcal{B},\omega_{\mathcal{B}})$,
the fundamental group of the neutralized Tannakian category
$(\mathcal{A}_{1}\times_{\mathcal{B}}\mathcal{A}_{2},\omega_{\mathcal{A}_{1}\times_{\mathcal{B}}\mathcal{A}_{2}})$
is the amalgamated free product $G_{1}*_{H}G_{2}$.

###### Proof.

Let $G$ denote the fundamental group of
$(\mathcal{A}_{1}\times_{\mathcal{B}}\mathcal{A}_{2},\omega_{\mathcal{A}_{1}\times_{\mathcal{B}}\mathcal{A}_{2}})$.
The $2$-commutative diagram

$\textstyle{(\mathcal{A}_{1}\times_{\mathcal{B}}\mathcal{A}_{2},\omega_{\mathcal{A}_{1}\times_{\mathcal{B}}\mathcal{A}_{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(P_{1},\chi_{1})}$$\scriptstyle{(P_{2},\chi_{2})}$$\textstyle{(\mathcal{A}_{1},\omega_{\mathcal{A}_{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathcal{A}_{2},\omega_{\mathcal{A}_{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathcal{B},\omega_{\mathcal{B}})}$
(3)

of neutralized Tannakian categories dualizes to a commutative diagram

$\textstyle{G}$$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\textstyle{G_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

of $k$-groups. Let

$\textstyle{G^{\prime}}$$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}_{1}}$$\textstyle{G_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}_{2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ (4)

be a commutative diagram of $k$-groups. As the morphisms between fundamental
groups are in bijection with isomorphism classes of tensor functors between
neutralized Tannakian categories (Lemma 1.7), there exists a $2$-commutative
diagram

$\textstyle{(\operatorname{Rep}(G^{\prime}),\omega_{G^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(Q_{1},\epsilon_{1})}$$\scriptstyle{(Q_{2},\epsilon_{2})}$$\textstyle{(\mathcal{A}_{1},\omega_{\mathcal{A}_{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathcal{A}_{2},\omega_{\mathcal{A}_{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathcal{B},\omega_{\mathcal{B}})}$
(5)

of neutralized Tannakian categories that dualizes to (4). By Proposition 1.10
there exists a morphism $((H,\eta),\beta_{1},\beta_{2})$ of $2$-commutative
diagrams from (5) to (3). The morphism
$(H,\eta)\colon(\operatorname{Rep}(G^{\prime}),\omega_{G^{\prime}})\to(\mathcal{A}_{1}\times_{\mathcal{B}}\mathcal{A}_{2},\omega_{\mathcal{A}_{1}\times_{\mathcal{B}}\mathcal{A}_{2}})$
dualizes to a morphism $\phi\colon G\to G^{\prime}$ of $k$-groups. The
$2$-commutativity of

$\textstyle{(\operatorname{Rep}(G^{\prime}),\omega_{G^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(H,\eta)}$$\scriptstyle{(Q_{j},\epsilon_{j})}$$\textstyle{(\mathcal{A}_{1}\times_{\mathcal{B}}\mathcal{A}_{2},\omega_{\mathcal{A}_{1}\times_{\mathcal{B}}\mathcal{A}_{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(P_{j},\chi_{j})}$$\textstyle{(\mathcal{A}_{j},\omega_{\mathcal{A}_{j}})}$

realized by $\beta_{j}$ implies that

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{G^{\prime}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
G_{j}}$$\scriptstyle{i_{j}^{\prime}}$$\scriptstyle{i_{j}}$ (6)

commutes for $j=1,2$. Finally, the uniqueness of $\phi$ (making (6) commute
for $j=1,2$) follows from the fact that $((H,\eta),\beta_{1},\beta_{2})$ is
unique up to an isomorphism. ∎

######  Corollary 1.12.

Let $H\to G_{1}$ and $H\to G_{2}$ be morphisms of $k$-groups. Then the
amalgamated free product $G_{1}*_{H}G_{2}$ exists.

###### Proof.

The restriction functors $\operatorname{Rep}(G_{i})\to\operatorname{Rep}(H)$
($i=1,2$) can be regarded as tensor functors
$(\operatorname{Rep}(G_{i}),\omega_{G_{i}})\to(\operatorname{Rep}(H),\omega_{H})$
between neutralized Tannakian categories. By Corollary 1.11 the fundamental
group of
$\operatorname{Rep}(G_{1})\times_{\operatorname{Rep}(H)}\operatorname{Rep}(G_{2})$
is $G_{1}*_{H}G_{2}$. ∎

##  2\. The Hopf algebra of the amalgamated free product

In this section we offer an explicit description of the Hopf algebra
$k[G_{1}*_{H}G_{2}]$ of the amalgamated free product of two morphisms $H\to
G_{1}$ and $H\to G_{2}$ of $k$-groups in terms of the Hopf algebras
$k[G_{1}]$, $k[G_{2}]$ and $k[H]$. In other words, we explicitly describe the
fibre product in the category of Hopf algebras. (Recall that all Hopf algebras
are assumed to be commutative.) We also discuss properties of amalgamated free
products, such as being connected or geometrically reduced.

###  2.1. Fibre products of Hopf algebras

To motivate our explicit construction of the fibre product of Hopf algebras,
let us revisit the construction of the amalgamated free product in the
category of groups from a more categorical/conceptual perspective. For now,
assume that $\phi_{1}\colon H\to G_{1}$ and $\phi_{2}\colon H\to G_{2}$ are
morphisms of abstract groups. The amalgamated free product $G_{1}*_{H}G_{2}$
(in the category of groups) can be constructed as follows: First form the
disjoint union $G_{1}\uplus G_{2}$ and then build the free monoid
$M(G_{1}\uplus G_{2})$ on the set $G_{1}\uplus G_{2}$. Finally, the
amalgamated free product $G_{1}*_{H}G_{2}$ is obtained from $M(G_{1}\uplus
G_{2})$, by quotienting out by the relations

$g_{1}\cdot g_{1}^{\prime}=g_{1}g_{1}^{\prime}\quad\forall\
g_{1},g_{1}^{\prime}\in G_{1},$ $g_{2}\cdot
g_{2}^{\prime}=g_{2}g_{2}^{\prime}\quad\forall\ g_{2},g_{2}^{\prime}\in
G_{2},$ $1_{G_{1}}=1_{M(G_{1}\uplus G_{2})},$ $1_{G_{2}}=1_{M(G_{1}\uplus
G_{2})},$ $\phi_{1}(h)=\phi_{2}(h)\quad\forall\ h\in H.$

Here the $\cdot$ is used to denote multiplication in $M(G_{1}\uplus G_{2})$.

This construction can be dualized. Let $k[G_{1}]\to k[H]$ and $k[G_{2}]\to
k[H]$ be morphisms of Hopf algebras. The disjoint union $G_{1}\uplus G_{2}$
corresponds to the product algebra $k[G_{1}]\oplus k[G_{2}]=k[G_{1}]\times
k[G_{2}]$. The free monoid is the left adjoint of the forgetful functor from
the category of monoids to the category of sets. In the Hopf algebra world, we
should thus consider the right adjoint $A\rightsquigarrow C(A)$ of the
forgetful functor from the category of bialgebras to the category of algebras.
Finally, taking the quotient of $M(G_{1}\uplus G_{2})$, corresponds to
choosing an appropriate subbialgebra $k[G_{1}*_{H}G_{2}]$ of $C(k[G_{1}]\times
k[G_{2}])$ (that turns out to be a Hopf algebra).

To carry out the above idea, we need an explicit description of the right
adjoint $A\rightsquigarrow C(A)$ of the forgetful functor from the category of
bialgebras to the category of algebras. To this end, let us recall the
construction of the _cofree coalgebra_ $C(V)$ on a $k$-vector space $V$. See
[Swe69, Section 6.4] or [Haz03]. It comes with a $k$-linear map $\pi\colon
C(V)\to V$ characterized by the following universal property: If $D$ is a
coalgebra and $\varphi\colon D\to V$ is a $k$-linear map, there exists a
unique morphism $\tilde{\varphi}\colon D\to C(V)$ of coalgebras making

$\textstyle{C(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{V}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\tilde{\varphi}}$

commutative. In other words, the functor $V\rightsquigarrow C(V)$ is right
adjoint to the forgetful functor from coalgebras to vector spaces. We consider
the tensor algebra $TV=k\oplus V\oplus V^{\otimes 2}\oplus\cdots$. However, we
disregard the usual (non-commutative) algebra structure on $TV$. Instead, we
define a coalgebra structure on $TV$. For every $n\in\mathbb{N}$ and $0\leq
i,j$ with $i+j=n$, we have an isomorphism $\psi_{i,j}\colon V^{\otimes n}\to
V^{\otimes i}\otimes V^{\otimes j}$. Here $V^{0\otimes}=k$ and
$V^{1\otimes}=V$. For $t\in TV$ homogeneous of degree $n$, i.e., $t\in
V^{\otimes n}\subseteq TV$, we define

$\Delta(t)=\sum_{i=0}^{n}\psi_{i,n-i}(t)\in\bigoplus_{i=0}^{n}V^{\otimes
i}\otimes V^{\otimes(n-i)}\subseteq TV\otimes TV.$

By linearity, this assignment extends to a $k$-linear map $\Delta\colon TV\to
TV\otimes TV$. Define $\epsilon\colon TV\to k$ as the projection onto the
degree zero component. It is straight forward to check that
$(TV,\Delta,\epsilon)$ is a coalgebra. However, this is not yet the cofree
coalgebra on $V$. To construct $C(V)$ we need to consider the completion
$\hat{T}V=k\times V\times V^{\otimes 2}\times\ldots$ of $TV$. Thinking of the
elements of $\hat{T}V$ as infinite sums, we have an inclusion
$TV\subseteq\hat{T}V$.

One cannot extend $\Delta\colon TV\to TV\otimes TV$ directly to a map
$\Delta\colon\hat{T}V\to\hat{T}V\otimes\hat{T}V$, because the natural
extension of $\Delta\colon TV\to TV\otimes TV$ is a map
$\Delta\colon\hat{T}V\to\hat{T}V\hat{\otimes}\hat{T}V$, where
$\hat{T}V\hat{\otimes}\hat{T}V=\prod_{i,j\in\mathbb{N}}V^{\otimes i}\otimes
V^{\otimes j}$. Note that $\hat{T}V\otimes\hat{T}V$ is a proper subset of
$\hat{T}V\hat{\otimes}\hat{T}V$: If we think of an element of
$\hat{T}V\hat{\otimes}\hat{T}V$ as an infinite matrix

$\begin{pmatrix}f_{00}&f_{01}&f_{02}&\ldots\\\ f_{10}&f_{11}&f_{12}&\ldots\\\
\vdots&\vdots&&\end{pmatrix}$ (7)

with $f_{ij}\in V^{\otimes i}\otimes V^{\otimes j}$, then the elements of
$\hat{T}V\otimes\hat{T}V$ are finite sums of elements of the form

$f\otimes g=\begin{pmatrix}f_{0}\otimes g_{0}&f_{0}\otimes g_{1}&f_{0}\otimes
g_{2}&\ldots\\\ f_{1}\otimes g_{0}&f_{1}\otimes g_{1}&f_{1}\otimes
g_{2}&\ldots\\\ \vdots&\vdots&&\end{pmatrix}$

with $f=(f_{0},f_{1},\ldots),\ g=(g_{0},g_{1},\ldots)\in\hat{T}V$. Note that
with respect to this matrix representation, the map
$\Delta\colon\hat{T}V\to\hat{T}V\hat{\otimes}\hat{T}V$ is given by

$\Delta(f)=\begin{pmatrix}\psi_{00}(f_{0})&\psi_{01}(f_{1})&\psi_{02}(f_{2})&\ldots\\\
\psi_{10}(f_{1})&\psi_{11}(f_{2})&\\\ \psi_{21}(f_{2})&&&\\\
\vdots&&&\ddots\end{pmatrix}$

for $f=(f_{0},f_{1},\ldots)\in\hat{T}V$. Set

$C(V)=\\{f\in\hat{T}V|\
\Delta(f)\in\hat{T}V\otimes\hat{T}V\subseteq\hat{T}V\hat{\otimes}\hat{T}V\\}.$

Following [Haz03], we call the elements of $\hat{T}V$ that belong to $C(V)$
_representative_.

