$\mathrm{Set}[\mathbb{T}]=\operatorname{PSh}({\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}}^{\mathrm{op}})=[\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c},\mathrm{Set}]\hbox
to0.0pt{.\hss}$

That is, if there is any presheaf site for $\mathrm{Set}[\mathbb{T}]$, then
the canonically given category
${\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}}^{\mathrm{op}}$
is also a presheaf site of definition for $\mathrm{Set}[\mathbb{T}]$. We note
that the category
$\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}$ is essentially
small, so that it can actually be used as a site. There is also a simple
description of the universal $\mathbb{T}$-model $M$ in
$[\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c},\mathrm{Set}]$,
namely, $M$ is the “tautological” model, interpreting a sort $A$ by the
functor

$\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}\to\mathrm{Set},\quad
M^{\prime}\mapsto\llbracket A\rrbracket_{M^{\prime}}\hbox to0.0pt{,\hss}$

and similarly for relation and function symbols [5, Theorem 6.1.1].

For a theory of presheaf type $\mathbb{T}$, the category
$\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})$ is finitely
accessible. This can be a useful criterion for showing that a theory is not of
presheaf type.

The basic source of theories of presheaf type is the following. Recall that a
geometric theory is a _Horn_ theory, if all of its axioms are of the form
$\phi\;\vdash_{\Gamma}\;\psi$ with $\phi$ and $\psi$ finite conjunctions of
atomic formulas. In particular, algebraic theories are Horn theories. Recall
also that in a _cartesian_ theory the formulas $\phi$ and $\psi$ can in
addition to finite conjunctions also contain existential quantifiers, but only
if these refer to unique existence, provably relative to all preceding axioms
in some chosen well-ordering. Then, all Horn theories and more generally all
cartesian theories are of presheaf type [5, Theorem 2.1.8]. For Horn theories,
there is even a good general description of the compact models.

######  Lemma 4.2.4.

Let $\mathbb{T}$ be a Horn theory. Then a model $M$ of $\mathbb{T}$ is compact
if and only if it is presented by some Horn formula (finite conjunction of
atomic formulas) in context $\vec{x}.\,\phi$, meaning that there is a natural
isomorphism

$\operatorname{Hom}(M,-)\cong\llbracket\phi\rrbracket_{-}\;:\;\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})\to\mathrm{Set}\hbox
to0.0pt{.\hss}$

Furthermore, every Horn formula in context presents some model.

###### Proof.

See [4, Lemma 4.3]. ∎

In other words, the compact models of a Horn theory are the ones presented by
finitely many generators (with specified sorts) and finitely many relations in
the form of atomic formulas.

For understanding the notion of rigid-topology quotients, we should recall the
Comparison Lemma, which allows to eliminate some of the objects of a site
under certain conditions. If $(C,J)$ is a site and $C^{\prime}\subseteq C$ is
any full subcategory, then we can restrict the Grothendieck topology $J$ to
$C^{\prime}$, letting a sieve $S$ on an object $c\in C^{\prime}$ be covering
for $J|_{C^{\prime}}$ if and only if the sieve generated by the arrows of $S$
in the bigger category $C$ is covering for $J$. In other words,
$J|_{C^{\prime}}$ is the biggest topology on $C^{\prime}$ such that the
inclusion $C^{\prime}\hookrightarrow C$ preserves covers. A _dense_ full
subcategory $C^{\prime}\subseteq C$ of a site $(C,J)$ is one such that every
object of $C$ can be covered by (a sieve generated by arrows from) objects in
$C^{\prime}$.

######  Theorem 4.2.5 (Comparison Lemma).

If $C^{\prime}$ is a dense full subcategory of a site $(C,J)$, then
restricting sheaves on $C$ to $C^{\prime}$ is an equivalence of categories

$\operatorname{Sh}(C^{\prime},J|_{C^{\prime}})\simeq\operatorname{Sh}(C).$

###### Proof.

See [7, Theorem C2.2.3]. ∎

Now suppose we want to add some axioms to a theory of presheaf type and would
like to show that the resulting theory is again of presheaf type. By the
duality between quotient theories and subtoposes, and the fact that the latter
correspond to Grothendieck topologies on any given site of definition, a
quotient extension $\mathbb{Q}$ of a theory of presheaf type $\mathbb{T}$
induces a topology on
${\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}}^{\mathrm{op}}$,
which we denote $J_{\mathbb{Q}}$. If there is a dense full subcategory of
${\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}}^{\mathrm{op}}$
with respect to $J_{\mathbb{Q}}$, such that $J_{\mathbb{Q}}$ becomes the
trivial topology when restricted to this subcategory, then by the Comparison
Lemma, $\mathbb{T}+\mathbb{Q}$ is again of presheaf type. This condition can
equivalently be expressed by saying that the _irreducible_ objects of
${\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}}^{\mathrm{op}}$,
which admit no covering sieve except the maximal sieve, are dense. A topology
with this property is called a _rigid_ topology.

######  Definition 4.2.6.

Let $\mathbb{T}$ be a theory of presheaf type. We say that a quotient
extension $\mathbb{Q}$ of $\mathbb{T}$ is a _rigid-topology quotient_ if the
Grothendieck topology $J_{\mathbb{Q}}$ on
${{\mathbb{T}\operatorname{-mod}(\mathrm{Set})}_{c}}^{\mathrm{op}}$ is rigid.

Note that for a quotient extension $\mathbb{Q}$ of an arbitrary geometric
theory $\mathbb{T}$, we would have to fix a site of definition for
$\mathrm{Set}[\mathbb{T}]$ before we can ask if the induced topology is rigid.

After proving that some $\mathbb{Q}$ is a rigid-topology quotient, we would
probably like to have a description of the compact models of
$\mathbb{T}+\mathbb{Q}$. From Lemma 4.2.2 it follows that the full subcategory

$(\mathbb{T}+\mathbb{Q})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})\subseteq\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})$

is closed under filtered colimits. With this, one easily sees that a
$(\mathbb{T}+\mathbb{Q})$-model which is compact as a $\mathbb{T}$-model is
also compact as a $(\mathbb{T}+\mathbb{Q})$-model (even without any presheaf
type conditions). The converse is part of the following result, which
simultaneously answers the question whether $\mathbb{Q}$ being a rigid-
topology quotient is necessary for $\mathbb{T}+\mathbb{Q}$ to be again of
presheaf type.

######  Lemma 4.2.7.

A quotient extension $\mathbb{Q}$ of a theory of presheaf type $\mathbb{T}$ is
a rigid-topology quotient if and only if $\mathbb{T}+\mathbb{Q}$ is again of
presheaf type and additionally every compact $(\mathbb{T}+\mathbb{Q})$-model
is also compact as a $\mathbb{T}$-model.

###### Proof.

See [5, Theorem 8.2.6]. ∎

So for example, take as $\mathbb{T}$ the theory consisting of a sort $A$ and
countably many constant symbols $c_{n}\,{:}\,A$, and let $\mathbb{Q}$ be the
axiom

$\top\;\vdash_{x,y\,{:}\,A}\;x=y\hbox to0.0pt{.\hss}$

Then both $\mathbb{T}$ and $\mathbb{T}+\mathbb{Q}$ are algebraic theories and
thus of presheaf type. But in the unique $\mathrm{Set}$-based model of
$\mathbb{T}+\mathbb{Q}$, the constants $c_{n}$ are all identified, which can
not happen in a compact model of $\mathbb{T}$, by Lemma 4.2.4. So $\mathbb{Q}$
is not a rigid-topology quotient.

###  4.3 Rigid-topology quotients

Let $M\in\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}$ be a
compact model of a theory of presheaf type and let

$\phi\;\vdash_{\vec{x}\,{:}\,\vec{A}}\;\psi$

be an axiom that we would like to add to $\mathbb{T}$. This axiom is satisfied
in the model $M$ if and only if we have
$\llbracket\phi\rrbracket_{M}\subseteq\llbracket\psi\rrbracket_{M}$. But in
any case, given some $\vec{x}\in\llbracket\phi\rrbracket_{M}$, we can consider
the collection of all arrows $f:M\to M^{\prime}$ in
$\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}$ that map
$\vec{x}$ into $\llbracket\psi\rrbracket_{M^{\prime}}$, which turns out to be
a cosieve on $M$, since any further model homomorphism $M^{\prime}\to
M^{\prime\prime}$ preserves truth of the geometric formula $\psi$. We now show
that these sieves in
${\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}}^{\mathrm{op}}$
generate the Grothendieck topology corresponding to the axiom
$\phi\;\vdash_{\vec{x}\,{:}\,\vec{A}}\;\psi$.

######  Theorem 4.3.1.

Let $\mathbb{T}$ be a theory of presheaf type and let
$\mathbb{Q}=\\{\,\phi\;\vdash_{\vec{x}\,{:}\,\vec{A}}\;\psi\,\\}$ be a
quotient extension of $\mathbb{T}$ adding a single axiom. Then the
Grothendieck topology $J_{\mathbb{Q}}$ on
${{\mathbb{T}\operatorname{-mod}(\mathrm{Set})}_{c}}^{\mathrm{op}}$ is
generated by the sieves (cosieves in
${\mathbb{T}\operatorname{-mod}(\mathrm{Set})}_{c}$)

$S_{\vec{x}}=S_{M,\vec{x},\psi}\vcentcolon=\\{\,f:M\to M^{\prime}\mid
f(\vec{x})\in\llbracket\psi\rrbracket_{M^{\prime}}\,\\}\hbox to0.0pt{,\hss}$

where $M\in\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}$ and
$\vec{x}=(x_{1},\dots,x_{n})\in\llbracket\phi\rrbracket_{M}\subseteq\llbracket
A_{1}\rrbracket_{M}\times\dots\times\llbracket A_{n}\rrbracket_{M}$.

###### Proof.

The axiom $\phi\;\vdash_{\vec{x}\,{:}\,\vec{A}}\;\psi$ is satisfied in a model
$M\in\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathcal{E})$ (in any topos) if
and only if the inclusion
$\llbracket\psi\land\phi\rrbracket_{M}\hookrightarrow\llbracket\phi\rrbracket_{M}$
of subobjects of $\llbracket A_{1}\rrbracket_{M}\times\dots\times\llbracket
A_{n}\rrbracket_{M}$ is an isomorphism. For the universal $\mathbb{T}$-model
in
$[\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c},\mathrm{Set}]$,
this is the inclusion

${\llbracket\psi\land\phi\rrbracket_{-}}$${\llbracket\phi\rrbracket_{-}}$

of functors
$\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}\to\mathrm{Set}$.
The classifying topos of $\mathbb{T}+\mathbb{Q}$ is therefore the greatest
subtopos (corresponding to the Grothendieck topology with the fewest covering
sieves) with the property that this inclusion is a _local isomorphism_ , that
is, it becomes an isomorphism when sheafified.

Now, let $M\in\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}$ and
let $\vec{x}\in\llbracket\phi\rrbracket_{M}$ be given, corresponding to an
arrow $\operatorname{Hom}(M,-)\to\llbracket\phi\rrbracket_{-}$. Then the
cosieve $S_{\vec{x}}$ on $M$, regarded as a subobject of
$\operatorname{Hom}(M,-)$ in
$[\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c},\mathrm{Set}]$,
is the pullback

${\llbracket\psi\land\phi\rrbracket_{-}}$${\llbracket\phi\rrbracket_{-}}$${S_{\vec{x}}}$${\operatorname{Hom}(M,-)\hbox
to0.0pt{.\hss}}$${\urcorner}$

Thus, if the top row is a local isomorphism, then the bottom row is a local
isomorphism for all $M$ and $\vec{x}$. But the converse is also true, since
any local section $\widetilde{x}\in a(\llbracket\phi\rrbracket_{-})(M)$ of the
sheafification of $\llbracket\phi\rrbracket_{-}$ is locally given by some
$\vec{x}\in\llbracket\phi\rrbracket_{M^{\prime}}$, which then locally lies in
$\llbracket\psi\land\phi\rrbracket_{-}$. From this, the statement follows,
because the covering sieves of a Grothendieck topology are precisely those
subobjects of representable presheaves which are local isomorphisms. ∎

######  Remark 4.3.2.

Theorem 4.3.1 is closely related to [5, Theorem 8.1.10], which describes the
topology $J_{\mathbb{Q}}$ by only one generating sieve per axiom, assuming
that the axioms are given in a certain form, basically consisting of formulas
that present compact models. While this yields much more concise descriptions
of the topology, it seems harder to apply to concrete theories, as it might be
nontrivial to determine whether a formula presents a model (this property is
characterized in [5, Theorem 6.1.13] as being an irreducible object of the
syntactic site). Also, for our main purpose of recognizing the topology
$J_{\mathbb{Q}}$ as rigid, the big number of covering sieves $S_{\vec{x}}$ in
Theorem 4.3.1 will actually be quite convenient and can therefore be seen as a
feature in this context.

If the quotient extension $\mathbb{Q}$ contains more than one axiom, the
topology $J_{\mathbb{Q}}$ is of course generated by the union of all the sets
of generators for the individual axioms. (So there is one generating cosieve
for every axiom, compact model and appropriate family of elements of the
model.) It should be noted that the collection of cosieves of the form
$S_{\vec{x}}$ satisfies the pullback-stability axiom of a Grothendieck
topology (in the terminology of [7, C2.1], it is a (sifted) coverage). Indeed,
if $g:M\to M^{\prime}$ is a homomorphism between compact models then the push-
forward of a cosieve $S_{\vec{x}}$ on $M$ for a family of elements
$\vec{x}\in\llbracket\phi\rrbracket_{M}$ is simply the cosieve
$S_{g(\vec{x})}$ on $M^{\prime}$, and we do have
$g(\vec{x})\in\llbracket\phi\rrbracket_{M^{\prime}}$.

As a first application of Theorem 4.3.1, we can give a short proof of the
following result.