One can show ([Haz03, Theorem 3.12]), that $\Delta(C(V))\subseteq C(V)\otimes
C(V)\subseteq\hat{T}V\otimes\hat{T}V$ and so $C(V)$ (together with the induced
structure maps) is a coalgebra. Let $\pi\colon C(V)\to V$ be the projection to
the degree one component. Then $\pi$ satisfies the universal property of the
cofree coalgebra ([Haz03, Theorem 3.14]). Indeed, if $(D,\mu,\varepsilon)$ is
a coalgebra with a $k$-linear map $\varphi\colon D\to V$, the unique
$\tilde{\varphi}\colon D\to C(V)$ such that $\pi\tilde{\varphi}=\varphi$ is
given by $\tilde{\varphi}(d)=(\varepsilon(d),\varphi(d),\varphi^{\otimes
2}(\mu_{2}(d)),\varphi^{\otimes 3}(\mu_{3}(d)),\ldots)\in C(V)$ for $d\in D$,
where $\mu_{i}\colon D\to D^{\otimes i}$ is the $(i-1)$-st iteration of $\mu$.

If $A$ is a $k$-algebra, then $C(A)$ naturally acquires the structure of a
$k$-algebra (Cf. [Swe69, p. 134]). Indeed, this follows from the universal
property: Let $m\colon A\otimes A\to A$ be the multiplication. As $C(A)\otimes
C(A)$ has the structure of a coalgebra (the product coalgebra), the map
$C(A)\otimes C(A)\xrightarrow{\pi\otimes\pi}A\otimes A\xrightarrow{m}A$ lifts
to a multiplication map $C(A)\otimes C(A)\to C(A)$. Similarly, as $k$ has the
trivial coalgebra structure, the unit $k\to A$ lifts to a unit $k\to C(A)$.
Explicitly, the unit $k\to C(A)$ is the map $k\to C(A),\
\lambda\mapsto(\lambda,\lambda,\lambda,\ldots)$ and multiplication in $C(A)$
is componentwise, using the multiplication in $A^{\otimes i}$. One sees that
$C(A)$ is a bialgebra and $\pi\colon C(A)\to A$ is a morphism of algebras.
Moreover, for every bialgebra $B$ with an algebra morphism $\varphi\colon B\to
A$, the unique $\tilde{\varphi}\colon B\to C(A)$ such that
$\tilde{\varphi}\pi=\varphi$ is a morphism of bialgebras. In other words,
$A\rightsquigarrow C(A)$ is the sought for right adjoint of the forgetful
functor from bialgebras to algebras.

Let $\alpha_{1}\colon k[G_{1}]\to k[H]$ and $\alpha_{2}\colon k[G_{2}]\to
k[H]$ be morphisms of Hopf algebras. We will construct their fibre product
$k[G_{1}]\times_{k[H]}k[G_{2}]=k[G_{1}*_{H}G_{2}]$ as a suitable subalgebra of
$C(k[G_{1}]\oplus k[G_{2}])$.

Note that for $n\geq 0$ the vector space $(k[G_{1}]\oplus k[G_{2}])^{\otimes
n}$ is the direct sum of the subspaces

$T_{i_{1},\ldots,i_{n}}=k[G_{i_{1}}]\otimes\ldots\otimes k[G_{i_{n}}]$

with $i_{1},\ldots,i_{n}\in\\{1,2\\}$. For $\ell\in\\{1,\ldots,n\\}$ define a
map

$\psi_{i_{1},\ldots,i_{n}}^{\ell}\colon T_{i_{1},\ldots,i_{n}}\to
T_{i_{1},\ldots,i_{\ell-1},i_{\ell},i_{\ell},i_{\ell+1},\ldots,i_{n}}$

by applying the comultiplication $\Delta_{i_{l}}\colon k[G_{i_{\ell}}]\to
k[G_{i_{\ell}}]\otimes k[G_{i_{\ell}}]$ on the $\ell$-th factor and the
identity on all the other factors. Similarly, define

$\delta_{i_{1},\ldots,i_{n}}^{\ell}\colon T_{i_{1},\ldots,i_{n}}\to
T_{i_{1},\ldots,i_{\ell-1},i_{\ell+1},\ldots,i_{n}}$

by applying the counit $\epsilon_{i_{\ell}}\colon k[G_{i_{\ell}}]\to k$ on the
$\ell$-th factor and the identity on all the other factors. We also define

$\theta^{\ell}_{i_{1},\ldots,i_{n}}\colon T_{i_{1},\ldots,i_{n}}\to
k[G_{i_{1}}]\otimes\ldots\otimes k[G_{i_{\ell-1}}]\otimes k[H]\otimes
k[G_{i_{\ell}}]\otimes\ldots\otimes k[G_{i_{n}}]$

by applying $\alpha_{i_{\ell}}\colon k[G_{i_{\ell}}]\to k[H]$ on the $\ell$-th
factor and the identity on all the other factors.

An element $f\in\hat{T}(k[G_{1}]\oplus k[G_{2}])$ can uniquely be written in
the form

$f=\sum_{n=0}^{\infty}\sum_{(i_{1},\ldots,i_{n})\in\\{1,2\\}^{n}}f_{i_{1},\ldots,i_{n}}$
(8)

with $f_{i_{1},\ldots,i_{n}}\in T_{i_{1},\ldots,i_{n}}$. Note that for $n=0$,
$T_{i_{1},\ldots,i_{n}}=k$ and in (8), the inner sum for $n=0$ has one summand
$f_{\emptyset}\in k$.

We say that $f$ is _multiplicatively coherent_ if

$\psi^{\ell}_{i_{1},\ldots,i_{n}}(f_{i_{1},\ldots,i_{n}})=f_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}}$
(9)

for all $n\geq 1$, $i_{1},\ldots,i_{n}\in\\{1,2\\}$ and
$\ell\in\\{1,\ldots,n\\}$. If

$\delta^{\ell}_{i_{1},\ldots,i_{n}}(f_{i_{1},\ldots,i_{n}})=f_{i_{1},\ldots,i_{\ell-1},i_{\ell+1},\ldots,i_{n}}$
(10)

for all $n\geq 1$, $i_{1},\ldots,i_{n}\in\\{1,2\\}$ and
$\ell\in\\{1,\ldots,n\\}$, then $f$ is _unitaly coherent_. If

$\theta^{\ell}_{i_{1},\ldots,i_{\ell-1},1,i_{\ell+1},\ldots,i_{n}}(f_{i_{1},\ldots,i_{\ell-1},1,i_{\ell+1},\ldots,i_{n}})=\theta^{\ell}_{i_{1},\ldots,i_{\ell-1},2,i_{\ell+1},\ldots,i_{n}}(f_{i_{1},\ldots,i_{\ell-1},2,i_{\ell+1},\ldots,i_{n}})$
(11)

for all $n\geq 1$, all $\ell\in\\{1,\ldots,n\\}$ and
$i_{1},\ldots,i_{\ell-1},i_{\ell+1},\ldots,i_{n}\in\\{1,2\\}$, then $f$ is
$H$-coherent. We call $f$ _coherent_ if it is multiplicatively, unitaly and
$H$-coherent.

If we think of $f$ as a function on the words in $G_{1}\uplus G_{2}$,
multiplicative coherence means that $f$ takes the same value on words that
only differ by replacing two adjacent elements from the same group with their
product. Unital coherence means that $f$ takes the same value on words that
only differ by deleting the identity element of $G_{1}$ or the identity
element of $G_{2}$. The meaning of $H$-coherence is that if $g_{1}\in G_{1}$
and $g_{2}\in G_{2}$ come from the same element in $H$, then the value of $f$
on a word does not change if a letter $g_{1}$ is replaced by $g_{2}$. We
define

$k[G_{1}*_{H}G_{2}]=\\{f\in C(k[G_{1}]\oplus k[G_{2}])|\ f\text{ is
coherent}\\}.$

As the $\psi^{\ell}_{i_{1},\ldots,i_{n}}$’s,
$\delta^{\ell}_{i_{1},\ldots,i_{n}}$’s and
$\theta_{i_{1},\ldots,i_{n}}^{\ell}$’s are morphisms of $k$-algebras, it is
clear that $k[G_{1}*_{H}G_{2}]$ is a subalgebra of $C(k[G_{1}]\oplus
k[G_{2}])$. On the other hand, it is not so clear that $k[G_{1}*_{H}G_{2}]$ is
a subcoalgebra of $C(k[G_{1}]\oplus k[G_{2}])$. To see this, we will work with
a different characterization of representative elements based on duality.

We briefly go back to the general setup where $V$ is a vector space over $k$.
We denote with $V^{*}=\operatorname{Hom}_{k}(V,k)$ the dual vector space of
functionals on $V$. Every element $f$ of $\hat{T}V$ defines a functional
$\operatorname{Fun}(f)\in(TV^{*})^{*}$ on $TV^{*}$. In detail, for every
$n\geq 0$ we have an evaluation map

$V^{\otimes n}\times(V^{*})^{\otimes n}\to k,\ (v,a)\mapsto a(v)$

determined by $(v_{1}\otimes\ldots\otimes v_{n},a_{1}\otimes\ldots\otimes
a_{n})\mapsto a_{1}(v_{1})\ldots a_{n}(v_{n})$. Then for
$f=(f_{0},f_{1},\ldots)\in\hat{T}V$ and $a=a_{0}+a_{1}+\ldots+a_{r}\in TV^{*}$
one sets $\operatorname{Fun}(f)(a)=\sum_{i=0}^{r}a_{i}(f_{i})$. The map
$\hat{T}V\to(TV^{*})^{*},\ f\mapsto\operatorname{Fun}(f)$ is injective. In a
similar fashion, any
$f\in\hat{T}V\hat{\otimes}\hat{T}V=\prod_{i,j\in\mathbb{N}}V^{\otimes
i}\otimes V^{\otimes j}$ defines a functional $\operatorname{Fun}(f)$ on
$TV^{*}\otimes TV^{*}$. The “comultiplication” $\Delta$ on $\hat{T}V$ is dual
to the usual (noncommutative) multiplication in the tensor algebra $TV^{*}$ in
the sense that

$\operatorname{Fun}(\Delta(f))(a\otimes b)=\operatorname{Fun}(f)(ab)$ (12)

for $f\in\hat{T}V$ and $a,b\in TV^{*}$.

For a functional $F$ on $TV^{*}$ and $b\in TV^{*}$ we define a functional
$R_{b}F$ on $TV^{*}$ by $(R_{b}F)(a)=F(ab)$ for all $a\in TV^{*}$. Similarly,
we define $L_{a}F$ by $(L_{a}F)(b)=F(ab)$. We abbreviate
$R_{b}f=R_{b}\operatorname{Fun}(f)$ and $L_{a}f=L_{a}\operatorname{Fun}(f)$.
Note that

$Rf=\\{R_{b}f|\ b\in TV^{*}\\}\text{ and }Lf=\\{L_{a}f|\ a\in TV^{*}\\}$

are subspaces of $(TV^{*})^{*}$. The following lemma is essentially Theorem
3.12 in [Haz03].

######  Lemma 2.1.

For $f\in\hat{T}V$ the following are equivalent:

  1. (i)

The element $f$ is representative.

  2. (ii)

The vector space $Rf$ is finite dimensional.

  3. (iii)

The vector space $Lf$ is finite dimensional.

Moreover, if these conditions are satisfied, then $\Delta(f)$ can be written
in the form $\Delta(f)=\sum_{i=1}^{r}g_{i}\otimes h_{i}$ with
$g_{i},h_{i}\in\hat{T}V$ representative and $\operatorname{Fun}(g_{i})\in Rf$
and $\operatorname{Fun}(h_{i})\in Lf$ for $i=1,\ldots,r$.

###### Proof.

Assume that $f$ is representative. So $\Delta(f)=\sum_{i=1}^{r}g_{i}\otimes
h_{i}$ for some $g_{i},h_{i}\in\hat{T}V$. Using (12) we find

$L_{a}f(b)=R_{b}f(a)=\operatorname{Fun}(f)(ab)=\sum_{i=0}^{r}\operatorname{Fun}(g_{i})(a)\operatorname{Fun}(h_{i})(b)$

for $a,b\in TV^{*}$. Thus $R_{b}f$ lies in the subspace generated by
$\operatorname{Fun}(g_{1}),\ldots,\operatorname{Fun}(g_{r})$ and $L_{a}f$ lies
in the subspace generated by
$\operatorname{Fun}(h_{1}),\ldots,\operatorname{Fun}(h_{r})$. Thus (i) implies
(ii) and (iii). The implication (ii)$\Rightarrow$(i) is shown in [Haz03,
Theorem 3.12] and the implication (iii)$\Rightarrow$(i) works similar to
(ii)$\Rightarrow$(i). The final statement of the lemma is contained in the
proof of [Haz03, Theorem 3.12]. See [Haz03, Remark 3.24]. ∎

We return to the discussion of the subalgebra $k[G_{1}*_{H}G_{2}]$ of
$C(k[G_{1}]\oplus k[G_{2}])$. For $n\geq 0$, the vector space
$(k[G_{1}]^{*}\oplus k[G_{2}]^{*})^{\otimes n}$ is the direct sum of the
subspaces

$T^{*}_{i_{1},\ldots,i_{n}}=k[G_{i_{1}}]^{*}\otimes\ldots\otimes
k[G_{i_{n}}]^{*}$

with $i_{1},\ldots,i_{n}\in\\{1,2\\}$.

The comultiplication $\Delta_{i}\colon k[G_{i}]\to k[G_{i}]\otimes k[G_{i}]$
dualizes to a map $(k[G_{i}]\otimes k[G_{i}])^{*}\to k[G_{i}]^{*}$.
Precomposing with $k[G_{i}]^{*}\otimes k[G_{i}]^{*}\to(k[G_{i}]\otimes
k[G_{i}])^{*}$, thus yields a map $\Delta_{i}^{*}\colon k[G_{i}]^{*}\otimes
k[G_{i}]^{*}\to k[G_{i}]^{*}$.

For $i_{1},\ldots,i_{n}\in\\{1,2\\}$ and $\ell\in\\{1,\ldots,n\\}$ we define a
map

${}^{*}\psi^{\ell}_{i_{1},\ldots,i_{n}}\colon
T^{*}_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}}\to
T^{*}_{i_{1},\ldots,i_{n}}$

by applying $\Delta_{i_{\ell}}^{*}$ on $k[G_{i_{\ell}}]^{*}\otimes
k[G_{i_{\ell}}]^{*}$ and the identity on all the other factors. Similarly, we
define a map

${}^{*}\delta^{\ell}_{i_{1},\ldots,i_{n}}\colon
T^{*}_{i_{1},\ldots,i_{\ell-1},i_{\ell+1},\ldots,i_{n}}\to
T^{*}_{i_{1},\ldots,i_{n}}$

by sending $a_{1}\otimes\ldots\otimes a_{n-1}$ to $a_{1}\otimes\ldots\otimes
a_{\ell-1}\otimes\epsilon_{i_{\ell}}\otimes a_{\ell},\ldots,a_{n-1}$. We also
define

${}^{*}\theta^{\ell}_{i_{1},\ldots,i_{n}}\colon
k[G_{i_{1}}]^{*}\otimes\ldots\otimes k[G_{i_{\ell-1}}]^{*}\otimes
k[H]^{*}\otimes k[G_{i_{\ell+1}}]^{*}\otimes\ldots\otimes
k[G_{i_{n}}]^{*}\longrightarrow T^{*}_{i_{1},\ldots,i_{n}}$

by applying $\alpha_{i_{\ell}}^{*}\colon k[H]^{*}\to k[G_{i_{\ell}}]^{*}$ on
the $\ell$-th factor and the identity on all the other factors.

A functional $F\in T(k[G_{1}]^{*}\oplus k[G_{2}]^{*})^{*}$ is called
_multiplicatively coherent_ if

$\textstyle{T^{*}_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{*}\psi^{\ell}_{i_{1},\ldots,i_{n}}}$$\scriptstyle{F}$$\textstyle{T^{*}_{i_{1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{k}$

commutes for every ${}^{*}\psi^{\ell}_{i_{1},\ldots,i_{n}}$. If

$\textstyle{T^{*}_{i_{1},\ldots,i_{\ell-1},i_{\ell+1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{{}^{*}\delta^{\ell}_{i_{1},\ldots,i_{n}}}$$\textstyle{T^{*}_{i_{1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{k}$

commutes for every ${}^{*}\delta^{\ell}_{i_{1},\ldots,i_{n}}$, then $F$ is
_unitaly coherent_. If

$\textstyle{k[G_{i_{1}}]^{*}\otimes\ldots\otimes k[G_{i_{\ell-1}}]^{*}\otimes
k[H]^{*}\otimes k[G_{i_{\ell+1}}]^{*}\otimes\ldots\otimes
k[G_{i_{n}}]^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{*}\theta^{\ell}_{i_{1},\ldots,i_{\ell-1},2,i_{\ell+1},\ldots,i_{n}}}$$\scriptstyle{{}^{*}\theta^{\ell}_{i_{1},\ldots,i_{\ell-1},1,i_{\ell+1},\ldots,i_{n}}}$$\textstyle{T^{*}_{i_{1}\ldots,i_{\ell-1},1,i_{\ell+1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{T^{*}_{i_{1}\ldots,i_{\ell-1},2,i_{\ell+1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{k}$

commutes for all $n\geq 1$, $\ell\in\\{1,\ldots,n\\}$ and
$i_{1},\ldots,i_{\ell-1},i_{\ell+1},\ldots,i_{n}\in\\{1,2\\}$, then $F$ is
$H$-coherent. Finally, $F$ is _coherent_ if it is multiplicatively, unitaly
and $H$/̄coherent.

The following lemma shows that whether or not $f\in\hat{T}(k[G_{1}]\oplus
k[G_{2}])$ is coherent can be tested on $\operatorname{Fun}(f)$.

######  Lemma 2.2.

Let $f\in\hat{T}(k[G_{1}]\oplus k[G_{2}])$. Then $f$ is coherent if and only
if $\operatorname{Fun}(f)$ is coherent.

###### Proof.

We first treat the case of multiplicative coherence. Fix $n\geq 1$,
${i_{1},\ldots,i_{n}}\in\\{1,2\\}^{n}$ and $\ell\in\\{1,\ldots,n\\}$. Let
$a_{1}\otimes\ldots\otimes a_{n+1}\in
T^{*}_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}}$. As
$\Delta_{i_{\ell}}^{*}(a_{\ell}\otimes a_{\ell+1})(h)=(a_{\ell}\otimes
a_{\ell+1})(\Delta_{i_{\ell}}(h))$ for $h\in k[G_{i_{\ell}}]$, we see that

${}^{*}\psi^{\ell}_{i_{1},\ldots,i_{n}}(a_{1}$
$\displaystyle\otimes\ldots\otimes a_{n+1})(h_{1}\otimes\ldots\otimes h_{n})=$
$\displaystyle=(a_{1}\otimes\ldots\otimes
a_{\ell-1}\otimes\Delta_{i_{\ell}}^{*}(a_{\ell}\otimes a_{\ell+1})\otimes
a_{\ell+2}\otimes\ldots\otimes a_{n+1})(h_{1}\otimes\ldots\otimes h_{n})=$
$\displaystyle=(a_{1}\otimes\ldots\otimes
a_{n+1})(h_{1}\otimes\ldots\otimes\Delta_{i_{\ell}}(h_{\ell})\otimes\ldots\otimes
h_{n})=$ $\displaystyle=(a_{1}\otimes\ldots\otimes
a_{n+1})(\psi^{\ell}_{i_{1},\ldots,i_{n}}(h_{1}\otimes\ldots\otimes h_{n}))$

for $h_{1}\otimes\ldots\otimes h_{n}\in T_{i_{1},\ldots,i_{n}}$. So
${}^{*}\psi^{\ell}_{i_{1},\ldots,i_{n}}(a)(h)=a(\psi^{\ell}_{i_{1},\ldots,i_{n}}(h))$
for all $a\in T^{*}_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}}$ and $h\in
T_{i_{1},\ldots,i_{n}}$.

Assume that $f$ is multiplicatively coherent. Then, writing $f$ as is in (8),
we have
${}^{*}\psi^{\ell}_{i_{1},\ldots,i_{n}}(a)(f_{i_{1},\ldots,i_{n}})=a(f_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}})$
for all $a\in T^{*}_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}}$. Thus the
diagram

$\textstyle{T^{*}_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{*}\psi^{\ell}_{i_{1},\ldots,i_{n}}}$$\scriptstyle{\operatorname{Fun}(f)}$$\textstyle{T^{*}_{i_{1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Fun}(f)}$$\textstyle{k}$

commutes and $\operatorname{Fun}(f)$ is multiplicatively coherent.

Conversely, assume that $\operatorname{Fun}(f)$ is multiplicatively coherent.
Then

$a(\psi^{\ell}_{i_{1},\ldots,i_{n}}(f_{i_{1},\ldots,i_{n}}))={{}^{*}\psi^{\ell}_{i_{1},\ldots,i_{n}}}(a)(f_{i_{1},\ldots,i_{n}})=a(f_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}})$

for all $a\in T^{*}_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}}$ and so
$\psi^{\ell}_{i_{1},\ldots,i_{n}}(f_{i_{1},\ldots,i_{n}})=f_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}}$,
i.e., $f$ is multiplicatively coherent.

The case of unital coherence is similar, using that
${}^{*}\delta^{\ell}_{i_{1},\ldots,i_{n}}(a)(h)=a(\delta^{\ell}_{i_{1},\ldots,i_{n}}(h))$
for $a\in T^{*}_{i_{1},\ldots,i_{\ell-1},i_{\ell+1},\ldots,i_{n}}$ and $h\in
T_{i_{1},\ldots,i_{n}}$.

For the case of $H$-coherence one uses that

${{}^{*}\theta}^{\ell}_{i_{1},\ldots,i_{\ell-1},i,i_{\ell+1},\ldots,i_{n}}(a)(h)=a(\theta^{\ell}_{i_{1},\ldots,i_{\ell-1},i,i_{\ell+1},\ldots,i_{n}}(h))$

for

$a\in k[G_{i_{1}}]^{*}\otimes\ldots\otimes k[G_{i_{\ell-1}}]^{*}\otimes
k[H]^{*}\otimes k[G_{i_{\ell+1}}]^{*}\otimes\ldots\otimes k[G_{i_{n}}]^{*},$

$h\in T_{i_{1},\ldots,i_{\ell-1},i,i_{\ell+1},\ldots,i_{n}}$ and $i=1,2$. ∎

The following lemma is the main reason, why the dual perspective is helpful
for showing that $k[G_{1}*_{H}G_{2}]$ is a subcoalgebra of $C(k[G_{1}]\oplus
k[G_{2}])$.