######  Corollary 4.3.3 (see [5, Theorem 8.2.5]).

Let $\mathbb{T}$ be a theory of presheaf type and $\mathbb{Q}$ a quotient
extension of $\mathbb{T}$. Then a compact model of $\mathbb{T}$ is
$J_{\mathbb{Q}}$-irreducible as an object of
${{\mathbb{T}\operatorname{-mod}(\mathrm{Set})}_{c}}^{\mathrm{op}}$ if and
only if it satisfies the axioms of $\mathbb{Q}$.

###### Proof.

A compact model $M$ of $\mathbb{T}$ satisfies an axiom
$\phi\;\vdash_{\vec{x}:\vec{A}}\;\psi$ if and only if for every family of
elements $\vec{x}\in\llbracket\phi\rrbracket_{M}$, the cosieve $S_{\vec{x}}$
from Theorem 4.3.1 is the maximal cosieve on $M$. This shows the “only if”
direction. For the “if” direction, one can check that saturating a collection
of sieves with respect to the transitivity condition of Grothendieck
topologies preserves the pullback-stability of the collection and can never
produce any nontrivial covers of an object if there were none before. ∎

######  Remark 4.3.4.

We can understand the cosieves $S_{\vec{x}}$ from Theorem 4.3.1 as
_operations_ correcting the failure of a single instance of one of the axioms
of $\mathbb{Q}$. Showing that $J_{\mathbb{Q}}$ is rigid then amounts to
providing an algorithm that turns any compact $\mathbb{T}$-model into a
$(\mathbb{T}+\mathbb{Q})$-model using these operations, as follows. Starting
with a compact $\mathbb{T}$-model $M$ which is not a model of
$\mathbb{T}+\mathbb{Q}$, we have to pick an axiom
$\phi\;\vdash_{\vec{x}:\vec{A}}\;\psi$ of $\mathbb{Q}$ and a family $\vec{x}$
of elements of (the appropriate sorts of) $M$ with $\phi(\vec{x})$ but not
$\psi(\vec{x})$. Then, if we are lucky, the cosieve $S_{\vec{x}}$ is generated
by a single arrow

$M\to M^{\prime}\hbox to0.0pt{,\hss}$

meaning that we have reduced the problem of covering $M$ by irreducibles to
covering $M^{\prime}$ by irreducibles by the transitivity property of the
Grothendieck topology $J_{\mathbb{Q}}$. If we can repeat this procedure a
finite number of times and arrive at a model that satisfies all axioms in
$\mathbb{Q}$, then we are done; we have covered $M$ by a single irreducible
object.

It can of course happen that $S_{\vec{x}}$ is not a principal cosieve, so that
we need multiple arrows

$M\to M^{\prime}_{i}$

to generate it. Then $S_{\vec{x}}$ should perhaps be seen as a
nondeterministic operation, that will turn $M$ into one of the
$M^{\prime}_{i}$, but we don’t know in advance which one it will be. In this
case, we have to deal with all $M^{\prime}_{i}$, and our algorithm must
terminate in the sense that any possible path of execution is finite. (On the
other hand, we can also be even _more_ lucky, in that $S_{\vec{x}}$ is the
empty cosieve on $M$. Then we are immediately done with the current branch.)

Here is a first illustration of this strategy.

######  Example 4.3.5.

Start with the theory $\mathbb{T}$ with two sorts and a function symbol $A\to
B$, and add to it the axiom

$\top\;\vdash_{y\,{:}\,B}\;\exists x{\,:\,}A.\;f(x)=y\hbox to0.0pt{,\hss}$

requiring $f$ to be surjective. We already know that the resulting theory is
of presheaf type, since it is Morita equivalent to the theory with just one
sort $A$ and an equivalence relation ${\sim}\subseteq A\times A$ by Example
2.4.2, and this is a Horn theory. But let us show that the surjectivity axiom
is a rigid-topology quotient of $\mathbb{T}$.

By Lemma 4.2.4, a model $M=(A\to
B)\in\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})$ is compact if
and only if both $A$ and $B$ are finite sets. Given such a compact model
$(A\to B)$ and an element $y\in B$, the cosieve $S_{y}$ on $(A\to B)$ from
Theorem 4.3.1 is generated by the single arrow

${(A\to B)}$${(A\sqcup\\{x\\}\to B)\hbox to0.0pt{,\hss}}$

where $x$ is sent to $y$ in the new model (which is clearly still compact),
since every model homomorphism

${A}$${B}$${A^{\prime}}$${B^{\prime}}$$\scriptstyle{f}$$\scriptstyle{g_{A}}$$\scriptstyle{g_{B}}$$\scriptstyle{f^{\prime}}$

with $\exists x\in A^{\prime}.\;f^{\prime}(x)=g_{B}(y)$ factors through
$(A\sqcup\\{x\\}\to B)$. The model $(A\to B)$ can thus be covered (in the
topology induced by the surjectivity axiom) by the model $(A\sqcup\\{x\\}\to
B)$. We can go on by creating preimages of other elements of $B$, until we
have covered $(A\to B)$ by a model satisfying the new axiom (since $B$ is
finite).

######  Lemma 4.3.6.

If $\mathbb{T}$ is of presheaf type, $\mathbb{Q}_{1}$ is a rigid-topology
quotient of $\mathbb{T}$ and $\mathbb{Q}_{2}$ is a rigid-topology quotient of
$\mathbb{T}+\mathbb{Q}_{1}$, then $\mathbb{Q}_{1}+\mathbb{Q}_{2}$ is a rigid-
topology quotient of $\mathbb{T}$.

###### Proof.

This follows easily from Lemma 4.2.7:
$\mathbb{T}+\mathbb{Q}_{1}+\mathbb{Q}_{2}$ is of presheaf type and every
compact $(\mathbb{T}+\mathbb{Q}_{1}+\mathbb{Q}_{2})$-model is compact as a
$(\mathbb{T}+\mathbb{Q}_{1})$-model and hence as a $\mathbb{T}$-model.

Note that arguing more directly by the transitivity of
$J_{\mathbb{Q}_{1}+\mathbb{Q}_{2}}$ is not so simple, as we would still have
to relate the topology induced by $\mathbb{Q}_{2}$ on
${(\mathbb{T}+\mathbb{Q}_{1})\operatorname{-mod}(\mathrm{Set})}_{c}$ with that
on ${\mathbb{T}\operatorname{-mod}(\mathrm{Set})}_{c}$. ∎

While we can assume, after choosing a representing set of axioms for the
quotient extension $\mathbb{Q}_{2}$, that its base theory is $\mathbb{T}$
instead of $\mathbb{T}+\mathbb{Q}_{1}$, the assumptions of Lemma 4.3.6 are not
at all the same as requiring $\mathbb{Q}_{1}$ and $\mathbb{Q}_{2}$ to be
rigid-topology quotients of $\mathbb{T}$ independently. In fact,
$\mathbb{Q}_{1}+\mathbb{Q}_{2}$ can fail to be a rigid-topology quotient of
$\mathbb{T}$ in this case, as we will see in Example 4.5.5. By the same method
as there, one can easily construct quotient extensions
$\mathbb{Q}_{1},\dots,\mathbb{Q}_{n}$ such that the sum of any proper subset
induces a rigid topology, but adding all $n$ even destroys presheaf type.

###  4.4 Syntactic constructions preserving presheaf type

We now investigate in which ways a theory of presheaf type can be extended
such that the resulting theory is again of presheaf type. At first sight, one
might hope for a result like “an algebraic (or Horn, or even cartesian)
extension preserves presheaf type”, but unfortunately, this is far too
optimistic. Indeed, if we have any geometric axiom
$\phi\;\vdash_{\vec{x}:\vec{A}}\;\psi$ we would like to add to a theory, we
can instead first perform an extension by definitions (which surely preserves
presheaf type), defining $R_{\phi},R_{\psi}\subseteq\vec{A}$ by $\phi$ and
$\psi$ respectively, and then add the Horn axiom
$R_{\phi}\;\vdash_{\vec{x}:\vec{A}}\;R_{\psi}$. Example 4.5.4 shows that also
an algebraic axiom can destroy presheaf type.

######  Corollary 4.4.1 (see [5, Theorem 8.2.8]).

Adding arbitrarily many axioms of the form

$\phi\;\vdash_{\vec{x}:\vec{A}}\;\bot$

to a theory of presheaf type is always a rigid-topology quotient.

###### Proof.

Let $\mathbb{T}$ be a theory of presheaf type,
$\mathbb{Q}=\\{\,\phi^{i}\;\vdash_{\vec{x}^{i}:\vec{A}^{i}}\;\bot\mid i\in
I\,\\}$ and $M\in{\mathbb{T}\operatorname{-mod}(\mathrm{Set})}_{c}$. If there
is some $i\in I$ and $\vec{x}\in\llbracket\phi^{i}\rrbracket_{M}$, then

$S_{\vec{x}}=\\{\,f:M\to M^{\prime}\mid
f(\vec{x})\in\llbracket\bot\rrbracket_{M^{\prime}}=\varnothing\,\\}$

is the empty cosieve on $M$ and we are done. But otherwise $M$ was already a
model of $\mathbb{T}+\mathbb{Q}$, i.e. irreducible itself. ∎

######  Corollary 4.4.2.

Adding finitely many axioms $\phi\;\vdash_{[]}\;\psi$ with empty context to a
theory of presheaf type is always a rigid-topology quotient.

###### Proof.

For one axiom $\phi\;\vdash_{[]}\;\psi$ with empty context, Theorem 4.3.1 says
that any compact model $M$ that satisfies the closed formula $\phi$ is covered
by all arrows (necessarily preserving truth of $\phi$) to compact models that
also satisfy $\psi$, as required. And any model that does not satisfy $\phi$
satisfies the new axiom anyway. For more than one axiom of this form, we can
use Lemma 4.3.6. ∎

######  Corollary 4.4.3.

Any propositional theory with finitely many axioms is of presheaf type.

###### Proof.

This immediately follows from Corollary 4.4.2, as a propositional theory has
no other contexts than the empty one. ∎

######  Remark 4.4.4.

Corollary 4.4.2 becomes false if we omit the word “finitely”. Indeed, it would
otherwise imply that all propositional theories are of presheaf type (see
Example 4.5.3 for one which is not). In particular, Lemma 4.3.6 becomes wrong
if instead of two consecutive rigid-topology quotients we consider the sum
$\mathbb{T}+\sum_{i\in\mathbb{N}}\mathbb{Q}_{i}$ of an infinite sequence of
quotient extensions where every $\mathbb{Q}_{n}$ is a rigid-topology quotient
of $\mathbb{T}+\sum_{i=0}^{n-1}\mathbb{Q}_{i}$.

######  Proposition 4.4.5.

Let $\mathbb{T}$ be a theory of presheaf type and $A$ a sort of $\mathbb{T}$.
Then the theory $\mathbb{T}+(c:A)$ obtained from $\mathbb{T}$ by introducing a
new constant symbol $c:A$ is also of presheaf type. Moreover, a model of
$\mathbb{T}+(c:A)$ in $\mathrm{Set}$ is compact if and only if the underlying
$\mathbb{T}$-model is compact.

###### Proof.

Let $\mathrm{Set}[\mathbb{T}]$ be a classifying topos for $\mathbb{T}$ with
universal model $M_{\mathbb{T}}$. Then $\mathbb{T}+(c:A)$ is classified by the
slice topos $\mathrm{Set}[\mathbb{T}]/\llbracket
A\rrbracket_{M_{\mathbb{T}}}$, see [8, Exercise X.3]. But if
$\mathrm{Set}[\mathbb{T}]=\operatorname{PSh}(C)$ is a presheaf topos, then we
can use the formula [8, Exercise III.8]

$\operatorname{PSh}(C)/X\simeq\operatorname{PSh}(\int_{C}X)$

for $X=\llbracket A\rrbracket_{M_{\mathbb{T}}}$ to see that $\mathbb{T}+(c:A)$
is of presheaf type.

For the compact models, we observe that
$(\mathbb{T}+(c:A))\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})$ is
equivalent, as a category over
$\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})$, to the category
$\int^{\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})}\llbracket
A\rrbracket_{-}$ of elements of the interpretation functor $\llbracket
A\rrbracket_{-}:\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})\to\mathrm{Set}$.
This functor preserves filtered colimits by Lemma 4.2.2, so Lemma 4.4.6 below
tells us exactly what we need. ∎

######  Lemma 4.4.6.

Let $C$ be a category with filtered colimits and let $F:C\to\mathrm{Set}$ be a
functor preserving filtered colimits. Then the category of elements of $F$,
$\int^{C}F$, has filtered colimits too, and the projection functor
$U:\int^{C}F\to C$ preserves and reflects compact objects.

###### Proof.

Let $(c_{i},x_{i})_{i\in I}$, where $x_{i}\in F(c_{i})$, be a filtered diagram
in $\int^{C}F$. It is clear that the $x_{i}$ all represent the same element in
$\operatorname*{colim}_{i}F(c_{i})\cong F(\operatorname*{colim}_{i}c_{i})$,
let $x^{*}\in F(\operatorname*{colim}_{i}c_{i})$ be this element. Then we have
a cocone on $(c_{i},x_{i})_{i\in I}$ with apex
$(\operatorname*{colim}_{i}c_{i},x^{*})$, and it is easy to check that this is
the colimit in $\int^{C}F$. This also shows that the functor $U:\int^{C}F\to
C$ preserves and reflects filtered colimits.