######  Lemma 2.3.

Let $F$ be a coherent functional on $T(k[G_{1}]^{*}\oplus k[G_{2}]^{*})$. Then
$R_{b}F$ is coherent for every $b\in T(k[G_{1}]^{*}\oplus k[G_{2}]^{*})$.
Similarly for $L_{a}F$.

###### Proof.

We first treat the case of multiplicative coherence. As the sum of
multiplicatively coherent functionals is multiplicatively coherent and
$R_{b_{1}+b_{2}}F=R_{b_{1}}F+R_{b_{2}}F$, we can reduce to the case that
$b=b_{1}\otimes\ldots\otimes b_{n}$ is a pure tensor in some
$T^{*}_{i_{1},\ldots,i_{n}}$. Moreover, as
$R_{b_{1}b_{2}}F=R_{b_{1}}(R_{b_{2}}F)$, we are reduced to the case $b\in
k[G_{i}]^{*}$ for some $i\in\\{1,2\\}$. Consider the diagram

$\textstyle{T^{*}_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{*}\psi^{\ell}_{i_{1},\ldots,i_{n}}}$$\textstyle{T^{*}_{i_{1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T^{*}_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}}\otimes
k[G_{i}]^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi}$$\scriptstyle{F}$$\textstyle{T^{*}_{i_{1},\ldots,i_{n}}\otimes
k[G_{i}]^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{k}$

where $\chi$ is the base change of ${}^{*}\psi^{\ell}_{i_{1},\ldots,i_{n}}$
and the two vertical arrows are multiplication with $b$ (using the
noncommutative multiplication of the tensor algebra). Then the top square
commutes. As $\chi={{}^{*}\psi^{\ell}_{i_{1},\ldots,i_{n},i}}$ and $F$ is
multiplicatively coherent, the lower triangle commutes. So the whole diagram
commutes, showing that $R_{b}F$ is multiplicatively coherent.

The argument for unital coherence and $H$-coherence is similar. The proof for
$L$ instead of $R$ is analogous. ∎

We can finally show that $k[G_{1}*_{H}G_{2}]$ is a subcoalgebra of
$C(k[G_{1}]\oplus k[G_{2}])$. Let $f\in k[G_{1}*_{H}G_{2}]$. We have to show
that

$\Delta(f)\in k[G_{1}*_{H}G_{2}]\otimes k[G_{1}*_{H}G_{2}]\subseteq
C(k[G_{1}]\oplus k[G_{2}])\otimes C(k[G_{1}]\oplus k[G_{2}]).$

From Lemma 2.1 we know that $\Delta(f)$ can be written as
$\Delta(f)=\sum_{i=1}^{r}g_{i}\otimes h_{i}$ with $g_{i},h_{i}\in
C(k[G_{1}]\oplus k[G_{2}])$,
$\operatorname{Fun}(g_{1}),\ldots,\operatorname{Fun}(g_{r})\in Rf$ and
$\operatorname{Fun}(h_{1}),\ldots,\operatorname{Fun}(h_{i})\in Lf$. Lemma 2.3
implies that the functionals
$\operatorname{Fun}(g_{1}),\ldots,\operatorname{Fun}(g_{r}),\operatorname{Fun}(h_{1}),\ldots,\operatorname{Fun}(h_{i})$
are coherent. So Lemma 2.2 implies that
$g_{1},\ldots,g_{r},h_{1},\ldots,h_{r}$ are coherent, i.e., they lie in
$k[G_{1}*_{H}G_{2}]$ as desired.

So $k[G_{1}*_{H}G_{2}]$ is a subbialgebra of $C(k[G_{1}]\oplus k[G_{2}])$. The
next step is to show that $k[G_{1}*_{H}G_{2}]$ is a Hopf algebra, i.e., we
need to find the antipode. Define

$S\colon\hat{T}(k[G_{1}]\oplus k[G_{2}])\to\hat{T}(k[G_{1}]\oplus k[G_{2}])$

by applying the antipodes $S_{i}\colon k[G_{i}]\to k[G_{i}]$ ($i=1,2$) on each
of the factors and then reversing the order of the factors. So $S\colon
T_{i_{1},\ldots,i_{n}}\to T_{i_{n},\ldots,i_{1}}$ is given by
$S(h_{1}\otimes\ldots\otimes h_{n})=S_{i_{n}}(h_{n})\otimes\ldots\otimes
S_{i_{1}}(h_{1})$. We would like to know that $S(C(k[G_{1}]\oplus
k[G_{2}]))\subseteq C(k[G_{1}]\oplus k[G_{2}])$.

Recall that for a vector space $V$, we defined
$\hat{T}V\hat{\otimes}\hat{T}V=\prod_{i,j\in\mathbb{N}}V^{\otimes i}\otimes
V^{\otimes j}$. Define
$\tau\colon\hat{T}V\hat{\otimes}\hat{T}V\to\hat{T}V\hat{\otimes}\hat{T}V$ by

$\tau\colon V^{\otimes i}\otimes V^{\otimes j}\to V^{\otimes j}\otimes V^{i},\
a\otimes b\mapsto b\otimes a.$

So in the matrix notation (7), the map $\tau$ transposes the matrix and then
applies the flip to every matrix entry. From this description it is clear that
$\tau$ maps $\hat{T}V\otimes\hat{T}V$ to $\hat{T}V\otimes\hat{T}V$.

For $f\in\hat{T}(k[G_{1}]\oplus k[G_{2}])$, a direct computation shows that
$\Delta(S(f))=\tau((S\otimes S)(\Delta(f)))$. So if $f\in C(k[G_{1}]\oplus
k[G_{2}])$, then $\Delta(S(f))\in\hat{T}(k[G_{1}]\oplus
k[G_{2}])\otimes\hat{T}(k[G_{1}]\oplus k[G_{2}])$, i.e., $S(f)\in
C(k[G_{1}]\oplus k[G_{2}])$. Thus $S$ restricts to a map $S\colon
C(k[G_{1}]\oplus k[G_{2}])\to C(k[G_{1}]\oplus k[G_{2}])$. The commutativity
of the diagrams

$\textstyle{T_{i_{1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{\ell}_{i_{1},\ldots,i_{n}}}$$\scriptstyle{S}$$\textstyle{T_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$$\textstyle{T_{i_{n},\ldots,i_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{n-\ell+1}_{i_{n},\ldots,i_{1}}}$$\textstyle{T_{i_{n},\ldots,i_{\ell},i_{\ell},\ldots,i_{1}}}$

$\textstyle{T_{i_{1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{\ell}_{i_{1},\ldots,i_{n}}}$$\scriptstyle{S}$$\textstyle{T_{i_{1},\ldots,i_{\ell-1},i_{\ell+1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$$\textstyle{T_{i_{n},\ldots,i_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{n-\ell+1}_{i_{1},\ldots,i_{n}}}$$\textstyle{T_{i_{n},\ldots,i_{\ell+1},i_{\ell-1},\ldots,i_{1}}}$

and

$\textstyle{T_{i_{1},\ldots,1,\ldots
i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$$\scriptstyle{\theta^{\ell}_{i_{1},\ldots,1,\dots,i_{n}}}$$\textstyle{k[G_{i_{1}}]\otimes\ldots\otimes
k[H]\otimes\ldots\otimes
k[G_{i_{n}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{i_{1},\ldots,2,\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{\ell}_{i_{1},\ldots,2,\dots,i_{n}}}$$\scriptstyle{S}$$\textstyle{T_{i_{n},\ldots,1,\ldots
i_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{n-\ell+1}_{i_{n},\ldots,1,\dots,i_{1}}}$$\textstyle{k[G_{i_{n}}]\otimes\ldots\otimes
k[H]\otimes\ldots\otimes
k[G_{i_{1}}]}$$\textstyle{T_{i_{n},\ldots,2,\ldots,i_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{n-\ell+1}_{i_{n},\ldots,2,\dots,i_{1}}}$

show that indeed $S$ restricts to a map $S\colon k[G_{1}*_{H}G_{2}]\to
k[G_{1}*_{H}G_{2}]$. To see that $S$ is an antipode for the bialgebra
$k[G_{1}*_{H}G_{2}]$, we need to show that

$\textstyle{k[G_{1}*_{H}G_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\scriptstyle{\epsilon}$$\textstyle{k[G_{1}*_{H}G_{2}]\otimes
k[G_{1}*_{H}G_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S\cdot\operatorname{id}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k[G_{1}*_{H}G_{2}]}$
(13)

commutes.

Let $f\in k[G_{1}*_{H}G_{2}]$. Note that the product in
$\hat{T}(k[G_{1}]\oplus k[G_{2}])$ is such that for $a\in
T_{i_{1},\ldots,i_{n}}$ and $b\in T_{j_{1},\ldots,j_{m}}$ we have $ab=0$,
unless $m=n$ and $i_{1}=j_{1},\ldots,i_{n}=j_{n}$. Thus, when trying to
determine the image of $\Delta(f)$ under $S\cdot\operatorname{id}$, only the
components of $\Delta(f)$ in $T_{i_{n},\ldots,i_{1}}\otimes
T_{i_{1},\ldots,i_{n}}\subseteq\hat{T}(k[G_{1}]\oplus
k[G_{2}])\hat{\otimes}\hat{T}(k[G_{1}]\oplus k[G_{2}])$ are relevant, as all
other components map to zero under $S\cdot\operatorname{id}$. So the component
$(S\cdot\operatorname{id})(\Delta(f))_{i_{1},\ldots,i_{n}}$ of
$(S\cdot\operatorname{id})(\Delta(f))$ in $T_{i_{1},\ldots,i_{n}}$, is the
image of the component $f_{i_{n},\ldots,i_{1},i_{1},\ldots,i_{n}}$ of $f$ in
$T_{i_{n},\ldots,i_{1},i_{1},\ldots,i_{n}}$ under the map

$\xi_{i_{1},\ldots,i_{n}}\colon T_{i_{n},\ldots,i_{1},i_{1},\ldots,i_{n}}\to
T_{i_{1},\ldots,i_{n}},\quad a_{n}\otimes\ldots\otimes a_{1}\otimes
b_{1}\otimes\ldots\otimes b_{n}\mapsto
S_{i_{1}}(a_{1})b_{1}\otimes\ldots\otimes S_{i_{n}}(a_{n})b_{n}.$

We will show, by induction on $n$, that
$\xi_{i_{1},\ldots,i_{n}}(f_{i_{n},\ldots,i_{1},i_{1},\ldots,i_{n}})=\epsilon(f)\in
k$ as desired. For $n=1$, $T_{i_{1},i_{1}}=k[G_{i_{1}}]\otimes k[G_{i_{1}}]$,
$T_{i_{1}}=k[G_{i_{1}}]$ and the commutative diagram

$\textstyle{k[G_{i_{1}}]\otimes
k[G_{i_{1}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{i_{1},i_{i}}=S_{i_{1}}\cdot\operatorname{id}}$$\textstyle{k[G_{i_{1}}]}$$\textstyle{k[G_{i_{1}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{1}_{i_{1}}=\Delta_{i_{1}}}$$\scriptstyle{\epsilon_{i_{1}}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

shows that $\xi_{i_{1},i_{1}}(f_{i_{1},i_{1}})=\epsilon_{i_{1}}(f_{i_{1}})$.
But for $n=1$, the condition
$\delta^{\ell}_{i_{1},\ldots,i_{n}}(f_{i_{1},\ldots,i_{n}})=f_{i_{1},\ldots,i_{\ell-1},i_{\ell+1},\ldots,i_{n}}$
means that $\epsilon_{i_{1}}(f_{i_{1}})=f_{\emptyset}=\epsilon(f)$, the degree
zero component of $f$. This resolves the $n=1$ case. For $n>1$, the diagram

$\textstyle{T_{i_{n},\ldots,i_{1},i_{1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{i_{1},\ldots,i_{n}}}$$\textstyle{T_{i_{n},\ldots,i_{1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{n}_{i_{n},\ldots,i_{1},\ldots,i_{n}}}$$\scriptstyle{\delta^{n}_{i_{n}\ldots,i_{1},\ldots,i_{n}}}$$\textstyle{T_{i_{1},\ldots,i_{n}}}$$\textstyle{T_{i_{n},\ldots,i_{2},i_{2},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{i_{2},\ldots,i_{n}}}$$\textstyle{T_{i_{2},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

commutes. (This is probably most conveniently seen by looking at what happens
with the tuples of group elements when considering the dual morphisms.) So, by
induction,

$\displaystyle\xi_{i_{1},\ldots,i_{n}}(f_{i_{n},\ldots,i_{1},i_{1},\ldots,i_{n}})$
$\displaystyle=\xi_{i_{1},\ldots,i_{n}}(\psi^{n}_{i_{n},\ldots,i_{1},\ldots,i_{n}}(f_{i_{n},\ldots,i_{1},\ldots,i_{n}}))=$
$\displaystyle=\xi_{i_{2},\ldots,i_{n}}(\delta^{n}_{i_{n},\ldots,i_{1},\ldots,i_{n}}(f_{i_{n},\ldots,i_{1},\ldots,i_{n}}))=\xi_{i_{2},\ldots,i_{n}}(f_{i_{n},\ldots,i_{2},i_{2},\ldots,i_{n}})=$
$\displaystyle=\epsilon(f).$

Thus (13) commutes and $k[G_{1}*_{H}G_{2}]$ is indeed a Hopf algebra.

The map $\pi\colon C(k[G_{1}]\oplus k[G_{2}])\to k[G_{1}]\oplus k[G_{2}]$ can
be restricted to $k[G_{1}*_{H}G_{2}]$ and composing with the projections on
$k[G_{1}]\oplus k[G_{2}]$ yields maps $p_{1}\colon k[G_{1}*_{H}G_{2}]\to
k[G_{1}]$ and $p_{2}\colon k[G_{1}*_{H}G_{2}]\to k[G_{1}]$.

It is clear that $p_{i}$ $(i=1,2)$ is a morphism of algebras. Compatibility
with the counit follows from (10) with $n=1$, whereas compatibility with the
comultiplication follows from (9) with $n=1$. The commutativity of the diagram

$\textstyle{k[G_{1}*_{H}G_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\textstyle{k[G_{1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{1}}$$\textstyle{k[G_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{2}}$$\textstyle{k[H]}$

follows from (11) with $n=1$.

######  Proposition 2.4.

The Hopf algebra $k[G_{1}*_{H}G_{2}]$ together with the projections
$p_{1},p_{2}$ is the fibre product of $\alpha_{1}\colon k[G_{1}]\to k[H]$ and
$\alpha_{2}\colon k[G_{2}]\to k[H]$ in the category of Hopf algebras.

###### Proof.

Let $k[G]$ be a Hopf algebra with morphisms $q_{1}\colon k[G]\to k[G_{1}]$ and
$q_{2}\colon k[G]\to k[G_{2}]$ of Hopf algebras such that
$\alpha_{1}q_{1}=\alpha_{2}q_{2}$. We then have a morphism
$\varphi=(q_{1},q_{2})\colon k[G]\to k[G_{1}]\oplus k[G_{2}]$ of $k$-algebras.
By the universal property of $C(k[G_{1}]\oplus k[G_{2}])$, there exists a
unique morphism $\tilde{\varphi}\colon k[G]\to C(k[G_{1}]\oplus k[G_{2}])$ of
bialgebras such that $\pi\tilde{\varphi}=\varphi$.

We claim that $\tilde{\varphi}$ maps $k[G]$ into $k[G_{1}*_{H}G_{2}]\subseteq
C(k[G_{1}]\oplus k[G_{2}])$. Let $f\in k[G]$. Then

$\tilde{\varphi}(f)=(\epsilon_{G}(f),\varphi(f),\varphi^{\otimes
2}(\Delta_{G}^{2}(f)),\varphi^{\otimes 3}(\Delta_{G}^{3}(f)),\ldots).$

The commutative diagram

$\textstyle{k[G]^{\otimes
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\otimes
n}}$$\textstyle{(k[G_{1}]\oplus k[G_{2}])^{\otimes
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{i_{1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{\ell}_{i_{1},\ldots,i_{n}}}$$\textstyle{k[G]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{G}^{n}}$$\scriptstyle{\Delta_{G}^{n+1}}$$\textstyle{k[G]^{\otimes(n+1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\otimes(n+1)}}$$\textstyle{(k[G_{1}]\oplus
k[G_{2}])^{\otimes(n+1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{i_{1},\ldots,i_{\ell},i_{\ell},\ldots,i_{n}}}$

shows that $\tilde{\varphi}(f)$ is multiplicatively coherent. The commutative
diagram

$\textstyle{k[G]^{\otimes
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\otimes
n}}$$\textstyle{(k[G_{1}]\oplus k[G_{2}])^{\otimes
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{i_{1},\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{\ell}_{i_{1},\ldots,i_{n}}}$$\textstyle{k[G]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{G}^{n}}$$\scriptstyle{\Delta_{G}^{n-1}}$$\textstyle{k[G]^{\otimes(n-1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\otimes(n-1)}}$$\textstyle{(k[G_{1}]\oplus
k[G_{2}])^{\otimes(n-1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{i_{1},\ldots,i_{\ell-1},i_{\ell+1},\ldots,i_{n}}}$

shows that $\tilde{\varphi}(f)$ is unitaly coherent. Finally, the commutative
diagram

$\textstyle{k[G]^{\otimes
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\otimes
n}}$$\textstyle{(k[G_{1}]\oplus k[G_{2}])^{\otimes
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{i_{1},\ldots,1,\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{\ell}_{i_{1},\ldots,1,\ldots,i_{n}}}$$\textstyle{k[G]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{G}^{n}}$$\scriptstyle{\Delta_{G}^{n}}$$\textstyle{k[G_{i_{1}}]\otimes\ldots\otimes
k[H]\otimes\ldots\otimes k[G_{i_{n}}]}$$\textstyle{k[G]^{\otimes
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\otimes
n}}$$\textstyle{(k[G_{1}]\oplus
k[G_{2}])^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{i_{1},\ldots,2,\ldots,i_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{\ell}_{i_{1},\ldots,2,\ldots,i_{n}}}$

shows that $\tilde{\varphi}(f)$ is $H$-coherent. So $\tilde{\varphi}(f)$ is
coherent and we have a well-defined morphism $\tilde{\varphi}\colon k[G]\to
k[G_{1}*_{H}G_{2}]$ with $p_{1}\tilde{\varphi}=q_{1}$ and
$p_{2}\tilde{\varphi}=q_{2}$.

If $\varphi^{\prime}\colon k[G]\to k[G_{1}*_{H}G_{2}]$ is another morphism
with $p_{1}\varphi^{\prime}=q_{1}$ and $p_{2}\varphi^{\prime}=q_{2}$, then
$\pi\varphi^{\prime}=\varphi$ and the universal property of $C(k[G_{1}]\oplus
k[G_{2}])$ yields $\varphi^{\prime}=\tilde{\varphi}$. ∎

###  2.2. Properties of amalgamated free products

In this section we discuss properties of amalgamated free products and their
compatibility with other constructions like proalgebraic completion or base
change. Throughout Section 2.2 we assume that $H\to G_{1}$ and $H\to G_{2}$
are two morphisms of $k$-groups.

######  Lemma 2.5.

The canonical morphism $G_{1}*G_{2}\to G_{1}*_{H}G_{2}$ is a quotient map.

###### Proof.

The canonical morphism $G_{1}*G_{2}\to G_{1}*_{H}G_{2}$ is dual to the
inclusion $k[G_{1}*_{H}G_{2}]\subseteq k[G_{1}*G_{2}]$ of subalgebras of
$C(k[G_{1}]\oplus k[G_{2}])$. ∎

######  Proposition 2.6.

If $G_{1}$ and $G_{2}$ are geometrically reduced, then $G_{1}*_{H}G_{2}$ is
geometrically reduced.

###### Proof.

According to Lemma 2.5 it suffices to show that $k[G_{1}*G_{2}]$ is
geometrically reduced. Let $\overline{k}$ denote the algebraic closure of $k$.
We have to show that $k[G_{1}*G_{2}]\otimes_{k}\overline{k}$ is reduced. Since
$k[G_{1}*G_{2}]\otimes_{k}\overline{k}$ is a subring of
$\hat{T}(k[G_{1}]\oplus k[G_{2}])\otimes_{k}\overline{k}$, it suffices to
shows that $\hat{T}(k[G_{1}]\oplus k[G_{2}])\otimes_{k}\overline{k}$ is
reduced. For $i=1,2$ we abbreviate
$\overline{k}[G_{i}]=k[G_{i}]\otimes_{k}\overline{k}$. As
$\hat{T}(k[G_{1}]\oplus k[G_{2}])\otimes_{k}\overline{k}$ injects into
$\hat{T}(\overline{k}[G_{1}]\oplus\overline{k}[G_{2}])$, it suffices to see
that $\hat{T}(\overline{k}[G_{1}]\oplus\overline{k}[G_{2}])$. But as a ring,
$\hat{T}(\overline{k}[G_{1}]\oplus\overline{k}[G_{2}])$ is the product of the
rings $(\overline{k}[G_{1}]\oplus\overline{k}[G_{2}])^{\otimes n}$. As
$\overline{k}[G_{1}]\oplus\overline{k}[G_{2}]=\overline{k}[G_{1}]\times\overline{k}[G_{2}]$
is reduced, also $(\overline{k}[G_{1}]\oplus\overline{k}[G_{2}])^{\otimes n}$
is reduced ([Bou90, Chapter V, §15, No. 5, Theorem 3]) and therefore
$\hat{T}(\overline{k}[G_{1}]\oplus\overline{k}[G_{2}])$ is reduced. ∎

######  Lemma 2.7.

The morphisms $G_{1}\to G_{1}*G_{2}$ and $G_{2}\to G_{1}*G_{2}$ are closed
embeddings.

###### Proof.

It suffices to show that $G_{1}(R)\to(G_{1}*G_{2})(R)$ is injective for every
$k$-algebra $R$. Set $\phi_{1}=\operatorname{id}\colon G_{1}\to G_{1}$ and let
$\phi_{2}\colon G_{2}\to G_{1}$ be the trivial morphism, i.e., the morphism
that factors through the trivial group. By the universal property of
$G_{1}*G_{2}$, there exists a morphism $\phi\colon G_{1}*G_{2}\to G_{1}$ such
that the composition $G_{1}\to G_{1}*G_{2}\xrightarrow{\phi}G_{1}$ is the
identity. In particular, $G_{1}(R)\to(G_{1}*G_{2})(R)$ is injective for every
$k$-algebra $R$. ∎

It may seem natural to expect that $G_{i}\to G_{1}*_{H}G_{2}$ ($i=1,2$) are
closed embeddings if $H\to G_{i}$ are closed embeddings, since the analogous
statement holds for abstract groups. However, the analogous statement does not
even hold for profinite groups ([RZ10, Example 9.2.10]) and therefore one
should not expect it for proalgebraic groups.

In a similar vein, still assuming that $H\to G_{i}$ are closed embeddings, one
may hope that the natural map
$G_{1}(R)*_{H(R)}G_{2}(R)\to(G_{1}*_{H}G_{2})(R)$ is injective for every
$k$-algebra $R$, where $G_{1}(R)*_{H(R)}G_{2}(R)$ is the amalgamated free
product of abstract groups. However, if $G_{1}(R)\to(G_{1}*_{H}G_{2})(R)$ is
not injective, then also $G_{1}(R)*_{H(R)}G_{2}(R)\to(G_{1}*_{H}G_{2})(R)$
cannot be injective, because $G_{1}(R)\to G_{1}(R)*_{H(R)}G_{2}(R)$ is
injective.