Let $(c,x)$ be a compact object of $\int^{C}F$, we want to show that $c$ is
compact in $C$. So let $(c_{i})_{i\in I}$ be a filtered diagram in $C$ and fix
a morphism

$f:c\to\operatorname*{colim}_{i\in I}c_{i}\hbox to0.0pt{,\hss}$

then we have to show that there is exactly one
$[g]\in\operatorname*{colim}_{i}\operatorname{Hom}(c,c_{i})$ that induces $f$.
If we want to use the compactness of $(c,x)$, we need to construct a filtered
diagram in $\int^{C}F$. Consider the element $f_{*}x\in
F(\operatorname*{colim}_{i}c_{i})\cong\operatorname*{colim}_{i}F(c_{i})$ and
pick a representative $x_{i_{0}}\in F(c_{i_{o}})$ for it. Since $I$ is
filtered, the comma category $(i_{0}\downarrow I)$ is also filtered and the
functor $(i_{0}\downarrow I)\to I$ is final. This means that we have
$\operatorname*{colim}_{i}c_{i}=\operatorname*{colim}_{i_{0}\to i}c_{i}$ and
$\operatorname*{colim}_{i}\operatorname{Hom}(c,c_{i})=\operatorname*{colim}_{i_{0}\to
i}\operatorname{Hom}(c,c_{i})$, so we can replace the diagram $(c_{i})_{i\in
I}$ with $(c_{i})_{(i_{0}\to i)\in(i_{0}\downarrow I)}$ and assume without
loss of generality that $i_{0}$ is initial in $I$. Then setting
$x_{i}\vcentcolon=({!}:i_{0}\to i)_{*}x_{i_{0}}\in F(c_{i})$ yields a filtered
diagram $(c_{i},x_{i})_{i\in I}$ in $\int^{C}F$. We also have

$\widetilde{f}:(c,x)\to\operatorname*{colim}_{i\in I}{(c_{i},x_{i})}$

with $U(\widetilde{f})=f$ by our original choice of $x_{i_{0}}$. So by
assumption, there is a $\widetilde{g}:(c,x)\to(c_{i},x_{i})$ for some $i$ that
induces $\widetilde{f}$. Then $g\vcentcolon=U(\widetilde{g})$ induces $f$. On
the other hand, if $g_{1}:c\to c_{i_{1}}$ and $g_{2}:c\to c_{i_{2}}$ both
induce $f$, then we have
$[{g_{1}}_{*}x]=[{g_{2}}_{*}x]=[x_{i_{0}}]\in\operatorname*{colim}_{i}F(c_{i})$,
in particular $[{g_{1}}_{*}x]=[x_{i_{1}}]$ and $[{g_{2}}_{*}x]=[x_{i_{2}}]$,
so we can prolong $g_{1}$ to some $g^{\prime}_{1}:c\to c_{i^{\prime}_{1}}$
such that ${g^{\prime}_{1}}_{*}x=x_{i^{\prime}_{1}}$, and similarly for
$g_{2}$, meaning that we have $\widetilde{g}_{1}:(c,x)\to(c_{i^{\prime}_{1}})$
and $\widetilde{g}_{2}:(c,x)\to(c_{i^{\prime}_{2}})$ that both induce
$\widetilde{f}$. Then we can conclude
$[\widetilde{g}_{1}]=[\widetilde{g}_{1}]\in\operatorname*{colim}_{i}\operatorname{Hom}((c,x),(c_{i},x_{i}))$,
and from this it follows that
$[g_{1}]=[g^{\prime}_{1}]=[g^{\prime}_{2}]=[g_{2}]\in\operatorname*{colim}_{i}\operatorname{Hom}(c,c_{i})$.

Now let $c\in C$ be compact and let $x\in F(c)$, we want to show that $(c,x)$
is compact in $\int^{C}F$. So let $(c_{i},x_{i})_{i\in I}$ be a filtered
diagram in $\int^{C}F$ and let

$\widetilde{f}:(c,x)\to\operatorname*{colim}_{i\in
I}(c_{i},x_{i})=(\operatorname*{colim}_{i\in I}c_{i},x^{*})$

be given. That is, we have $f:c\to\operatorname*{colim}_{i}c_{i}$ with
$f_{*}x=x^{*}$. This $f$ is induced by some $g:c\to c_{i}$, implying that
$[g_{*}x]=x^{*}\in\operatorname*{colim}_{i}F(c_{i})$, so it is also induced by
some $g^{\prime}:c\to c_{i^{\prime}}$ with $g^{\prime}_{*}x=x_{i^{\prime}}$.
This means that we have
$\widetilde{g}:(c,x)\to(c_{i^{\prime}},x_{i^{\prime}})$ inducing
$\widetilde{f}$. And for $\widetilde{g}_{1}$, $\widetilde{g}_{2}$ both
inducing $\widetilde{f}$, we immediately see that
$[U(\widetilde{g}_{1})]=[U(\widetilde{g}_{2})]\in\operatorname*{colim}_{i}\operatorname{Hom}(c,c_{i})$
and therefore
$[\widetilde{g}_{1}]=[\widetilde{g}_{2}]\in\operatorname*{colim}_{i}\operatorname{Hom}((c,x),(c_{i},x_{i}))$.
∎

######  Example 4.4.7.

From Example 2.4.4 we know that the theory $\underline{A}$, where we have just
imported a set $A$, is Morita equivalent to the empty theory (over the empty
signature). In particular, it is of presheaf type and we can apply Proposition
4.4.5 to conclude that the theory $c:\underline{A}$, where we have added one
more constant symbol is also of presheaf type. It is classified by
$\mathrm{Set}/A$.

It is also Morita equivalent to a disjoint disjunction of an $A$-indexed
family of proposition symbols,

$\top\;\vdash_{[]}\;\bigvee_{a\in A}p_{a}\qquad\text{and}\qquad p_{a}\land
p_{a^{\prime}}\;\vdash_{[]}\;\bot\quad(\text{for $a\neq a^{\prime}\in
A$})\hbox to0.0pt{,\hss}$

by identifying $p_{a}$ with $c=c_{a}$. This way, we can easily see that the
theory is of presheaf type using Corollary 4.4.2 for the first axiom and
Corollary 4.4.1 for the $A$-indexed family of negated axioms.

######  Example 4.4.8.

Consider the theory of a surjective function $f:A\twoheadrightarrow B$, which
is of presheaf type by Example 2.4.2 or Example 4.3.5. By Proposition 4.4.5,
it is still of presheaf type after adding a constant symbol $c:B$.

When we translate this extension to the theory of an equivalence relation
${\sim}\subseteq A\times A$, we obtain a relation symbol $R\subseteq A$ with
the axioms

$R(x)\land x\sim x^{\prime}\;\vdash_{x,x^{\prime}\,{:}\,A}\;R(x^{\prime})\hbox
to0.0pt{,\hss}\qquad R(x)\land
R(x^{\prime})\;\vdash_{x,x^{\prime}\,{:}\,A}\;x\sim x^{\prime}\hbox
to0.0pt{,\hss}\qquad\top\;\vdash_{[]}\;\exists x{\,:\,}A.\;R(x)\hbox
to0.0pt{.\hss}$

The existential quantifier in the last axiom is not cartesian, but the context
of the axiom is empty (and all other axioms of the theory are Horn), so we
again find that the theory is of presheaf type using Corollary 4.4.2. However,
if the sort $A$ and the equivalence relation $\sim$ are instead given as part
of a theory of presheaf type which is not cartesian, then the Horn axioms on
$R$ can not be argued away so easily, but we can still introduce the quotient
$B=A/{\sim}$ and add a constant symbol $c:B$ without destroying presheaf type.

###  4.5 Some counterexamples

We now give a number of examples of non-presheaf-type theories, to show that
the above positive results are optimal in certain aspects and to illustrate
the various techniques useful for recognizing such theories. Our first example
shows that even a very simple, finite theory can fail to be of presheaf type.

######  Example 4.5.1.

Consider the theory

$\mathbb{T}=\left\\{\leavevmode\hbox to35.3pt{\vbox
to17.75pt{\pgfpicture\makeatletter\hbox{\hskip 8.05554pt\lower-10.67067pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.05554pt}{-7.07637pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
8.05554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.05554pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{} {{}{{}{}}{}}{{}{{}{}}{}}{{}{}}{{}}
{{}{{}{}}{}}{{{}}{{}}}{{}}{{}{{}{}}{}}{{{}}{{}}}{ {}{}{}}{}{{}}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{}{}{}}{}{}{}{}{{}}{}{{}}{}{}{}{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{8.25554pt}{-2.36447pt}\pgfsys@curveto{21.99718pt}{1.31795pt}{21.99718pt}{-10.47069pt}{8.64186pt}{-6.89178pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.96593}{0.25883}{-0.25883}{-0.96593}{9.83955pt}{-7.21272pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.96593}{0.25883}{-0.25883}{-0.96593}{8.44872pt}{-6.84003pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{20.9145pt}{-6.32635pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{f}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right\\}$

consisting of one sort $A$, a unary function symbol $f:A\to A$ and one axiom
$\top\;\vdash_{y:A}\;\exists x{\,:\,}A.\;f(x)=y$ stating that $f$ is
surjective. We show that ${\mathbb{T}\operatorname{-mod}(\mathrm{Set})}$ is
not finitely accessible, implying that $\mathbb{T}$ is not of presheaf type.

Specifically, we show that the model $(\mathbb{Z},+1)$, where $A=\mathbb{Z}$
and $f(n)=n+1$, is not a filtered colimit of compact models. Consider the
models

$M_{n}=\Big{(}\,\\{\,(x,y)\in\mathbb{Z}^{2}\mid
y\leq\min(n,-x)\,\\}\,,\,(x,y)\mapsto(x+1,\min(y,-x-1))\,\Big{)}\hbox
to0.0pt{,\hss}$

which can be pictured like this.

${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\mathclap{\dots}}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\mathclap{\dots}}$${\mathclap{\smash{\ddots}}}$

The inclusions $M_{n}\hookrightarrow M_{n+1}$ (adding another row on top) are
model homomorphisms, and we have a homomorphism

$(\mathbb{Z},+1)\to M_{\omega}\vcentcolon=\operatorname*{colim}_{n}M_{n},\quad
k\mapsto(k,-k)\hbox to0.0pt{,\hss}$

that does not factor through any $M_{n}$. This shows that $(\mathbb{Z},+1)$ is
not compact. But any map $M\to(\mathbb{Z},+1)$ from a nonempty model $M$ is
necessarily surjective, so the composite $M\to(\mathbb{Z},+1)\to M_{\omega}$
does not factor through any $M_{n}$ either, showing that $(\mathbb{Z},+1)$ is
not a colimit of compact models.

Another method to show that a theory has “too few compact models” to be of
presheaf type is illustrated by the next example. In it, we see that adding
countably many constant symbols of the same sort can destroy presheaf type (in
contrast to Proposition 4.4.5). An even more drastic version of this statement
will be one of the conclusions of Example 4.5.3.

######  Example 4.5.2.

Start with the theory of a surjective function $f:A\twoheadrightarrow B$ as in
Example 4.4.8, but then add countably many constant symbols
$c_{0},c_{1},\ldots:B$ instead of just a single one. We show that this theory
$\mathbb{T}$ is not of presheaf type by investigating its compact models.

$A\twoheadrightarrow B\ni c_{0},c_{1},\dots$

Let $\mathbb{T}^{\prime}$ be the theory of inhabited sets, with one sort $A$
and the axiom $\top\;\vdash_{[]}\;(\exists x{\,:\,}A.\;\top)$. Then we can
regard $\mathbb{T}$ as an extension of $\mathbb{T}^{\prime}$ and the forgetful
functor

$U:\mathbb{T}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})\to\mathbb{T}^{\prime}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set}),\quad(A\twoheadrightarrow
B)\mapsto A$

has a right adjoint $A^{\prime}\mapsto(A^{\prime}\twoheadrightarrow\\{*\\})$,
which preserves filtered colimits by their description in Lemma 4.2.2. Thus,
$U$ preserves compact objects by Lemma 4.2.3. But the compact objects of
$\mathbb{T}^{\prime}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})$ are the
finite nonempty sets, so, in summary, for any compact model
$(A\twoheadrightarrow B)$ of $\mathbb{T}$, $A$ is a finite set. This means
that $B$ is finite too, from which we can in particular conclude that the
geometric formula

$\phi\vcentcolon=\quad\bigvee_{i\neq j}(c_{i}=c_{j})$

is satisfied in all compact models. If $\mathbb{T}$ was of presheaf type, this
would mean that $\phi$ is satisfied in the universal model and therefore
provable in $\mathbb{T}$. This is not the case, as witnessed by the model
$(\mathbb{N}\twoheadrightarrow\mathbb{N})$ with $c_{n}=n$.

The following example will allow us to draw several conclusions. The technique
used here is to compare a theory to another theory with the same
$\mathrm{Set}$-based models which is known to be of presheaf type.

######  Example 4.5.3.

Given two sets $A$ and $B$, there are (at least) two different theories “of a
map from $A$ to $B$”. One is

$\langle f:\underline{B^{A}}\rangle\hbox to0.0pt{,\hss}$

where we import (see Example 2.4.4) the set $B^{A}$ of functions from $A$ to
$B$ and add a constant symbol $f$ of this sort; it is of presheaf type as in
Example 4.4.7. The other is

$\langle f:\underline{A}\to\underline{B}\rangle\hbox to0.0pt{,\hss}$

where we import $A$ and $B$ separately and add a unary function symbol as
indicated. For both theories, the category of models in $\mathrm{Set}$ is (up
to equivalence) the discrete category with $B^{A}$ objects. So if $\langle
f:\underline{A}\to\underline{B}\rangle$ was of presheaf type too, then this
would imply that the two theories are Morita-equivalent.

Specialize to the case $A=\mathbb{N}$ and $B=2=\\{0,1\\}$. Then $\langle
f:\underline{2^{\mathbb{N}}}\rangle$ is classified by
$\mathrm{Set}/2^{\mathbb{N}}$, or equivalently, by $\operatorname{Sh}(X)$,
where $X$ is the discrete space with $2^{\mathbb{N}}$ points. But by [8,
Exercise VIII.10], $\langle f:\underline{\mathbb{N}}\to\underline{2}\rangle$
is classified by $\operatorname{Sh}(2^{\mathbb{N}})$, where $2^{\mathbb{N}}$
is the Cantor space. Since these two sober topological spaces are not
homeomorphic, $\langle f:\underline{\mathbb{N}}\to\underline{2}\rangle$ is not
of presheaf type.