For a $k$-group $G$, let $G^{\operatorname{et}}$ be the maximal pro-étale
quotient. It is canonically isomorphic to the $k$/̄group $\pi_{0}(G)$ of
connected components of $G$. Every morphism of $k$-groups $H\to
G^{\operatorname{et}}$ factors uniquely $H\to H^{\operatorname{et}}\to
G^{\operatorname{et}}$. Hence, for $k$-groups $H,G_{1},G_{2}$ and $G$, the
following two commutative diagrams are equivalent

$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\operatorname{et}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\operatorname{et}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G}$
and
$\textstyle{H^{\operatorname{et}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\operatorname{et}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\operatorname{et}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\;.}$

Hence we have a canonical identification

$G^{\operatorname{et}}_{1}\ast_{H}G^{\operatorname{et}}_{2}=G^{\operatorname{et}}_{1}\ast_{H^{\operatorname{et}}}G^{\operatorname{et}}_{2}\;.$
(14)

######  Proposition 2.8.

Given morphisms $\alpha_{1}:H\to G_{1}$ and $\alpha_{2}:H\to G_{2}$ of
$k$-groups, the canonical morphism

$G_{1}\ast_{H}G_{2}\longrightarrow
G^{\operatorname{et}}_{1}\ast_{H}G^{\operatorname{et}}_{2}=G^{\operatorname{et}}_{1}\ast_{H^{\operatorname{et}}}G^{\operatorname{et}}_{2}$

induces an isomorphism

$(G_{1}\ast_{H}G_{2})^{\operatorname{et}}\xrightarrow{\sim}(G^{\operatorname{et}}_{1}\ast_{H^{\operatorname{et}}}G^{\operatorname{et}}_{2})^{\operatorname{et}}\;.$
(15)

Equivalently we have an isomorphism

$\pi_{0}(G_{1}\ast_{H}G_{2})\xrightarrow{\sim}\pi_{0}(\pi_{0}(G_{1})\ast_{\pi_{0}(H)}\pi_{0}(G_{2}))\;.$
(16)

###### Proof.

Consider commutative diagrams of $k$-groups where $E$ is pro-étale

$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\alpha_{2}}$$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\,.}$

We say that the diagram is a pro-étale push-out, if for every other such
diagram with pro-étale $E^{\prime}$ there is a unique morphism $E\to
E^{\prime}$ for which the obvious diagrams commute. A pro-étale push-out is
unique up to a unique isomorphism. It is straightforward to check that both
sides of (15) give rise to pro-étale push-outs $E$. Hence (15) is an
isomorphism. ∎

######  Remark 2.9.

It is not true that the amalgamated product of pro-étale $k$-groups is again
pro-étale. Consider the representations

$\mathbb{Z}/4\to\operatorname{GL}_{2}(\mathbb{Q})\;,\;1\mapsto\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\right)\quad\text{and}\quad\mathbb{Z}/6\to\operatorname{GL}_{2}(\mathbb{Q})\;,\;1\mapsto\left(\begin{smallmatrix}1&-1\\\
1&0\end{smallmatrix}\right)\;.$

Their images are known to generate the infinite subgroup
$\operatorname{SL}_{2}(\mathbb{Z})$ of $\operatorname{GL}_{2}(\mathbb{Q})$. We
have corresponding representations of $\mathbb{Q}$-groups
$(\mathbb{Z}/4)_{\mathbb{Q}}\to\operatorname{GL}_{2,\mathbb{Q}}$ and
$(\mathbb{Z}/6)_{\mathbb{Q}}\to\operatorname{GL}_{2,\mathbb{Q}}$ and hence an
algebraic representation of the free product

$(\mathbb{Z}/4)_{\mathbb{Q}}\ast(\mathbb{Z}/6)_{\mathbb{Q}}\xrightarrow{\rho}\operatorname{GL}_{2,\mathbb{Q}}\;.$

Its image on $\mathbb{Q}$-valued points contains
$\operatorname{SL}_{2}(\mathbb{Z})$ because the latter is the image of the
composition

$\mathbb{Z}/4\ast\mathbb{Z}/6\longrightarrow((\mathbb{Z}/4)_{\mathbb{Q}}\ast(\mathbb{Z}/6)_{\mathbb{Q}})(\mathbb{Q})\xrightarrow{\rho(\mathbb{Q})}\operatorname{GL}_{2}(\mathbb{Q})\;.$

The algebraic quotients of pro-étale group schemes are finite. Hence
$(\mathbb{Z}/4)_{\mathbb{Q}}\ast(\mathbb{Z}/6)_{\mathbb{Q}}$ cannot be pro-
étale.

######  Corollary 2.10.

If $G_{1},G_{2}$ are connected $k$-groups, then $G_{1}\ast_{H}G_{2}$ is
connected as well.

###### Proof.

By Proposition 2.8, we have

$\pi_{0}(G_{1}\ast_{H}G_{2})=\pi_{0}(\pi_{0}(G_{1})\ast_{\pi_{0}(H)}\pi_{0}(G_{2}))\;.$

An affine group scheme $G$ (over a field $k$) is connected if and only if
$\pi_{0}(G)=\operatorname{Spec}k$ is the trivial group. Since the amalgamated
product of two trivial $k$-groups is the trivial $k$-group by the universal
property, the assertion follows. ∎

We next show that free products are compatible with the formation of _free
proalgebraic groups_. To this end, let us recall the definition of the free
proalgebraic group $\Gamma_{X}$ on a set $X$ ([Wib20, Section 2.5]). Let
$\overline{k}$ denote the algebraic closure of $k$ and let $G$ be a $k$-group.
A map $X\to G(\overline{k})$ _converges to $1$_ if for every quotient map
$G\to H$ with $H$ algebraic (i.e., of finite type over $k$) the composition
$X\to G(\overline{k})\to H(\overline{k})$ maps all but finitely many elements
of $X$ to the identity element of $H(\overline{k})$. The free proalgebraic
group $\Gamma_{X}$ on $X$ comes equipped with a map
$X\to\Gamma_{X}(\overline{k})$ that converges to $1$ and satisfies the
following universal property: If $X\to G(\overline{k})$ is a map converging to
$1$, then there exists a unique morphism $\Gamma_{X}\to G$ of $k$-groups such
that

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma_{X}(\overline{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G(\overline{k})}$

commutes.

######  Proposition 2.11.

Let $X$ and $Y$ be sets, then $\Gamma_{X}*\Gamma_{Y}\simeq\Gamma_{X\uplus Y}$.

###### Proof.

We will show that $\Gamma_{X\uplus Y}$ satisfies the universal property of
$\Gamma_{X}*\Gamma_{Y}$. As the compositions $X\to X\uplus Y\to\Gamma_{X\uplus
Y}(\overline{k})$ and $Y\to X\uplus Y\to\Gamma_{X\uplus Y}(\overline{k})$
converge to $1$, we have induced morphisms $\Gamma_{X}\to\Gamma_{X\uplus Y}$
and $\Gamma_{Y}\to\Gamma_{X\uplus Y}$. Given morphisms $\Gamma_{X}\to G$ and
$\Gamma_{Y}\to G$ of $k$-groups, we can define a map $X\uplus Y\to
G(\overline{k})$ by sending an element of $X$ to its image under
$X\to\Gamma_{X}(\overline{k})\to G(\overline{k})$ and an element of $Y$ to its
image under $Y\to\Gamma_{X}(\overline{k})\to G(\overline{k})$. Then $X\uplus
Y\to G(\overline{k})$ converges to $1$ and therefore induces a morphism
$\Gamma_{X\uplus Y}\to G$. The diagrams

$\textstyle{\Gamma_{X\uplus
Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G}$$\textstyle{\Gamma_{X\uplus
Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G}$$\textstyle{\Gamma_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(17)

commute because their restrictions to $X$ respectively $Y$ commute. On the
other hand, any two morphisms $\Gamma_{X\uplus Y}\to G$ making (17) commute,
will have the same restriction to $X\uplus Y$ and therefore are equal. ∎

We next show that amalgamated free products are compatible with _proalgebraic
completion_. For a discrete group $\Gamma$, let $\Gamma^{\operatorname{alg}}$
be the proalgebraic completion (or proalgebraic hull) of $\Gamma$ over $k$. It
is a $k$-group equipped with a morphism
$\Gamma\to\Gamma^{\operatorname{alg}}(k)$ with the universal property that any
linear representation $\Gamma\to\operatorname{GL}(V)$ on a finite dimensional
$k$-vector space $V$ comes from an algebraic representation of
$\Gamma^{\operatorname{alg}}\to\operatorname{GL}_{V}$ by composition

$\Gamma\longrightarrow\Gamma^{\operatorname{alg}}(k)\longrightarrow\operatorname{GL}_{V}(k)=\operatorname{GL}(V)\;.$

The $k$-group $\Gamma^{\operatorname{alg}}$ may be obtained as the Tannakian
dual of the category $\operatorname{Rep}(\Gamma)$ of representations of
$\Gamma$ on finite dimensional $k$-vector spaces, with its canonical fibre
functor. For more background on proalgebraic completions see e.g., [BLMM02,
Section 2.1].

######  Proposition 2.12.

Given morphisms of discrete groups $\Gamma\to\Gamma_{1}$ and
$\Gamma\to\Gamma_{2}$, the following canonical morphism is an isomorphism:

$(\Gamma_{1}\ast_{\Gamma}\Gamma_{2})^{\operatorname{alg}}\xrightarrow{\sim}\Gamma^{\operatorname{alg}}_{1}\ast_{\Gamma^{\operatorname{alg}}}\Gamma^{\operatorname{alg}}_{2}\;.$

###### Proof.

We have equivalences (Section 1)

$\displaystyle\operatorname{Rep}(\Gamma^{\operatorname{alg}}_{1}\ast_{\Gamma^{\operatorname{alg}}}\Gamma^{\operatorname{alg}}_{2})$
$\displaystyle=\operatorname{Rep}(\Gamma^{\operatorname{alg}}_{1})\times_{\operatorname{Rep}(\Gamma^{\operatorname{alg}})}\operatorname{Rep}(\Gamma^{\operatorname{alg}}_{2})$
$\displaystyle=\operatorname{Rep}(\Gamma_{1})\times_{\operatorname{Rep}(\Gamma)}\operatorname{Rep}(\Gamma_{2})$
$\displaystyle=\operatorname{Rep}(\Gamma_{1}\ast_{\Gamma}\Gamma_{2})=\operatorname{Rep}((\Gamma_{1}\ast_{\Gamma}\Gamma_{2})^{\operatorname{alg}})\;.$

The assertion now follows from Tannakian duality. ∎

Contrary to what one might expect, the amalgamated free product may not be
compatible with base change. Indeed, if $H\to G_{1}$ and $H\to G_{2}$ are
morphisms of $k$-groups and $K$ is a field extension of $k$, the commutative
diagram

$\textstyle{(G_{1}*_{H}G_{2})_{K}}$$\textstyle{G_{1,K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{2,K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

yields a morphism $G_{1,K}*_{H_{K}}G_{2,K}\to(G_{1}*_{H}G_{2})_{K}$. The dual
morphism is the restriction of $\widehat{T}(k[G_{1}]\oplus
k[G_{2}])\otimes_{k}K\to\widehat{T}((k[G_{1}]\oplus k[G_{2}])\otimes_{k}K)$.
The latter map is always injective but rarely surjective ([Bou98, Chapter II,
§3.7, Corollary 3]). So $(G_{1}*_{H}G_{2})_{K}$ is a quotient of
$G_{1,K}*_{H_{K}}G_{2,K}$. To construct a concrete example, where the morphism
$G_{1,K}*_{H_{K}}G_{2,K}\to(G_{1}*_{H}G_{2})_{K}$ fails to be an isomorphism,
we need some preparation.