Another way to see that $\langle f:\underline{2^{\mathbb{N}}}\rangle$ and
$\langle f:\underline{\mathbb{N}}\to\underline{2}\rangle$ are not Morita
equivalent is as follows. First, as in Example 4.4.7, a constant symbol in an
imported set, such as $f:\underline{2^{\mathbb{N}}}$, can be replaced by a
disjoint disjunction of $2^{\mathbb{N}}$ proposition symbols. On the other
hand, the function symbol $f:\underline{\mathbb{N}}\to\underline{2}$ can be
replaced by countably many constant symbols $c_{n}:\underline{2}$ and
therefore by countably many decidable propositions (disjoint binary
disjunctions). In this way, one sees that the models of both theories in any
topos $\mathcal{E}$ form discrete categories, namely, $\langle
f:\underline{2^{\mathbb{N}}}\rangle$ classifies decompositions of
$\mathcal{E}$ into $2^{\mathbb{N}}$ clopen subtoposes, while $\langle
f:\underline{\mathbb{N}}\to\underline{2}\rangle$ classifies countable families
of compositions into two clopen subtoposes. Now, we can consider the
topological spaces $X_{n}\vcentcolon=\\{0,\dots,n\\}$ (with the discrete
topology) and $X_{\omega}\vcentcolon=\mathbb{N}\cup\\{\infty\\}$, the one-
point compactification of $\mathbb{N}$, and the continuous maps

$X_{\omega}\to X_{n},\quad x\mapsto\min(x,n)\hbox to0.0pt{.\hss}$

Then we see that every model of $\langle f:\underline{2^{\mathbb{N}}}\rangle$
in $X_{\omega}$ is (isomorphic to) the pullback of a model in some $X_{n}$,
because any open set of $X_{\omega}$ containing $\infty$ is cofinite, but this
is not the case for $\langle f:\underline{\mathbb{N}}\to\underline{2}\rangle$,
because we can choose a family of partitions like
$X_{\omega}=\\{0,\dots,n\\}\sqcup\\{n+1,\dots,\infty\\}$.

In summary, we can record the following.

  1. (i)

Adding a single unary function symbol (such as
$f:\underline{\mathbb{N}}\to\underline{2}$) to a theory of presheaf type (even
a theory that is classified by $\mathrm{Set}$) can destroy presheaf type.

  2. (ii)

Adding countably many constant symbols (such as $c_{n}:\underline{2}$) to a
theory of presheaf type (even a theory that is classified by $\mathrm{Set}$)
can destroy presheaf type.

  3. (iii)

A propositional theory (such as the theory of a countable family of decidable
propositions, with $p_{n}\lor q_{n}$ and $p_{n}\land
q_{n}\;\vdash_{[]}\;\bot$) can fail to be of presheaf type. (There are of
course many other examples of this.)

  4. (iv)

Adding countably many closed geometric formulas (such as $p_{n}\lor q_{n}$ in
the previous item, where all other axioms were negated ones, which can’t
destroy presheaf type) can destroy presheaf type.

######  Example 4.5.4.

Let us show that adding an algebraic axiom can destroy presheaf type. We do
this by showing that adding an algebraic axiom is, up to Morita equivalence,
at least as expressive as adding an arbitrary family $\phi_{i}$ of closed
geometric formulas as axioms. We can then for example use $\phi_{n}=(p_{n}\lor
q_{n})$ to obtain the theory of countably many decidable propositions from a
theory with only negated axioms, as in example 4.5.3.

So let $\mathbb{T}$ be a theory with closed geometric formulas $\phi_{i}$,
$i\in I$. Import the set $I\sqcup\\{*\\}$. We can define (by a geometric
formula) an equivalence relation $\sim$ on $\underline{I\sqcup\\{*\\}}$ such
that $c_{i}\sim c_{*}\;\dashv\vdash_{[]}\;\phi_{i}$. Introduce the quotient
$\underline{I\sqcup\\{*\\}}\twoheadrightarrow A$ by $\sim$. These equivalence
extensions allow us to formulate the algebraic axiom

$\top\;\vdash_{x,y:A}\;x=y\hbox to0.0pt{,\hss}$

which is syntactically equivalent (as a quotient extension) to
$\top\;\vdash_{x:\underline{I\sqcup\\{*\\}}}\;x\sim c_{*}$, and therefore to
the set of axioms $\\{\,\top\;\vdash_{[]}\;\phi_{i}\mid i\in I\,\\}$.

######  Example 4.5.5.

Here we give an example of two rigid-topology quotients $\mathbb{Q}_{1}$,
$\mathbb{Q}_{2}$ of the same theory of presheaf type $\mathbb{T}$ such that
$\mathbb{Q}_{1}+\mathbb{Q}_{2}$ is not a rigid-topology quotient again and
$\mathbb{T}+\mathbb{Q}_{1}+\mathbb{Q}_{2}$ is not even of presheaf type. Start
with the theory $\mathbb{T}$ of a sequence of maps, infinite to the left:

$\dots\xrightarrow{f_{2}}A_{2}\xrightarrow{f_{1}}A_{1}\xrightarrow{f_{0}}A_{0}.$

The compact $\mathrm{Set}$-models of $\mathbb{T}$ are those with finitely many
elements in total. Consider the quotient extension $\mathbb{Q}_{1}$ requiring
every second map to be surjective:

$\top\;\vdash_{y:A_{n}}\;\exists
x{\,:\,}A_{n+1}.\;f_{n}(x)=y\qquad\qquad\text{for $n$ even}.$

The topology on
${{\mathbb{T}\operatorname{-mod}(\mathrm{Set})}_{c}}^{\mathrm{op}}$ induced by
these axioms is rigid: Given $M=(\dots\to A_{1}\to
A_{0})\in{\mathbb{T}\operatorname{-mod}(\mathrm{Set})}_{c}$ and an element
$y\in A_{n}$ not in the image of $f_{n}$, the cosieve $S_{y}$ is generated by
the single model homomorphism

${\dots}$${A_{n+2}}$${A_{n+1}}$${A_{n}}$${\dots}$${A_{0}}$${\dots}$${A_{n+2}}$${A_{n+1}\sqcup\\{x\\}}$${A_{n}}$${\dots}$${A_{0}\hbox
to0.0pt{,\hss}}$

where we have freely added a preimage $x$ of $y$. After finitely many steps,
we have covered $M$ by a compact $\mathbb{T}$-model that satisfies
$\mathbb{Q}_{1}$.

However, denoting by $\mathbb{Q}_{2}$ the quotient extension requiring $f_{n}$
to be surjective for $n$ odd, which is a rigid-topology quotient for the same
reason, the theory

$\mathbb{T}^{\prime}\vcentcolon=\mathbb{T}+\mathbb{Q}_{1}+\mathbb{Q}_{2}$

is _not_ of presheaf type. To see this, we show that
${\mathbb{T}^{\prime}\operatorname{-mod}(\mathrm{Set})}$ is not finitely
accessible. In fact, we can show that the only compact object of
${\mathbb{T}^{\prime}\operatorname{-mod}(\mathrm{Set})}$ is the empty model
$(\dots\to\varnothing\to\varnothing)$, by considering the following sequence
of inclusions.

${\dots}$${\\{0\\}}$${\\{0\\}}$${\\{0\\}}$${\\{0\\}}$${=}$${M^{0}}$${\dots}$${\\{0,1\\}}$${\\{0,1\\}}$${\\{0,1\\}}$${\\{0\\}}$${=}$${M^{1}}$${\dots}$${\\{0,1,2\\}}$${\\{0,1,2\\}}$${\\{0,1\\}}$${\\{0\\}}$${=}$${M^{2}}$${\vdots}$${\vdots}$$\scriptstyle{1\mapsto
0}$$\scriptstyle{2\mapsto 1}$$\scriptstyle{1\mapsto 0}$

The colimit of this diagram is a model $M^{\omega}$ with
$A^{\omega}_{n}=\\{0,\dots,n\\}$. If $M=(\dots\to A_{1}\to A_{0})$ is any
nonempty model, then $A_{n}$ is nonempty for all $n$, so the model
homomorphism $M\to M^{\omega}$ sending any $x\in A_{n}$ to $n\in
A^{\omega}_{n}$ does not factor through any of the models in the diagram,
showing that $M$ is not compact.

Finally, it seems appropriate to show that the theory of local rings, which
played an important role in Subsection 3.7 and will do so again in Section 5,
is not of presheaf type. Equivalently, this shows that the big Zariski topos
$(\operatorname{Spec}\mathbb{Z})_{\mathrm{Zar}_{\mathrm{fp}}}$ is not
equivalent to a presheaf topos.

######  Proposition 4.5.6.

The theory of local rings, $\mathrm{Ring}+(\mathrm{loc})$ (see Definition
3.7.1), is not of presheaf type.

###### Proof.

The argument given here resembles the one given in [5, Section 9.4] showing
that the usual geometric theory of fields is not of presheaf type. Assume that
the theory of local rings is of presheaf type, that is, it is classified by
$[\mathrm{locRing}_{c},\mathrm{Set}]$ with the tautological local ring as
universal model. We aim to reveal a subterminal object of
$[\mathrm{locRing}_{c},\mathrm{Set}]$ which is not definable by a closed
geometric formula. Note that every local ring $A$ is a
$\mathbb{Z}_{(p)}$-algebra for some prime number $p$, since at most one prime
number can be non-invertible as an element of $A$. This means that the join of
the subterminal objects

$U_{p}\vcentcolon=\operatorname{Hom}(\mathbb{Z}_{(p)},-):\mathrm{locRing}_{c}\to\mathrm{Set}$

is $\bigvee_{p}U_{p}=1$. Now, if there were geometric formulas $\phi_{p}$ such
that $\llbracket\phi_{p}\rrbracket=U_{p}$ (i.e. $\phi_{p}$ expresses that all
prime numbers except $p$ are invertible) for all $p$, then the disjunction
$\bigvee_{p}\phi_{p}$, being valid in the universal model, would be provable
in the theory of local rings. But since this is a coherent theory, a finite
disjunction $\bigvee_{p\in P}\phi_{p}$ with $P$ a finite set of primes would
also have to be provable by [5, Theorem 10.8.6 (iii)]. This is not the case,
as $\bigvee_{p\in P}\phi_{p}$ is falsified by the local ring
$\mathbb{Z}_{(p^{\prime})}$ for any $p^{\prime}\notin P$. ∎

##  5 Syntactic presentations for crystalline toposes

###  5.1 Introduction

In this section, we determine a syntactic presentation of the big crystalline
toposes used for studying crystalline cohomology in algebraic geometry. This
fulfills a promise made by Wraith in 1979 in [12, p. 743], where he writes:
“It is my belief that a great many of the toposes occurring in algebraic
geometry can be conveniently described in terms of the theories they classify.
This is certainly so in the case of étale and crystalline toposes, for
example.” This optimistic statement is however not accompanied by any hint of
what the classified theory might be, and no answer has been given since.

The reason why the question is not as simple as for the big Zariski topos, for
which the answer has been known since the infancy of the field of classifying
toposes, is that much more data is involved in its construction. To wit, the
crystalline topos depends on two schemes, a scheme morphism between them, and
a certain structure known as a divided powers structure, or PD structure on
one of them.

In the affine case, a PD structure on a ring $A$ of positive characteristic
$p$ assigns to an element $a\in A$ another element of $A$, acting as a
replacement for the expression

$\frac{1}{n}a^{n}\hbox to0.0pt{,\hss}$

which can never be taken literally when $p$ divides $n$. The intuition why
such a structure might be useful for cohomological techniques is that in
positive characteristic $p$, taking the derivative of a polynomial, or a power
series,

$t^{n}\mapsto nt^{n-1}\hbox to0.0pt{,\hss}$

is never a surjective map, as there is no integral of $t^{p-1}$, and this
leads to a failure of the Poincaré Lemma. But if divided powers of $t$ are
available, then the integral does exist.

For us, this means that the syntactic presentation of the crystalline topos
will involve two rings and a geometric theory formulation of a PD structure.
To handle this somewhat complex theory, we make heavy use of the techniques
around theories of presheaf type developed in Section 4.

###  5.2 Background on divided power rings

Let us collect a number of properties and constructions of divided power rings
that will be relevant. A _PD structure_ or _divided powers structure_ on an
ideal $I$ of a ring $A$ (always commutative and unitary) is a family of
functions

$\gamma_{n}:I\to I,\quad n\geq 1$

such that $x^{n}=n!\gamma_{n}(x)$ for every $x\in I$, compatible with natural
operations on the ideal $I$ and with each other, as specified in Definition
5.3.1 below. For example, the zero ideal $(0)$ of any ring carries the trivial
PD structure $\gamma_{n}(0)=0$, while any ideal of a $\mathbb{Q}$-algebra $A$
admits a unique PD structure given by $\gamma_{n}(a)=\frac{a^{n}}{n!}$.
Slightly more interestingly, the maximal ideal $(p)$ of $\mathbb{Z}_{(p)}$
admits a unique PD structure, as the prime factor $p$ occurs at most $n-1$
times in $n!$, so $\frac{p^{n}}{n!}\in(p)\subseteq\mathbb{Z}_{(p)}$ for all
$n\geq 1$.

An ideal equipped with a PD structure is called a _PD ideal_ and a ring
equipped with a PD ideal (that is, a model in $\mathrm{Set}$ of the theory
$\mathrm{Ring}+\mathrm{Ideal}+\mathrm{PD}$, see Definition 5.3.1 below) is a
_PD ring_. This also defines homomorphisms of PD rings. We sometimes write a
PD ring $(A,I,\gamma)$ as $(\gamma\curvearrowright I\triangleleft A)$.