Assume, for now, that $k$ is an algebraically closed field of characteristic
zero. Let $C_{2}$ denote the (abstract) group with two elements. We first show
that the set of isomorphism classes of irreducible finite dimensional
$k$-linear representations of $C_{2}*C_{2}$ has cardinality $|k|$. Note that
to specify a representation of $C_{2}$ on a $k$-vector space $V$ is equivalent
to specifying a direct sum decomposition $V=V_{1}\oplus V_{-1}$, where the
non-trivial element of $C_{2}$ acts trivially on $V_{1}$ and by
multiplications with $-1$ on $V_{-1}$. To specify a representation of
$C_{1}*C_{2}$ is thus equivalent to specifying two direct sum decompositions
$V=V_{1}\oplus V_{-1}$ and $V=V^{\prime}_{1}\oplus V^{\prime}_{-1}$. The
corresponding representation of $C_{2}*C_{2}$ is irreducible if and only if
$V_{i}\cap V^{\prime}_{j}=0$ for all $i,j\in\\{1,-1\\}$. For $a\in
k\smallsetminus\\{0,1\\}$ let $V_{a}=k^{2}$ be the representation of
$C_{2}*C_{2}$ determined by $V_{1}=k\begin{pmatrix}1\\\ 0\end{pmatrix}$,
$V_{-1}=k\begin{pmatrix}0\\\ 1\end{pmatrix}$,
$V_{1}^{\prime}=k\begin{pmatrix}1\\\ 1\end{pmatrix}$ and
$V_{-1,a}^{\prime}=k\begin{pmatrix}1\\\ a\end{pmatrix}$. The representations
$V_{a}$ are irreducible and pairwise non-isomorphic. Indeed, if $a,b\in
k\smallsetminus\\{0,1\\}$, and $f\colon V_{a}\to V_{b}$ is an isomorphism,
then $f$ must stabilize $V_{1}$ and $V_{-1}$, i.e., $f$ is given by a diagonal
matrix. As $f$ also stabilizes $V_{1}^{\prime}$, $f$ is in fact given by a
scalar matrix. As $f$ maps $V^{\prime}_{-1,a}$ into $V^{\prime}_{-1,b}$, we
find that $a=b$.

Since the $V_{a}$’s are irreducible and pairwise non-isomorphic as
representations of $C_{1}*C_{2}$, the $V_{a}$’s are also irreducible and
pairwise non-isomorphic as representations of
$(C_{2}*C_{2})^{\operatorname{alg}}=(C_{2}*C_{2})^{\operatorname{alg},k}$.
Every (finite dimensional) representation of a $k$-group $G$ embeds into
$k[G]^{n}$ for some $n\geq 1$ ([Wat79a]). It follows that every irreducible
representation of $G$ embeds into $k[G]$. In particular, every $V_{a}$ embeds
into $k[(C_{2}*C_{2})^{\operatorname{alg}}]$. As a sum of pairwise non-
isomorphic irreducible representations is always a direct sum, it follows that
the dimension $\dim_{k}k[(C_{2}*C_{2})^{\operatorname{alg}}]$ of
$k[(C_{2}*C_{2})^{\operatorname{alg}}]$ as a $k$-vector space is at least
$|k|$.

On the other hand, the set $S$ of all morphisms $\rho\colon
C_{2}*C_{2}\to\operatorname{GL}_{n(\rho)}(k)$ has at most cardinality $|k|$
and $(C_{2}*C_{2})^{\operatorname{alg}}$ can be realized as a closed subgroup
scheme of $\prod_{\rho\in S}\operatorname{GL}_{n(\rho),k}$ (see [BLMM02,
Section 2.1]). So the dimension of $k[(C_{2}*C_{2})^{\operatorname{alg}}]$ is
at most $|k|$. Therefore $\dim_{k}k[(C_{2}*C_{2})^{\operatorname{alg}}]=|k|$.

Let $\mu_{2}=\mu_{2,k}$ denote the $k$-group of second roots of unity, i.e.,
$\mu_{2}(R)=\\{g\in R^{\times}|\ g^{2}=1\\}$ for every $k$-algebra $R$. Using
Proposition 2.12 we find

$\mu_{2}*\mu_{2}=C_{2}^{\operatorname{alg}}*C_{2}^{\operatorname{alg}}=(C_{2}*C_{2})^{\operatorname{alg}}.$

So $\dim_{k}k[\mu_{2}*\mu_{2}]=|k|$. Therefore, if $k\subseteq K$ is an
inclusion of algebraically closed fields of characteristic zero with
$|k|<|K|$, then $\mu_{2,K}*\mu_{2,K}$ and $(\mu_{2,k}*\mu_{2,k})_{K}$ cannot
be isomorphic because the coordinate ring of $\mu_{2,K}*\mu_{2,K}$ has
$K$-dimension $|K|$, whereas the coordinate ring of
$(\mu_{2,k}*\mu_{2,k})_{K}$ has $K$-dimension $|k|$.

##  3\. Local systems and a Seifer-van Kampen Theorem

The main goal of this short section is to establish an analog of the Seifert-
van Kampen theorem for the proalgebraic fundamental group.

Let $X$ be a topological space. We denote with
$\mathbb{Loc}(X)=\mathbb{Loc}_{k}(X)$ the category of local systems of finite
dimensional $k$-vector spaces on $X$. An object $\mathcal{E}$ of
$\mathbb{Loc}(X)$ is thus a sheaf of $k$-vector spaces on $X$, such that every
point $x\in X$ has an open neighborhood $U$ with
$\mathcal{E}|_{U}\simeq\underline{k^{r}}$, where $\underline{k^{r}}$ is the
constant sheaf on $U$ associated to the vector space $k^{r}$ (and $r\geq 0$
may depend on $U$).

For an open subset $U$ of $X$ and $\mathcal{E}\in\mathbb{Loc}(X)$, the
restriction $\mathcal{E}|_{U}$ is in $\mathbb{Loc}(U)$. Indeed
$\mathcal{E}\rightsquigarrow\mathcal{E}|_{U}$ is a functor
$\mathbb{Loc}(X)\to\mathbb{Loc}(U)$.

######  Lemma 3.1.

Let $X$ be a topological space and let $U_{1},U_{2}$ be open subsets of $X$
such that $X=U_{1}\cup U_{2}$. Then the functor
$\mathcal{E}\rightsquigarrow(\mathcal{E}|_{U_{1}},\mathcal{E}|_{U_{2}},\operatorname{id}_{\mathcal{E}|_{U_{1}\cap
U_{2}}})$ from $\mathbb{Loc}(X)$ to
$\mathbb{Loc}(U_{1})\times_{\mathbb{Loc}(U_{1}\cap U_{2})}\mathbb{Loc}(U_{2})$
is an equivalence of categories.

###### Proof.

As in Example 1.2 this is just the sheaf property. See [Sta23, Tag 00AK]. ∎

The espace étale of any $\mathcal{E}$ in $\mathbb{Loc}(X)$ may be viewed as a
vector bundle over $X$ with locally constant transition functions in
$\operatorname{GL}_{r}(k)$ for some $r$ which may vary. Equivalently, if we
give $k$ the discrete topology, these are the vector bundles with continuous
$\operatorname{GL}_{r}(k)$-valued transition functions. The stalk
$\mathcal{E}_{x}$ of $\mathcal{E}$ in a point $x\in X$ is isomorphic to the
fibre in $x$ of the corresponding vector bundle. We have
$\mathcal{E}_{x}=k^{r}$ for some $r=r(x)\geq 0$. If $X$ is connected, these
ranks are independent of $x$ and $\mathbb{Loc}_{k}(X)$ is a neutral Tannakian
category. It is neutralized by the fibre functor
$\omega_{x}:\mathbb{Loc}(X)\to\operatorname{Vec}_{k}$, which sends local
systems and their morphisms to the respective stalks, c.f. [Den, Proposition
2.1]. In [Den], the _proalgebraic fundamental group_ $\pi_{k}(X,x)$ of a
connected pointed topological space $(X,x)$ was defined to be the Tannakian
dual of $(\mathbb{Loc}_{k}(X),\omega_{x})$. Thus we have

$\mathbb{Loc}_{k}(X)=\operatorname{Rep}(\pi_{k}(X,x))\;.$

The previous constructions immediately give the following result:

######  Theorem 3.2.

Let $U_{1},U_{2}$ be open subsets of a topological space $X$ such that
$X=U_{1}\cup U_{2}$ and assume that $X,U_{1},U_{2}$ and $U_{1}\cap
U_{2}\neq\emptyset$ are all connected. Then, for any point $x\in U_{1}\cap
U_{2}$ the commutative diagram of $k$-groups

$\textstyle{\pi_{k}(U_{1}\cap
U_{2},x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{k}(U_{1},x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{k}(U_{2},x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{k}(X,x)}$
(18)

is a push-out diagram in the category of $k$-groups, i.e. we have an
isomorphism

$\pi_{k}(X,x)\xrightarrow{\sim}\pi_{k}(U_{1},x)\times_{\pi_{k}(U_{1}\cap
U_{2},x)}\pi_{k}(U_{2},x)\;.$

###### Proof.

By Lemma 3.1 the following diagram is a fibre product of categories

$\textstyle{\mathbb{Loc}_{k}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Loc}_{k}(U_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Loc}_{k}(U_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Loc}_{k}(U_{1}\cap
U_{2})\;.}$

Note that all (restriction) functors commute with the respective fibre
functors $\omega_{x}$ on the respective categories. Hence the diagram of
representation categories induced by (18) is a fibre product as well

$\textstyle{\mathbb{Rep}(\pi_{k}(X,x))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Rep}(\pi_{k}(U_{1},x))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Rep}(\pi_{k}(U_{2},x))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Rep}(\pi_{k}(U_{1}\cap
U_{2},x))\;.}$

By Corollary 1.11 in the situation of Corollary 1.12 it follows that (18) is a
push-out diagram. ∎

##  4\. The proalgebraic fundamental group of a topological space is perfect

Throughout Section 4 we assume that $k$ is a perfect field of characteristic
$p>0$. An affine group scheme $G=\operatorname{Spec}A$ over $k$ is _perfect_
if the $p$-power map $(\;)^{p}$ is bijective on $A$. In this section we give
Tannakian criteria with applications for when the Tannaka dual group of a
neutralized Tannakian category $(\mathcal{T},\omega)$ over $k$ is reduced
resp. perfect. The main tool is the functor $\operatorname{Fr}_{+}$ from
[Cou20] and our criteria are implicitely contained in [Cou20]. For convenience
we give short self-contained proofs of the required properties of
$\operatorname{Fr}_{+}$. In fact, there are two versions of
$\operatorname{Fr}_{+}$ depending on the choice of a subgroup
$S\subseteq\mathcal{S}_{p}$, the symmetric group on $p$ letters, and either
can be used in the following. The two choices are $S=\mathcal{S}_{p}$ and
$S=\langle\sigma\rangle=\mathbb{Z}/p$ where $\sigma$ is the cyclic permutation
$\sigma=(1,2,\ldots,p)$. For an object $X$ of $\mathcal{T}$ there is a natural
action of $S$ on $X^{\otimes p}$ and $\operatorname{Fr}_{+}(X)$ is the image
in $\mathcal{T}$ of the composion:

$H^{0}(S,X^{\otimes p})\hookrightarrow X^{\otimes
p}\overset{[\,]}{\twoheadrightarrow}H_{0}(S,X^{\otimes p})\;.$

Here one sets

$H^{0}(S,X^{\otimes
p})=\ker\Big{(}(\sigma-\operatorname{id})_{\sigma}:X^{\otimes
p}\to\prod_{\sigma\in S}X^{\otimes p}\Big{)}$

and

$H_{0}(S,X^{\otimes
p})=\operatorname{coker}\Big{(}\sum_{\sigma}(\sigma-\operatorname{id})_{\sigma}:\bigoplus_{\sigma\in
S}X^{\otimes p}\to X^{\otimes p}\Big{)}\;.$

For a morphism $\varphi$ in $\mathcal{T}$ there is a canonical induced
morphism $\operatorname{Fr}_{+}(\varphi)$ and $\operatorname{Fr}_{+}$ becomes
a functor.

Let $\mathcal{T}^{(p)}$ be the category with the same objects as $\mathcal{T}$
and morphisms

$\operatorname{Hom}_{\mathcal{T}^{(p)}}(X,Y)=\operatorname{Hom}_{\mathcal{T}}(X,Y)\otimes_{k,(\,)^{p}}k\;.$

These are $k$-vector spaces via multiplication on the second factor. We view
$\operatorname{Fr}_{+}$ as a functor from $\mathcal{T}^{(p)}$ to $\mathcal{T}$
by setting
$\operatorname{Fr}_{+}(\varphi\otimes\lambda)=\lambda\operatorname{Fr}_{+}(\varphi)$.
Note that this is well defined since for $\mu\in k$ we have
$\operatorname{Fr}_{+}(\mu\varphi)=\mu^{p}\operatorname{Fr}_{+}(\varphi)$.
Thus $\operatorname{Fr}_{+}:\mathcal{T}^{(p)}\to\mathcal{T}$ is compatible
with scalar multiplication. It is not difficult to see that
$\operatorname{Fr}_{+}$ is also additive and hence $k$-linear. E.g. for
$S=\mathcal{S}_{p}$ and morphisms $\varphi,\psi:X\to Y$ we have
$\operatorname{Fr}_{+}(\varphi+\psi)=\operatorname{Fr}_{+}(\varphi)+\operatorname{Fr}_{+}(\psi)$
because as morphisms from $H_{0}(\mathcal{S}_{p},X^{\otimes p})$ to
$H_{0}(\mathcal{S}_{p},Y^{\otimes p})$ we have:

$[(\varphi+\psi)^{\otimes p}]=\sum^{p}_{i=0}{p\choose i}[\varphi^{\otimes
i}\otimes\psi^{\otimes p-i}]=[\varphi^{\otimes p}]+[\psi^{\otimes p}]\;.$

Define a faithful tensor functor
$\omega^{(p)}:\mathcal{T}^{(p)}\to\operatorname{Vec}_{k}$ by setting
$\omega^{(p)}(X)=\omega(X)\otimes_{k,(\,)^{p}}k$ and
$\omega^{(p)}(\varphi\otimes\lambda)=\omega(\varphi)\otimes\lambda$. Then
$(\mathcal{T}^{(p)},\omega^{(p)})$ becomes a neutralized Tannakian category
over $k$. If $G=\operatorname{Spec}A$ is the Tannakian dual of
$(\mathcal{T},\omega)$, then $G^{(p)}=\operatorname{Spec}A^{(p)}$ with
$A^{(p)}=A\otimes_{k,(\,)^{p}}k$ may be identified with the Tannakian dual of
$(\mathcal{T}^{(p)},\omega^{(p)})$ because a $k$-linear $G^{(p)}$-action on
$\omega^{(p)}(X)$ can be identified with a $k$-linear $G$-action on
$\omega(X)$.

Let $F_{\operatorname{rel}}:G\to G^{(p)}$ be the relative Frobenius morphism.
It is the $k$-linear morphism corresponding to the map of $k$-algebras
$F^{\sharp}_{\operatorname{rel}}:A^{(p)}\to A$ sending $a\otimes\lambda$ to
$\lambda a^{p}$. The following facts can be found in [Cou20]:

######  Proposition 4.1.

  1. (i)

For $V$ in $\operatorname{Vec}_{k}$ the natural map

$t(V):V\otimes_{k,(\,)^{p}}k\xrightarrow{\sim}\operatorname{Fr}_{+}(V)\;,\;v\otimes\lambda\longmapsto\lambda[v\otimes\ldots\otimes
v]$

is a $k$-linear isomorphism.

  2. (ii)

The maps in a) induce a natural isomorphism of functors
$\omega^{(p)}=\omega\circ\operatorname{Fr}_{+}:\mathcal{T}^{(p)}\to\operatorname{Vec}_{k}$
and the functor $\operatorname{Fr}_{+}:\mathcal{T}^{(p)}\to\mathcal{T}$ can be
identified with the functor
$\operatorname{Rep}(F_{\operatorname{rel}}):\operatorname{Rep}(G^{(p)})\to\operatorname{Rep}(G)$
between the corresponding representation categories.

  3. (iii)

$\operatorname{Fr}_{+}$ is an exact tensor functor.

###### Proof.

(i) For $S=\mathcal{S}_{p}$ the map $t(V)$ is additive by the binomial
theorem. Additivity in the case $S=\langle\sigma\rangle=\mathbb{Z}/p$ is also
not difficult to see. Compatibility with scalar multiplications is obvious so
that $t(V)$ is $k$-linear. Since $t(k)$ is an isomorphism and the functors
$\\_\otimes_{k,(\,)^{p}}k$ and $\operatorname{Fr}_{+}$ are additive on
$\operatorname{Vec}_{k}$, it follows that $t(V)$ is an isomorphism for all
$V$.
(ii) For $X$ in
$\operatorname{Ob}\,\mathcal{T}^{(p)}=\operatorname{Ob}\,\mathcal{T}$ the
isomorphisms $t(\omega(X))$ from a) give the first of the isomorphisms
$\omega^{(p)}=\operatorname{Fr}_{+}\circ\,\omega=\omega\circ\operatorname{Fr}_{+}$.
The second one follows since $\omega$ is an exact $k$-linear
$\otimes$-functor. The $G^{(p)}$-action $\rho^{(p)}$ on $\omega^{(p)}(X)$
corresponds to a $G$-action $\rho$ on $V=\omega(X)$. By functoriality, we get
an induced $G$-action $\operatorname{Fr}_{+}(\rho)$ on
$\operatorname{Fr}_{+}(V)$ and via the isomorphism $t(V)$ a $G$-action
$\rho^{\prime}$ on $\omega^{(p)}(X)=V\otimes_{k,(\,)^{p}}k$. We have to show
that $\rho^{\prime}=\rho^{(p)}\circ F_{\operatorname{rel}}$. Let $\Delta:V\to
V\otimes A$ be the $A$-comodule map corresponding to $\rho$ and
$\Delta_{\operatorname{Fr}_{+}(V)}$ the induced $A$-comodule map on
$\operatorname{Fr}_{+}(V)$. For $v\in V$ set $\Delta(v)=\sum_{i}v_{i}\otimes
a_{i}$. Then we have on $v\otimes 1\in V\otimes_{k,(\,)^{p}}k$

$\displaystyle(\Delta_{\operatorname{Fr}_{+}(V)}\circ t(V))(v\otimes 1)$
$\displaystyle=\Delta_{\operatorname{Fr}_{+}(V)}([v\otimes\ldots\otimes v])$
$\displaystyle=\sum_{i_{1},\ldots,i_{p}}[v_{i_{1}}\otimes\ldots\otimes
v_{ip}]\otimes a_{i_{1}}\cdots a_{i_{p}}$
$\displaystyle=\sum_{i}[v_{i}\otimes\ldots\otimes v_{i}]\otimes a^{p}_{i}\;.$

The last equation holds because if the stabilizer $S_{\underline{i}}$ of
$\underline{i}=(i_{1},\dots,i_{p})$ in
$S=\langle\sigma\rangle=\mathbb{Z}/p\subseteq\mathcal{S}_{p}$ is not $S$, i.e.
if not all $i_{1},\ldots,i_{p}$ are equal, then $S_{\underline{i}}=1$ and we
sum $|S|=p$-times the same element in
$A\otimes\operatorname{Fr}_{+}(V)\subseteq A\otimes H_{0}(S,V^{\otimes p})$
which gives zero. For $S=\mathcal{S}_{p}$ this is all the more true since
$H_{0}(\mathcal{S}_{p},V^{\otimes p})$ is a quotient of
$H_{0}(\mathbb{Z}/p,V^{\otimes p})$. On the other hand, we have

$\displaystyle((t(V)\otimes\operatorname{id}_{A})\circ(\operatorname{id}\otimes
F^{\sharp}_{\operatorname{rel}})\circ\Delta^{(p)})(v\otimes 1)$
$\displaystyle=((t(V)\otimes\operatorname{id}_{A})\circ(\operatorname{id}\otimes
F^{\sharp}_{\operatorname{rel}}))\Big{(}\sum_{i}(v_{i}\otimes
1)\otimes(a_{i}\otimes 1)\Big{)}$
$\displaystyle=(t(V)\otimes\operatorname{id}_{A})\Big{(}\sum_{i}(v_{i}\otimes
1)\otimes a^{p}_{i}\Big{)}=\sum_{i}[v_{i}\otimes\ldots\otimes v_{i}]\otimes
a^{p}_{i}\;.$

Thus we have checked that $\rho^{\prime}=\rho^{(p)}\circ
F_{\operatorname{rel}}$ and b) is proved.
(iii) These assertions follow from the interpretation of
$\operatorname{Fr}_{+}$ as $\operatorname{Rep}(F_{\operatorname{rel}})$ in
(ii). ∎

######  Corollary 4.2.

For a neutralized Tannakian category $(\mathcal{T},\omega)$ over a perfect
field $k$ of characteristic $p>0$ let $G$ be its Tannakian dual.

  1. (i)

The group scheme $G$ is reduced if and only if the functor
$\operatorname{Fr}_{+}:\mathcal{T}^{(p)}\to\mathcal{T}$ is fully faithful and
every subobject of $\operatorname{Fr}_{+}(T)$ for some $T$ in
$\operatorname{Ob}\,\mathcal{T}^{(p)}$ is isomorphic to
$\operatorname{Fr}_{+}(T^{\prime})$ for some subobject $T^{\prime}$ of $T$.

  2. (ii)

The group scheme $G$ is perfect if and only if $\operatorname{Fr}_{+}$ is an
equivalence of categories.

###### Proof.

(i) Note that $G$ is reduced if and only if $(\;)^{p}$ is injective on $A$.
This condition is equivalent to the map
$F^{\sharp}_{\operatorname{rel}}:A^{(p)}\to A$ being injective, i.e. to the
relative Frobenius $F_{\operatorname{rel}}:G\to G^{(p)}$ being faithfully
flat, c.f. [Wat79b, 14.1 Theorem]. By [DM82, Proposition 2.21] and Proposition
4.1, (ii) this translates into the stated properties of
$\operatorname{Fr}_{+}=\operatorname{Rep}(F_{\operatorname{rel}})$.
(ii) $G$ is perfect if and only if $(\;)^{p}$ is an automorphism of $A$ as an
$\mathbb{F}_{p}$-algebra. Since $A^{(p)}\cong A$ as $\mathbb{F}_{p}$-algebras
this is equivalent to $F^{\sharp}_{\operatorname{rel}}$ being an isomorphism
of $k$-algebras i.e. to $F_{\operatorname{rel}}$ being an isomorphism of
$k$-group schemes. By Tannakian duality and Proposition 4.1 (ii) this is
equivalent to
$\operatorname{Fr}_{+}=\operatorname{Rep}(F_{\operatorname{rel}})$ being an
equivalence of categories. ∎

######  Corollary 4.3.

Let $k$ be a perfect field of positive characteristic $p$.

  1. (i)