Not every ideal $J\subseteq I$ contained in the PD ideal of a PD ring
$(A,I,\gamma)$ is closed under the $\gamma_{n}$. But we can always close it up
in one step, that is, the ideal generated by $\gamma_{n}(a)$ for $n\geq 1$ and
$a\in J$ is a sub-PD-ideal of $I$, and it actually suffices to take
$\gamma_{n}(a)$ for $a$ in any generating set of the ideal $J$. More
generally, for any ideal $J$ of $A$, we let $\overline{J}$ be the ideal
generated by $J$ and $\\{\,\gamma_{n}(a)\mid n\geq 1,a\in J\cap I\,\\}$ and
call it the _PD saturation_ of the ideal $J$ (in the PD ring $(A,I,\gamma)$).
There is an induced PD structure on the ideal $I/(J\cap I)$ of the ring $A/J$
if and only if $J$ is PD saturated ($J=\overline{J}$) [9, Tag 07H2], and this
constitutes a bijective correspondence between PD saturated ideals of $A$ and
isomorphism classes of maps of PD rings
$(A,I,\gamma)\to(A^{\prime},I^{\prime},\gamma^{\prime})$ such that both $A\to
A^{\prime}$ and $I\to I^{\prime}$ are surjective.

For a PD ring $(A,I,\gamma)$ and an element $a\in A$, there is a unique PD
structure $\gamma_{a}$ on the ideal $I_{a}$ of $A_{a}=A[a^{-1}]$ such that
$A\to A_{a}$ is a homomorphism of PD rings, namely
$\gamma_{a,n}(\frac{a^{\prime}}{a^{k}})=\frac{\gamma_{n}(a^{\prime})}{a^{nk}}$.
The new PD ring has the expected universal property: a homomorphism
$(\gamma_{a}\curvearrowright I_{a}\triangleleft
A_{a})\to(\gamma^{\prime}\curvearrowright I^{\prime}\triangleleft A^{\prime})$
is the same as one $(\gamma\curvearrowright I\triangleleft
A)\to(\gamma^{\prime}\curvearrowright I^{\prime}\triangleleft A^{\prime})$
that makes $a$ invertible in $A^{\prime}$.

The _PD envelope_ construction [9, Tag 07H8] is a left adjoint functor for the
forgetful functor

$(\mathrm{Ring}+\mathrm{Ideal}+\mathrm{PD})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})\to(\mathrm{Ring}+\mathrm{Ideal})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})\hbox
to0.0pt{,\hss}$

and more generally, for a fixed PD ring $(K,I_{K},\gamma_{K})$, a left adjoint
for the forgetful functor

${({K\text{-}\mathrm{Alg}}+\mathrm{Ideal}_{I_{K}}+\mathrm{PD}_{\gamma_{K}})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})}$${({K\text{-}\mathrm{Alg}}+\mathrm{Ideal}_{I_{K}})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})}$${(K,I_{K},\gamma_{K})/(\mathrm{Ring}+\mathrm{Ideal}+\mathrm{PD})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})}$${(K,I_{K})/(\mathrm{Ring}+\mathrm{Ideal})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})\hbox
to0.0pt{.\hss}}$

Forming the PD envelope $(A^{\prime},I^{\prime},\gamma^{\prime})$ of $(A,I)$
(possibly over some $(K,I_{K},\gamma_{K})$) not only “enlarges” the ideal $I$
as necessary but also the ring $A$. However, there is an isomorphism $A/I\cong
A^{\prime}/I^{\prime}$, induced by the unit of the adjunction
$\eta_{(A,I)}:(A,I)\to(A^{\prime},I^{\prime})$.

The polynomial ring $A[X]$ over a PD ring $A=(A,I,\gamma)$ inherits a PD
structure on the ideal $I[X]$ (generated by the elements of $I$), such that PD
maps $A[X]\to A^{\prime}$ over $(A,I,\gamma)$ correspond to elements of
$A^{\prime}$. The same is true for an arbitrary set of polynomial variables
instead of a single one. But there is a separate notion of _PD polynomial
algebra_ [9, Tag 07H4], denoted $A\langle X\rangle$, which is freely generated
as an $A$-module by $1$ and the divided powers $\gamma_{n}(X)$ of $X$ (instead
of the ordinary powers $X^{n}$). The PD ideal of $A\langle X\rangle$ is
generated by the elements of $I$ and the $\gamma_{n}(X)$, and PD maps
$A\langle X\rangle\to A^{\prime}$ over $(A,I,\gamma)$ correspond to elements
of the PD ideal $I^{\prime}$ of $A^{\prime}$. Equivalently, $A\langle
X\rangle$ is the PD envelope of the ideal $I[X]+(X)$ in $A[X]$ over
$(A,I,\gamma)$. Again, an arbitrary set of variables works just as well.

###  5.3 Relevant geometric theories

Here, we define the various theories and theory extensions from which the
syntactic presentation of the crystalline topos will be built. We do this
right away, since it will also be of some use in the definition of the
crystalline topos itself. In particular, by giving our formal definition of a
PD structure in the form of an extension of geometric theories, it can
immediately be used to define PD structures on schemes as well.

######  Definition 5.3.1.

  1. (i)

We denote $\mathrm{Ideal}$ the extension of the theory $\mathrm{Ring}$ (see
Definition 3.7.1) consisting of a relation symbol $I\subseteq A$ with these
axioms:

$0\in I,\qquad x\in I\land y\in I\;\vdash_{x,y:A}\;x+y\in I,\qquad x\in
I\;\vdash_{\lambda,x:A}\;\lambda x\in I\hbox to0.0pt{.\hss}$

  2. (ii)

In the theory $\mathrm{Ring}+\mathrm{Ideal}$, we can require the ideal to be a
nil ideal by adding the axiom:

$(\mathrm{nil})\vcentcolon=\qquad x\in
I\;\vdash_{x:A}\;\bigvee_{n\in\mathbb{N}}(x^{n}=0)\hbox to0.0pt{.\hss}$

  3. (iii)

We denote $\mathrm{PD}\text{-}\mathrm{Ideal}$ the following extension of the
theory $\mathrm{Ring}$. First introduce a sort $S_{I}$, a function symbol
$\iota:S_{I}\to A$ with

$\iota(x)=\iota(y)\;\vdash_{x,y:S_{I}}\;x=y$

and function symbols $0:S_{I}$, ${+}:S_{I}\times S_{I}\to S_{I}$ and
${\cdot}:A\times S_{I}\to S_{I}$ together with equational axioms stating that
$\iota$ is an $A$-module homomorphism. Then add function symbols
$\gamma_{n}:S_{I}\to S_{I}$ for all $n\geq 1$ and these equational axioms (see
[9, Tag 07GL]):

     * •

$\gamma_{1}(x)=x$

     * •

$\gamma_{n}(x+y)=\gamma_{n}(x)+\gamma_{n}(y)+\sum\limits_{i+j=n,\;i,j\geq
1}\iota(\gamma_{i}(x))\gamma_{j}(y)$

     * •

$\gamma_{n}(\lambda x)=\lambda^{n}\gamma_{n}(x)$

     * •

$\iota(\gamma_{m}(x))\gamma_{n}(x)=\binom{m+n}{m}\gamma_{m+n}(x)$

     * •

$\gamma_{m}(\gamma_{n}(x))=\frac{(mn)!}{m!(n!)^{m}}\gamma_{mn}(x)$

  4. (iv)

To be able to talk about a PD structure on an existing ideal, we define the
extension $\mathrm{PD}$ of $\mathrm{Ring}+\mathrm{Ideal}$ to be
$\mathrm{PD}\text{-}\mathrm{Ideal}$ plus the axiom

$\exists x{\,:\,}S_{I}.\;\iota(x)=y\;\dashv\vdash_{y:A}\;y\in I\hbox
to0.0pt{.\hss}$

Note that $\mathrm{Ideal}+\mathrm{PD}$ and $\mathrm{PD}\text{-}\mathrm{Ideal}$
are Morita equivalent extensions of the theory $\mathrm{Ring}$, since the
axioms in $\mathrm{Ideal}$ are redundant here, and the relation symbol $I$ is
definable by the formula $\exists x{\,:\,}S_{I}.\;\iota(x)=y$. We prefer to
write $\mathrm{Ring}+\mathrm{Ideal}+\mathrm{PD}$ because we think of the
divided powers as an additional structure on an ideal. Also note that
$\mathrm{PD}$ is equivalent to a localic extension of
$\mathrm{Ring}+\mathrm{Ideal}$, as the divided power structure could
alternatively be implemented as relation symbols
$\widetilde{\gamma}_{n}\subseteq A\times A$ expressing partial functions from
$A$ to $A$. The main purpose of the sort $S_{I}$ above is to be able to write
the axioms in a more readable equational style.

######  Definition 5.3.2.

  1. (i)

Similarly to the extension ${K\text{-}\mathrm{AlgStr}}$ of $\mathrm{Ring}$
from Definition 3.7.1, for a $K$-algebra $R$ we denote
${R\text{-}\mathrm{AlgStr}}_{K}$ the extension of ${K\text{-}\mathrm{Alg}}$
adding an $R$-algebra structure compatible with the given $K$-algebra
structure. If necessary, we write ${R\text{-}\mathrm{AlgStr}}_{K}(A)$ to
indicate the sort $A$ to which the $R$-algebra structure is added.

  2. (ii)

For a ring $K$ and a $K$-algebra $R$, we denote
$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}$ the theory
${K\text{-}\mathrm{Alg}}+{R\text{-}\mathrm{Alg}}$ extended by a function
symbol $f:A\to B$ (where $A$ is the $K$-algebra and $B$ is the $R$-algebra)
and equational axioms expressing that $f$ is a $K$-algebra homomorphism. A
model can be pictured as

${K}$${R}$${A}$${B\hbox to0.0pt{.\hss}}$$\scriptstyle{f}$

When we treat $K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}$ as if it
contained ${K\text{-}\mathrm{Alg}}+\mathrm{Ideal}$, we intend to use the
kernel of $f$ as the ideal, that is, $x\in I$ is defined as $f(x)=0$.

  3. (iii)

We set
$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}\vcentcolon=K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}+(\mathrm{surj})$,
where

$(\mathrm{surj})\vcentcolon=\qquad\top\;\vdash_{y:B}\;\exists
x{\,:\,}A.\;f(x)=y\hbox to0.0pt{.\hss}$

  4. (iv)

For a ring $K$ and an ideal $I_{K}\subseteq K$, we denote
$\mathrm{Ideal}_{I_{K}}$ the extension of ${K\text{-}\mathrm{Alg}}$ consisting
of the extension $\mathrm{Ideal}$ and the additional axioms $c_{\lambda}\in I$
for all $\lambda\in I_{K}$.

  5. (v)

For a PD ring $(K,I_{K},\gamma_{K})$, we denote $\mathrm{PD}_{\gamma_{K}}$ the
extension of ${K\text{-}\mathrm{Alg}}+\mathrm{Ideal}_{I_{K}}$ consisting of
the extension $\mathrm{PD}$ and the additional axioms (for every $\lambda\in
I_{K}$)

$\iota(x)=c_{\lambda}\;\vdash_{x:S_{I}}\;\iota(\gamma_{n}(x))=c_{\gamma_{K,n}(\lambda)}\hbox
to0.0pt{.\hss}$

We can write
$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}+\mathrm{PD}_{\gamma_{K}}$
if $I_{K}$ vanishes in $R$, since
$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}$ then proves
$f(c_{\lambda})=0$ for $\lambda\in I_{K}$.

Similarly to the economical version of the theory ${K\text{-}\mathrm{Alg}}$
mentioned in Remark 3.7.2, we can use a presentation $R=K[X_{i}]/(r_{j})$ of
$R$ as a $K$-algebra to implement the $R$-algebra structure of $B$ in the
theory $K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}$. That is,
$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}$ is equivalent to the
extension of $K\text{-}\mathrm{Alg}\text{-}K\text{-}\mathrm{Alg}$ where we
only add constant symbols $c_{i}:B$ and axioms $r_{j}(c_{i})=0$, using the
available $K$-algebra structure to interpret $r_{j}(c_{i})$ as a term of sort
$B$. This will be used in Propositions 5.5.1 and 5.7.3.

######  Remark 5.3.3.

The theories ${K\text{-}\mathrm{Alg}}$,
${K\text{-}\mathrm{Alg}}+\mathrm{Ideal}_{I_{K}}$ and
${K\text{-}\mathrm{Alg}}+\mathrm{Ideal}_{I_{K}}+\mathrm{PD}_{\gamma_{K}}$ have
in common that their categories of $\mathrm{Set}$-models are equivalent to
certain slice categories.

$\displaystyle{K\text{-}\mathrm{Alg}}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})$
$\displaystyle\simeq
K/(\mathrm{Ring}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set}))$
$\displaystyle({K\text{-}\mathrm{Alg}}+\mathrm{Ideal}_{I_{K}})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})$
$\displaystyle\simeq(K,I_{K})/((\mathrm{Ring}+\mathrm{Ideal})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set}))$
$\displaystyle({K\text{-}\mathrm{Alg}}+\mathrm{Ideal}_{I_{K}}+\mathrm{PD}_{\gamma_{K}})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})$
$\displaystyle\simeq(K,I_{K},\gamma_{K})/((\mathrm{Ring}+\mathrm{Ideal}+\mathrm{PD})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set}))$

We even have

$(K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})\simeq(K\to
R)/(\mathrm{Ring}^{\to}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set}))$

if we use a theory $\mathrm{Ring}^{\to}$ of ring homomorphisms. We could in
fact define these theories by a general construction that turns a geometric
theory $\mathbb{T}$ and a $\mathrm{Set}$-model $M$ of $\mathbb{T}$ into a
theory of “$M$-algebras”. This is the same construction that is used by
Blechschmidt in [4, Definition 3.1], who however applies it internally in the
classifying topos of $\mathbb{T}$, to the universal model of $\mathbb{T}$
instead of a $\mathrm{Set}$-based model. But for our present goals, we need to
be able to combine the above theories and extensions in flexible ways, so that
such a formulation does not help us much.

###  5.4 Definition of the big crystalline topos

Before we can define the crystalline topos, we need to introduce a number of
notions involving PD structures on schemes. Recall that a closed embedding
$U\to T$ defined by a quasi-coherent ideal sheaf
$\mathcal{I}\subseteq\mathcal{O}_{T}$ is a _thickening_ (in the terminology of
[9, Tag 04EX]), that is, $U\to T$ is a homeomorphism, if and only if every
local section of $\mathcal{I}$ is locally nilpotent, which in turn means
simply that the internal ideal $\mathcal{I}$ of the internal ring
$\mathcal{O}_{T}$ in $\operatorname{Sh}(T)$ satisfies the axiom
$(\mathrm{nil})$.

######  Definition 5.4.1 (see [9, Tag 07I1]).

  1. (i)

A _PD scheme_ is a scheme $S$ together with a quasi-coherent ideal sheaf
$\mathcal{I}\subseteq\mathcal{O}_{S}$ and a PD structure on $\mathcal{I}$,
that is, a model extension along $\mathrm{PD}$ of the
$(\mathrm{Ring}+\mathrm{Ideal})$-model
$(\mathcal{I}\triangleleft\mathcal{O}_{S})$ in $\operatorname{Sh}(S)$. A
_morphism_ of PD schemes
$(S,\mathcal{I},\gamma)\to(S^{\prime},\mathcal{I}^{\prime},\gamma^{\prime})$
is a morphism of schemes $f:S\to S^{\prime}$ such that
$f^{\sharp}:f^{-1}\mathcal{O}_{S^{\prime}}\to\mathcal{O}_{S}$ is a
$(\mathrm{Ring}+\mathrm{Ideal}+\mathrm{PD})$-model homomorphism. In
particular, it induces a morphism between the closed subschemes of $S$ and
$S^{\prime}$ defined by $\mathcal{I}$ respectively $\mathcal{I}^{\prime}$,

${S^{\prime}}$${V(\mathcal{I}^{\prime})}$${S}$${V(\mathcal{I})\hbox
to0.0pt{.\hss}}$

  2. (ii)

A _PD thickening_ is a PD scheme $(T,\mathcal{I},\gamma)$ such that the closed
embedding $V(\mathcal{I})\to T$ is a thickening, that is, the
$(\mathrm{Ring}+\mathrm{Ideal}+\mathrm{PD})$-model
$(\mathcal{O}_{T},\mathcal{I},\gamma)$ in $\operatorname{Sh}(T)$ fulfills the
axiom $(\mathrm{nil})$. PD thickenings are sometimes denoted $(U,T,\gamma)$,
where $U=V(\mathcal{I})$.

  3. (iii)

If $(S,\mathcal{I}_{S},\gamma_{S})$ is a PD scheme and $X$ is a scheme over
the closed subscheme $S_{0}\vcentcolon=V(\mathcal{I}_{S})$ of $S$, then a PD
thickening _over_ $S$ _and_ $X$ is a PD thickening $(T,\mathcal{I},\gamma)$
over $(S,\mathcal{I}_{S},\gamma_{S})$ together with a morphism of schemes
$V(\mathcal{I})\to X$ over $S$.

${S}$${S_{0}}$${X}$${T}$${V(\mathcal{I})}$

A _morphism_ of PD thickenings over $S$ and $X$ is a morphism of PD schemes
over $S$ such that the induced morphism $V(\mathcal{I})\to
V(\mathcal{I}^{\prime})$ respects the structure morphisms to $X$.

The PD thickenings over $S$ and $X$ will be the objects of the crystalline
site of $X/S$. Most important to us is that if $S$ and $X$ are affine and we
also require $T$ to be affine, then we can nicely describe the resulting
category by a geometric theory.

######  Lemma 5.4.2.

The full subcategory of the category of PD thickenings where the underlying
scheme $T$ is affine is equivalent to

${(\mathrm{Ring}+\mathrm{Ideal}+\mathrm{PD}+(\mathrm{nil}))\operatorname{-mod}(\mathrm{Set})}^{\mathrm{op}}\hbox
to0.0pt{.\hss}$

For a fixed PD ring $(K,I_{K},\gamma_{K})$ and a $K/I_{K}$-algebra $R$, the
full subcategory of the category of PD thickenings over
$S=\operatorname{Spec}(K)$ and $X=\operatorname{Spec}(R)$ where the underlying
scheme $T$ is affine is equivalent to

${(K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}_{\gamma_{K}}+(\mathrm{nil}))\operatorname{-mod}(\mathrm{Set})}^{\mathrm{op}}\hbox
to0.0pt{.\hss}$

###### Proof.

The category
${(\mathrm{Ring}+\mathrm{Ideal})\operatorname{-mod}(\mathrm{Set})}$ is of
course dual to the category of affine schemes equipped with a quasi-coherent
ideal sheaf, via $(I\triangleleft A)\mapsto(\operatorname{Spec}A,\tilde{I})$.
One can check that the closed embedding
$\operatorname{Spec}(A/I)\to\operatorname{Spec}A$ is a thickening if and only
if every element of $I$ is nilpotent. Similarly, a PD structure on $I$
uniquely determines a PD structure on $\tilde{I}$, and morphisms respecting
one correspond to morphisms respecting the other [2, p. 3.18].

For the second part, the theory extension $\mathrm{PD}_{\gamma_{K}}$
(requiring a PD structure on the kernel of $f:A\to B$ extending $\gamma_{K}$)
makes sense because the PD ideal $I_{K}$ vanishes in $R$, so the theory
$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}$ proves
$f(c_{\lambda})=0$ for all $\lambda\in I_{K}$. The rest follows from the first
part. ∎

If $(T,\mathcal{I},\gamma)$ is a PD thickening and $T^{\prime}\subseteq T$ is
an open subscheme, then
$(T^{\prime},\mathcal{I}|_{T^{\prime}},\gamma|_{T^{\prime}})$ is again a PD
thickening (as restricting to $T^{\prime}$ is pulling back along the geometric
morphism $\operatorname{Sh}(T^{\prime})\to\operatorname{Sh}(T)$) and there is
a canonical inclusion morphism
$(T^{\prime},\mathcal{I}|_{T^{\prime}},\gamma|_{T^{\prime}})\to(T,\mathcal{I},\gamma)$.
This allows us to define a Zariski topology on the category of PD thickenings
over $S$ and $X$.

######  Definition 5.4.3.

Let $(S,\mathcal{I}_{S},\gamma_{S})$ be a PD scheme and let $X$ be a scheme
over $S_{0}\vcentcolon=V(\mathcal{I}_{S})$. The _big crystalline site_
$\mathrm{Cris}(X/S)$ is the category of PD thickenings over $S$ and $X$,
endowed with the Zariski topology, that is, a sieve on
$(T,\mathcal{I},\gamma)$ is covering if and only if it contains all the arrows
$(T_{i},\mathcal{I}|_{T_{i}},\gamma|_{T_{i}})\to(T,\mathcal{I},\gamma)$ for
some open cover $T=\bigcup_{i}T_{i}$ of $T$.

If we want to consider the topos $\operatorname{Sh}(\mathrm{Cris}(X/S))$ of
sheaves on the crystalline site, there remains an issue of size. There is no
way to talk about the collection of presheaves on a big category such as
$\mathrm{Cris}(X/S)$ (or generally, about the collection of functions between
two proper classes) in ZFC set theory. If the category in question is
essentially small, then one can circumvent this by choosing a skeleton. More
generally, if $J$ is a Grothendieck topology on a category $\mathcal{C}$ such
that there is a (small) dense set $S\subseteq\mathcal{C}$ of objects of
$\mathcal{C}$, then one can write

$\operatorname{Sh}(\mathcal{C},J)\vcentcolon=\operatorname{Sh}(S,J|_{S})$

(where $J|_{S}$ is the greatest topology on $S$ such that
$S\hookrightarrow\mathcal{C}$ preserves covers). This is justified because if
$S^{\prime}\subseteq\mathcal{C}$ is another dense set of objects, then by the
Comparison Lemma, there are canonical equivalences

$\operatorname{Sh}(S,J|_{S})\simeq\operatorname{Sh}(S\cup S^{\prime},J|_{S\cup
S^{\prime}})\simeq\operatorname{Sh}(S^{\prime},J|_{S^{\prime}})\hbox
to0.0pt{.\hss}$

However, the site $\mathrm{Cris}(X/S)$ does not admit a dense set of objects.
To define a crystalline topos, a choice of some class
$\mathcal{C}\subseteq\mathrm{Cris}(X/S)$ which does admit a dense subset is
therefore necessary. We leave this choice as an explicit parameter to the
definition for now, but we will specialize to a particular class $\mathcal{C}$
in Definition 5.7.1.

To avoid an ambiguity in the topology on the resulting site, we require
$\mathcal{C}$ to be closed under taking open subschemes $T^{\prime}\subseteq
T$ (with the induced PD structure on $T^{\prime}$). While there is always the
induced topology $J_{\mathrm{Zar}}|_{\mathcal{C}}$, for which a sieve on
$T\in\mathcal{C}$ is covering if and only if it generates a covering sieve in
$\mathrm{Cris}(X/S)$, if $\mathcal{C}$ does not contain all open subschemes of
$T$, then there is no guarantee that such a sieve actually contains a
“distinguished” Zariski cover by open subschemes. With this assumption, the
two conditions become equivalent.

######  Definition 5.4.4.

Let $S$, $X$ be as before and let $\mathcal{C}\subseteq\mathrm{Cris}(X/S)$ be
a class of objects admitting a dense subset and closed under taking open
subschemes. Then we denote $\mathrm{Cris}_{\mathcal{C}}(X/S)$ the full
subcategory on these objects endowed with the Zariski topology as above. And
the topos

$(X/S)_{\mathrm{Cris}_{\mathcal{C}}}\vcentcolon=\operatorname{Sh}(\mathrm{Cris}_{\mathcal{C}}(X/S))$

is the _big crystalline topos_ of $X$ over $(S,\mathcal{I}_{S},\gamma_{S})$
(defined using the objects in $\mathcal{C}$).

There is a structure sheaf on $\mathrm{Cris}_{\mathcal{C}}(X/S)$ (see [9, Tag
07IN]),

$\mathcal{O}=\mathcal{O}_{(X/S)_{\mathrm{Cris}_{\mathcal{C}}}}\;:\;\mathrm{Cris}_{\mathcal{C}}(X/S)^{\mathrm{op}}\to\mathrm{Set},\quad(T,\mathcal{I},\gamma)\mapsto\mathcal{O}_{T}(T)\hbox
to0.0pt{.\hss}$

As in the case of the Zariski topos, this sheaf carries a ring structure. But
here, there is more. We have a second ring object

$\mathcal{O}^{\prime}=\mathcal{O}^{\prime}_{(X/S)_{\mathrm{Cris}_{\mathcal{C}}}}\;:\;\mathrm{Cris}_{\mathcal{C}}(X/S)^{\mathrm{op}}\to\mathrm{Set},\quad(T,\mathcal{I},\gamma)\mapsto\mathcal{O}_{U}(U)=\mathcal{O}_{V(\mathcal{I})}(V(\mathcal{I}))\hbox
to0.0pt{,\hss}$

sometimes denoted $\mathcal{O}_{X}$. Furthermore, there is a surjective ring
homomorphism $\mathcal{O}\to\mathcal{O}^{\prime}$, and its kernel

$\mathcal{J}=\mathcal{J}_{(X/S)_{\mathrm{Cris}_{\mathcal{C}}}}\;:\;\mathrm{Cris}_{\mathcal{C}}(X/S)^{\mathrm{op}}\to\mathrm{Set},\quad(T,\mathcal{I},\gamma)\mapsto\mathcal{I}(T)$

carries a canonical PD structure. In summary, we have a model

$(\mathcal{O},\mathcal{J})\;\in\;(\mathrm{Ring}+\mathrm{Ideal}+\mathrm{PD})\text{-}\allowbreak\mathrm{mod}((X/S)_{\mathrm{Cris}_{\mathcal{C}}})\hbox
to0.0pt{.\hss}$

An extension of this model (along a localic theory extension) will turn out to
be the universal model of a theory classified by
$(X/S)_{\mathrm{Cris}_{\mathcal{C}}}$, at least for an appropriate choice of
$\mathcal{C}$.

The Zariski topology on $\mathrm{Cris}(X/S)$ (or
$\mathrm{Cris}_{\mathcal{C}}(X/S)$) in particular ensures that every object
can be covered by objects $(T,\mathcal{I},\gamma)$ with $T$ affine, in other
words, the affine objects are dense. Thus we can use the Comparison Lemma to
obtain an equivalent site consisting of affine objects, which we have already
described as a category in Lemma 5.4.2. The following lemma complements this
with a description of the topology on the new site.

######  Lemma 5.4.5.

Let $(K,I_{K},\gamma_{K})$ be a PD ring and let $R$ be a $K/I_{K}$-algebra.
The topology induced via the Comparison Lemma on
$(K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}_{\gamma_{K}}+(\mathrm{nil}))\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})^{\mathrm{op}}$
regarded as a (dense) full subcategory of
$\mathrm{Cris}(\operatorname{Spec}R/\operatorname{Spec}K)$ is the following: A
cosieve on an object

$(\gamma\curvearrowright I\triangleleft A\twoheadrightarrow B)$

is covering if and only if it contains the canonical arrows to

$(\gamma_{a_{i}}\curvearrowright I_{a_{i}}\triangleleft
A_{a_{i}}\twoheadrightarrow B_{a_{i}})$

for some finite family of elements $a_{i}\in A$ with
$(a_{1},\dots,a_{n})=(1)$.

###### Proof.

We first note that the model $(\gamma_{a_{i}}\curvearrowright
I_{a_{i}}\triangleleft A_{a_{i}}\twoheadrightarrow B_{a_{i}})$ is well-defined
— the ring map $A_{a_{i}}\to B_{a_{i}}$ is surjective and has kernel
$I_{a_{i}}$ since localization is exact, and $I_{a_{i}}$ is still a nil ideal
— and corresponds to the open subscheme $D(a_{i})$ of the PD scheme
$\operatorname{Spec}A_{a_{i}}$. A cosieve on $(\gamma\curvearrowright
I\triangleleft A\twoheadrightarrow B)$ is covering for the induced topology if
it generates a covering sieve in
$\mathrm{Cris}(\operatorname{Spec}R/\operatorname{Spec}K)$. But such a sieve
contains (the morphisms of PD thickenings over $\operatorname{Spec}K$ and
$\operatorname{Spec}R$ induced by) an open cover of $\operatorname{Spec}A$ if
and only if it contains a cover by standard opens $D(a_{i})$, which in turn
cover $\operatorname{Spec}A$ if and only if $(a_{1},\dots,a_{n})=(1)$. ∎

###  5.5 Preliminary presheaf type results

We saw in Lemma 5.4.2 that the crystalline site of affine schemes
$S=\operatorname{Spec}K$, $X=\operatorname{Spec}R$ is closely tied to the
theory

$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}_{\gamma_{K}}+(\mathrm{nil})\hbox
to0.0pt{.\hss}$

Our strategy for proving a classification result on the crystalline topos is
to show, under appropriate assumptions, that this theory is of presheaf type,
then choose the class of objects of the crystalline site in such a way that
the affine objects turn out to be exactly the compact models of the theory,
and finally add the quotient extension $(\mathrm{loc})$ to produce the Zariski
topology.

Right now, we want to approach the presheaf type part by looking at somewhat
simpler theories. While $K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}$
is of presheaf type simply because it is an algebraic theory, the situation
already becomes more interesting for
$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}$, as the axiom
$(\mathrm{surj})$ is not cartesian. We also include the extension
$\mathrm{PD}$ but assume the PD structure on $K$ to be trivial and forget
about $(\mathrm{nil})$ for now.

The following two propositions can also serve as an illustration of the
different ways in which one might try to decompose the same theory into parts
to show that it is of presheaf type, and how this can lead to results of
different strengths.

######  Proposition 5.5.1.

Let $R$ be a finitely presented $K$-algebra. Then the theory

$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}$

is of presheaf type.

###### Proof.

The cartesian theory ${K\text{-}\mathrm{Alg}}+\mathrm{Ideal}+\mathrm{PD}$ is
Morita-equivalent to
$K\text{-}\mathrm{Alg}\text{-}K\text{-}\mathrm{Quot}+\mathrm{PD}$ by
introducing a sort $B$ for the quotient ring $A/I$, similarly to Example
2.4.2. So the latter theory is of presheaf type and we are only missing the
$R$-algebra structure on the $K$-algebra $B$. Since $R\cong
K[X_{1},\dots,X_{n}]/(r_{1},\dots,r_{m})$ is a finitely presented $K$-algebra,
the extension ${R\text{-}\mathrm{AlgStr}}_{K}(B)$ is equivalent to finitely
many constant symbols $c_{i}\,{:}\,B$, $i=1,\dots,n$, and finitely many axioms
in the empty context $\top\;\vdash_{[]}\;r_{j}(\vec{c})=0$, $j=1,\dots,m$.
Thus we are done by Proposition 4.4.5 and Corollary4.4.2. ∎

We can reduce the assumption on $R$ to finite type instead of finite
presentation if we handle the axiom $(\mathrm{surj})$ in a different way. For
this, we need a lemma.

######  Lemma 5.5.2.

Let $M=(\gamma\curvearrowright\mathfrak{a}\triangleleft
A\xrightarrow{f}B)\in\mathcal{C}={(K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}+\mathrm{PD})\operatorname{-mod}(\mathrm{Set})}$
and $b\in B$. Then there is a universal triple $(M^{\prime},a^{\prime},g)$,
where
$M^{\prime}=(\gamma^{\prime}\curvearrowright\mathfrak{a}^{\prime}\triangleleft
A^{\prime}\xrightarrow{f^{\prime}}B^{\prime})\in\mathcal{C}$, $a^{\prime}\in
A^{\prime}$ and $g=(g_{A},g_{B}):M\to M^{\prime}$, such that
$f^{\prime}(a^{\prime})=g_{B}(b)$; that is, the following diagram in
$[\mathcal{C},\mathrm{Set}]$ is a pullback:

${\operatorname{Hom}(M^{\prime},-)}$${\operatorname{Hom}(M,-)}$${\llbracket
A\rrbracket_{-}}$${\llbracket
B\rrbracket_{-}}$$\scriptstyle{g^{*}}$$\scriptstyle{a^{\prime}}$${\lrcorner}$$\scriptstyle{b}$$\scriptstyle{\llbracket
f\rrbracket_{-}}$

Furthermore, $g_{B}:B\to B^{\prime}$ is an isomorphism.

###### Proof.

Consider the $A$-algebra homomorphism
$\widetilde{f}:\widetilde{A}\vcentcolon=A[X]\to B$ with $\widetilde{f}(X)=b$
and set $\widetilde{\mathfrak{a}}\vcentcolon=\ker\widetilde{f}$. Let
$\gamma^{\prime}\curvearrowright\mathfrak{a}^{\prime}\triangleleft A^{\prime}$
be the PD envelope (over $\gamma$) of the $(\mathfrak{a}\triangleleft
A)$-algebra $\widetilde{\mathfrak{a}}\triangleleft\widetilde{A}$. In
particular, we have $\widetilde{A}/\widetilde{\mathfrak{a}}\cong
A^{\prime}/\mathfrak{a}^{\prime}$ (as $A$-algebras) via the unit
$\eta_{\widetilde{\mathfrak{a}}\triangleleft\widetilde{A}}:(\widetilde{\mathfrak{a}}\triangleleft\widetilde{A})\to(\mathfrak{a}^{\prime}\triangleleft
A^{\prime})$ of the adjunction, so that $\widetilde{f}$ induces an $A$-algebra
homomorphism $f^{\prime}:A^{\prime}\to B$ with $\ker
f^{\prime}=\mathfrak{a}^{\prime}$. We have thus defined a model
$M^{\prime}\in\mathcal{C}$ and an arrow $g:M\to M^{\prime}$, where $g_{A}:A\to
A^{\prime}$ is just the homomorphism of PD-rings from the PD envelope
construction and $g_{B}=\mathrm{id}_{B}$. Setting
$a^{\prime}\vcentcolon=\eta_{\widetilde{\mathfrak{a}}\triangleleft\widetilde{A}}(X)$
satisfies $f^{\prime}(a^{\prime})=b$.

To check the universal property, let $h:M\to
M^{\prime\prime}=(\gamma^{\prime\prime}\curvearrowright\mathfrak{a}^{\prime\prime}\triangleleft
A^{\prime\prime}\xrightarrow{f^{\prime\prime}}B^{\prime\prime})$ be given. An
element $a^{\prime\prime}\in A^{\prime\prime}=\llbracket
A\rrbracket_{M^{\prime\prime}}$ with
$f^{\prime\prime}(a^{\prime\prime})=h_{B}(b)$ is the same thing as an
$A$-algebra homomorphism $\widetilde{A}\to A^{\prime\prime}$ fitting in the
square

${\widetilde{A}}$${B}$${A^{\prime\prime}}$${B^{\prime\prime}\hbox
to0.0pt{.\hss}}$$\scriptstyle{\widetilde{f}}$$\scriptstyle{h_{B}}$$\scriptstyle{f^{\prime\prime}}$

But then it is automatic that $\widetilde{\mathfrak{a}}$ is sent to
$\mathfrak{a}^{\prime\prime}$, so there is a unique homomorphism of PD-rings
$(\gamma^{\prime}\curvearrowright\mathfrak{a}^{\prime}\triangleleft
A^{\prime})\to(\gamma^{\prime\prime}\curvearrowright\mathfrak{a}^{\prime\prime}\triangleleft
A^{\prime\prime})$ over $A$ sending $a^{\prime}$ to $a^{\prime\prime}$, and it
makes the square

${A^{\prime}}$${B}$${A^{\prime\prime}}$${B^{\prime\prime}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{h_{B}}$$\scriptstyle{f^{\prime\prime}}$

commute, as required for a model homomorphism $M^{\prime}\to
M^{\prime\prime}$. ∎

######  Proposition 5.5.3.

Let $R$ be a finitely generated $K$-algebra. Then the theory

$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}$

is of presheaf type.

###### Proof.

The theory $K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}+\mathrm{PD}$
(with a PD structure on the kernel of the $K$-algebra homomorphism $A\to B$)
is of presheaf type because it is cartesian. We show that the missing axiom
$(\mathrm{surj})$ is a rigid-topology quotient.

Let a compact model

$M=(\gamma\curvearrowright\mathfrak{a}\triangleleft A\xrightarrow{f}B)$

of $K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}+\mathrm{PD}$ and an
element $b\in B$ be given. Consider the model homomorphism $g:M\to M^{\prime}$
from Lemma 5.5.2. We first note that $M^{\prime}$ is also compact, since
$\operatorname{Hom}(M^{\prime},-)$ is a finite limit of functors preserving
filtered colimits. (For $\llbracket A\rrbracket_{-}$ and $\llbracket
B\rrbracket_{-}$ preserving filtered colimits, see Lemma 4.2.2.) Secondly, a
map $h:M\to M^{\prime\prime}$ factors through $M^{\prime}$ (not necessarily
uniquely) if and only if $h_{B}$ maps $b$ to something in the image of
$f^{\prime\prime}$. This means that the cosieve on $M$ generated by $g$ (in
the category
${(K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}+\mathrm{PD})\operatorname{-mod}(\mathrm{Set})}_{c}$)
is exactly the cosieve $S_{b}$ as in Theorem 4.3.1 and thus $M$ is
$J_{(\mathrm{surj})}$-covered by $M^{\prime}$. And lastly, the subset
$f^{\prime}(A^{\prime})\subseteq B^{\prime}=B$ contains of course both $f(A)$
and the element $b$.

Now, since $M$ is compact, $B$ is a finitely presented $R$-algebra. This
follows from Lemma 4.2.3 by observing that the right adjoint of the forgetful
functor in the following diagram preserves filtered colimits.

${(K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}+\mathrm{PD})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})}$${({R\text{-}\mathrm{Alg}})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})}$${(\gamma\curvearrowright\mathfrak{a}\triangleleft
A\to B)}$${B}$${((0)\triangleleft
B^{\prime}\xrightarrow{=}B^{\prime})}$${B^{\prime}}$

$\vdash$

Thus, $B$ is also a finitely generated $K$-algebra by assumption. But if
$b_{1},\dots,b_{n}\in B$ are $K$-algebra generators of $B$, then we can
successively apply Lemma 5.5.2 to them to obtain a covering arrow $M\to M_{n}$
(using the transitivity of the topology $J_{(\mathrm{surj})}$) such that
$b_{i}\in f_{n}(A_{n})$ for all $i$, implying that $f_{n}$ is surjective. ∎

######  Remark 5.5.4.

Propositions 5.5.1 and 5.5.3 become wrong if we drop the finiteness assumption
entirely. For example, take $K=\mathbb{Q}$, $R=\mathbb{C}$. The extension
$\mathrm{PD}$ is an equivalence extension of
$\mathbb{Q}\text{-}\mathrm{Alg}\text{-}\mathbb{C}\text{-}\mathrm{Quot}$
because the $\gamma_{n}$ are definable by

$\gamma_{n}(x)=y\;\dashv\vdash_{x,y\,{:}\,S_{I}}\;\frac{1}{n!}\iota(x)^{n}=\iota(y)\hbox
to0.0pt{,\hss}$

using the available $\mathbb{Q}$-algebra structure. And
$\mathbb{Q}\text{-}\mathrm{Alg}\text{-}\mathbb{C}\text{-}\mathrm{Quot}$ has
too few compact $\mathrm{Set}$-based models to be of presheaf type, similarly
to the situation in Example 4.5.2.

Namely, we can apply Lemma 4.2.3 to the pair of adjoint functors

${(\mathbb{Q}\text{-}\mathrm{Alg}\text{-}\mathbb{C}\text{-}\mathrm{Quot})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})}$${({\mathbb{Q}\text{-}\mathrm{Alg}})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})}$${(A\twoheadrightarrow
B)}$${A}$${(A^{\prime}\twoheadrightarrow 0)}$${A^{\prime}}$

$\vdash$

to see that if $A\twoheadrightarrow B$ is a compact model of
$\mathbb{Q}\text{-}\mathrm{Alg}\text{-}\mathbb{C}\text{-}\mathrm{Quot}$, then
$A$ is a finitely generated $\mathbb{Q}$-algebra. Then the
$\mathbb{C}$-algebra $B$ is also a finitely generated $\mathbb{Q}$-algebra,
which implies $B=0$ for cardinality reasons. But there are certainly non-
compact models with $B\neq 0$ (for example, let $A$ be freely generated by the
elements of $B$), so the sequent

$\top\;\vdash_{[]}\;1_{B}=0_{B}$

can not be provable in
$\mathbb{Q}\text{-}\mathrm{Alg}\text{-}\mathbb{C}\text{-}\mathrm{Quot}$,
meaning that the compact $\mathrm{Set}$-based models are not jointly
conservative and therefore
$\mathbb{Q}\text{-}\mathrm{Alg}\text{-}\mathbb{C}\text{-}\mathrm{Quot}$ is not
of presheaf type (see [5, Theorem 6.1.1]).

We can now prove a “lazy version” of our classification result, where we
simply assume the crystalline site to contain exactly the desired objects.
More precisely, we even have to assume that a suitable class of objects
$\mathcal{C}\subseteq\mathrm{Cris}(X/S)$ exists, which is not trivial given
the additional assumption that $\mathcal{C}$ is closed under taking open
subschemes. The axiom $(\mathrm{nil})$ is made redundant here by an additional
assumption on $K$.

######  Corollary 5.5.5.

Let $K$ be a ring of nonzero characteristic, regard $K$ as a PD ring with the
trivial PD ideal $(0)$, and let $R$ be a finitely generated $K$-algebra.
Assume there is a class
$\mathcal{C}\subseteq\mathrm{Cris}(\operatorname{Spec}R/\operatorname{Spec}K)$
admitting a dense subset and closed under open subschemes, such that an affine
object belongs to $\mathcal{C}$ if and only if the corresponding model of
$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}+(\mathrm{nil})$
is compact. Then the topos
$(\operatorname{Spec}R/\operatorname{Spec}K)_{\mathrm{Cris}_{\mathcal{C}}}$
classifies the theory

$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}+(\mathrm{nil})+(\mathrm{loc})\hbox
to0.0pt{.\hss}$

###### Proof.

If $n=0$ in $K$ for some $n\geq 1$, then for any PD ring $(A,I,\gamma)$ over
$K$ and any $a\in I$, we have $a^{n}=n!\gamma_{n}(a)=0$. This calculation can
be carried out within the theory
$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}$, so that the
axiom $(\mathrm{nil})$ is redundant. Thus by Proposition 5.5.3,
$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}+(\mathrm{nil})$
is of presheaf type. By our assumption on $\mathcal{C}$, the underlying
category of the site of affine objects for
$(\operatorname{Spec}R/\operatorname{Spec}K)_{\mathrm{Cris}_{\mathcal{C}}}$ is

${(K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}+(\mathrm{nil}))\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}}^{\mathrm{op}}\hbox
to0.0pt{,\hss}$

the site for the presheaf topos classifying this theory. The only difference
is the Zariski topology as described in Lemma 5.4.5. By Theorem 4.3.1, this is
precisely the topology corresponding to the quotient extension
$(\mathrm{loc})$, as each arrow

${(\gamma\curvearrowright I\triangleleft A\twoheadrightarrow
B)}$${(\gamma_{a_{i}}\curvearrowright I_{a_{i}}\triangleleft
A_{a_{i}}\twoheadrightarrow B_{a_{i}})}$

is the universal arrow that sends $a_{i}\in A$ to something invertible. ∎

######  Remark 5.5.6.

When adding $(\mathrm{loc})$ to a theory like
$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Alg}$, we mean to impose the
axioms on the $K$-algebra $A$, not on the $R$-algebra $B$. However, in the
presence of $(\mathrm{surj})$ and $(\mathrm{nil})$, the two options are in
fact syntactically equivalent. This is because an elementary calculation using
these axioms shows

$\operatorname{inv}(f(x))\;\dashv\vdash_{x\,{:}\,A}\;\operatorname{inv}(x)\hbox
to0.0pt{.\hss}$

###  5.6 Finiteness conditions for PD schemes

The set of objects that we include in the crystalline site will be determined
by the compactness condition on models of the appropriate geometric theory.

######  Lemma 5.6.1.

Let $(K,I_{K},\gamma_{K})$ be a PD ring. The compact models of
${K\text{-}\mathrm{Alg}}+\mathrm{Ideal}_{I_{K}}+\mathrm{PD}_{\gamma_{K}}$ are
those PD rings over $K$ which are of the form

$K\langle
X_{1},\dots,X_{m}\rangle[Y_{1},\dots,Y_{n}]/\overline{(r_{1},\dots,r_{k})}\hbox
to0.0pt{.\hss}$

###### Proof.

While the theory $\mathrm{Ring}+\mathrm{Ideal}+\mathrm{PD}$ as defined in
Definition 5.3.1 is cartesian, the equivalent theory
$\mathrm{Ring}+\mathrm{PD}\text{-}\mathrm{Ideal}$ is even a Horn theory. We
can similarly construct a Horn theory equivalent to
${K\text{-}\mathrm{Alg}}+\mathrm{Ideal}_{I_{K}}+\mathrm{PD}_{\gamma_{K}}$ by
adding constants $\widetilde{c}_{\lambda}:S_{I}$ in addition to
$c_{\lambda}:A$ for every $\lambda\in I_{K}$, with
$\iota(\widetilde{c}_{\lambda})=c_{\lambda}$.

By Lemma 4.2.4, the compact models of a Horn theory are those presented by
some Horn formula in context. In our case, the model presented by the formula
$\top$ in the context $\vec{x}:{S_{I}}^{m},\vec{y}:A^{n}$ is
$K\langle\vec{X}\rangle[\vec{Y}]$. And since our theory has no relation
symbols and the inclusion $\iota:S_{I}\to A$ is required to be injective, any
atomic formula in this context is provably equivalent to an equality of two
terms of sort $A$, and therefore also to one of the form $r=0$, where $r$ is a
term representing an element of $K\langle\vec{X}\rangle[\vec{Y}]$. Thus we are
done, as the universal arrow out of $K\langle\vec{X}\rangle[\vec{Y}]$ that
kills $r_{1},\dots,r_{n}$ is $K\langle\vec{X}\rangle[\vec{Y}]\to
K\langle\vec{X}\rangle[\vec{Y}]/\overline{(\vec{r})}$. ∎

We now want to globalize this property of PD rings over $(K,I,\gamma)$ to be
able to apply it to morphisms of PD schemes. This parallels the treatment of
morphisms of finite type and morphisms of finite presentation as for example
in [9, Tag 01T0] and [9, Tag 01TO].

######  Definition 5.6.2.

We say a homomorphism of PD rings is _of finite PD type_ if it is of the form

$(K,I_{K},\gamma_{K})\to K\langle\vec{X}\rangle[\vec{Y}]/\rho$

for finite sets of variables $\vec{X}=X_{1},\dots,X_{n}$ and
$\vec{Y}=Y_{1},\dots,Y_{m}$, where $\rho=\overline{\rho}$ is any PD saturated
ideal of $K\langle\vec{X}\rangle[\vec{Y}]$. We say it is _of finite PD
presentation_ if it is of the form $(K,I_{K},\gamma_{K})\to
K\langle\vec{X}\rangle[\vec{Y}]/\overline{(\vec{r})}$ for finitely many
elements $r_{1},\dots,r_{k}\in K\langle\vec{X}\rangle[\vec{Y}]$.

######  Lemma 5.6.3.

A homomorphism of PD rings $(K,I_{K},\gamma_{K})\to(A,I,\gamma)$ is of finite
PD type if and only if there are elements $x_{1},\dots,x_{n}\in I$ and
$y_{1},\dots,y_{m}\in A$ such that $A$ is generated as a $K$-algebra by
$\gamma_{k}(x_{i})$ and $y_{i}$, and the ideal $I$ is generated by the image
of $I_{K}$ and $\gamma_{k}(x_{i})$.

###### Proof.

The choice of elements $x_{i}\in I$ and $y_{j}\in A$ corresponds to a
homomorphism $f:K\langle\vec{X}\rangle[\vec{Y}]\to A$. The kernel of $f$ is
always a PD saturated ideal. This $f$ is surjective if and only if $A$ is
generated as a $K$-algebra by $\gamma_{k}(x_{i})$ and $y_{j}$. And in this
case, the induced map between the PD ideals of
$K\langle\vec{X}\rangle[\vec{Y}]$ and $A$ is surjective if and only if $I$ is
generated by the image of $I_{K}$ and the $\gamma_{k}(x_{i})$. ∎

######  Lemma 5.6.4.

Homomorphisms of finite PD presentation have the following properties.

  1. (i)

If $R\to A\to A^{\prime}$ are homomorphisms of PD rings with $R\to A$ and
$R\to A^{\prime}$ of finite PD presentation, then $A\to A^{\prime}$ is also of
finite PD presentation.

  2. (ii)

If $\rho$ is a PD saturated ideal of a PD ring $R$ and $R\to R/\rho$ is of
finite PD presentation, then there are finitely many elements $r_{i}\in R$
such that $\rho=\overline{(\vec{r})}$.

###### Proof.

  1. (i)

For any PD ring $R$ and any element $r\in R$, we have
$R[X]/\overline{(X-r)}=R[X]/(X-r)\cong R$, but we also have $R\langle
X\rangle/\overline{(X-r)}\cong R$ as long as $r$ lies in the PD ideal of $R$.
Now, let $A=R\langle\vec{X}\rangle[\vec{Y}]/\overline{(\vec{r})}$,
$A^{\prime}=R\langle\vec{X}^{\prime}\rangle[\vec{Y}^{\prime}]/\overline{(\vec{r}^{\prime})}$,
$f:A\to A^{\prime}$, and choose $x_{i}$ in the PD ideal of
$R\langle\vec{X}^{\prime}\rangle[\vec{Y}^{\prime}]$ and $y_{i}\in
R\langle\vec{X}^{\prime}\rangle[\vec{Y}^{\prime}]$ such that $f(X_{i})=x_{i}$
and $f(Y_{i})=y_{i}$ in $A^{\prime}$. Then we can calculate:

$\displaystyle A^{\prime}$
$\displaystyle=A^{\prime}\langle\vec{X}\rangle[\vec{Y}]/\overline{(X_{i}-x_{i},Y_{i}-y_{i})}$
$\displaystyle=A^{\prime}\langle\vec{X}\rangle[\vec{Y}]/\overline{(X_{i}-x_{i},Y_{i}-y_{i},\vec{r})}$
$\displaystyle=A\langle\vec{X}^{\prime}\rangle[\vec{Y}^{\prime}]/\overline{(X_{i}-x_{i},Y_{i}-y_{i},\vec{r}^{\prime})}\hbox
to0.0pt{.\hss}$

  2. (ii)

Let $R\langle\vec{X}\rangle[\vec{Y}]/\overline{(\vec{r})}\cong R/\rho$. By
choosing elements in $I_{R}$ respectively $R$ corresponding to the $X_{i}$ and
$Y_{i}$, this isomorphism is induced by a PD map
$f:R\langle\vec{X}\rangle[\vec{Y}]\to R$. But then we have
$\rho=\overline{(f(r_{i}))}$. ∎

######  Lemma 5.6.5.

Homomorphisms of PD rings of finite type and homomorphisms of PD rings of
finite presentation are both stable under composition.

###### Proof.

Let $(K,I,\gamma)$ be a PD ring. We have

$\Big{(}K\langle\vec{X}\rangle[\vec{Y}]/\overline{(\vec{r})}\Big{)}\langle\vec{X}^{\prime}\rangle[\vec{Y}^{\prime}]/\overline{(\vec{r}^{\prime})}=K\langle\vec{X},\vec{X}^{\prime}\rangle[\vec{Y},\vec{Y}^{\prime}]/\overline{(\vec{r},\vec{r}^{\prime\prime})}\hbox
to0.0pt{,\hss}$

where $r^{\prime\prime}_{i}$ is a lift of $r^{\prime}_{i}$ to
$K\langle\vec{X},\vec{X}^{\prime}\rangle[\vec{Y},\vec{Y}^{\prime}]$. Here,
$\vec{r}$ and $\vec{r}^{\prime}$ can either be finite lists or arbitrary set-
indexed families. ∎

######  Lemma 5.6.6.

Homomorphisms of PD rings of finite type and homomorphisms of PD rings of
finite presentation are both stable under base change (pushout in the category
of PD rings).

###### Proof.

The pushout of $K\langle\vec{X}\rangle[\vec{Y}]/\overline{(\vec{r})}$ along
$f:(K,I,\gamma)\to(K^{\prime},I^{\prime},\gamma^{\prime})$ is
$K^{\prime}\langle\vec{X}\rangle[\vec{Y}]/\overline{(f(\vec{r}))}$. ∎

######  Lemma 5.6.7.

Being of finite PD type and being of finite PD presentation are _local_
properties of homomorphisms of PD rings, in the sense of [9, Tag 01SR], that
is: Let $R=(R,I_{R},\gamma_{R})$ and $A=(A,I,\gamma)$ be PD rings.

  1. (a)

If $R\to A$ is of finite PD type (presentation) and $r\in R$, then $R_{r}\to
A_{r}$, with the induced PD structures, is of finite PD type (presentation).

  2. (b)

If $r\in R$, $a\in A$ and $R_{r}\to A$ is of finite PD type (presentation),
then $R\to A_{a}$ is of finite PD type (presentation).

  3. (c)

If $R\to A$ is a homomorphism of PD rings, $a_{i}\in A$ are elements such that
$(a_{1},\dots,a_{n})=(1)$, and $R\to A_{a_{i}}$ is of finite PD type
(presentation) for every $i$, then $R\to A$ is of finite PD type
(presentation).

###### Proof.

  1. (a)

This is a special case of Lemma 5.6.6.

  2. (b)

This follows from Lemma 5.6.5 and the fact that $R_{r}\cong
R[X]/\overline{(rX-1)}$, and the same for $A$.

  3. (c)

Let the $R\to A_{a_{i}}$ be of finite PD type. For every $i$, choose elements
$x_{i,j}a_{i}^{-m_{i,j}}\in I_{a_{i}}$ and $y_{i,j}a_{i}^{-n_{i,j}}\in
A_{a_{i}}$ (with $x_{i,j}\in I$ and $y_{i,j}\in A$) according to Lemma 5.6.3.
Then $A_{a_{i}}$ is generated as an $R$-algebra by
$\gamma_{k}(x_{i,j}a_{i}^{-m_{i,j}})=\gamma_{k}(x_{i,j})a_{i}^{-km_{i,j}}$ and
$y_{i,j}a_{i}^{-n_{i,j}}$. Let $\sum_{i}c_{i}a_{i}=1$ in $A$ and set
$\widetilde{A}\vcentcolon=R[a_{i},c_{i},\gamma_{k}(x_{i,j}),y_{i,j}]\subseteq
A$. Then we see that for every element $a\in A$ and every $i$, we have
$a_{i}^{N}a\in\widetilde{A}$ for $N$ big enough, which means that we also have
$a=(\sum_{i}c_{i}a_{i})^{N}a\in\widetilde{A}$ for $N$ big enough, so
$\widetilde{A}=A$. We also have (by Lemma 5.6.3) that $I_{a_{i}}$ is generated
as an ideal by the elements of $I_{R}$ and the
$\gamma_{k}(x_{i,j})a_{i}^{-km_{i,j}}$. But then if $\widetilde{I}\subseteq A$
is the ideal generated by $I_{R}$ and all $\gamma(x_{i,j})$, we have
$\widetilde{I}_{a_{i}}=I_{a_{i}}$ for all $i$, which implies
$\widetilde{I}=I$. Thus, $R\to A$ is of finite PD type by the other direction
of Lemma 5.6.3.

Now let the $R\to A_{a_{i}}$ be of finite PD presentation. We already know
$A=R\langle\vec{X}\rangle[\vec{Y}]/\rho$ for a PD saturated ideal $\rho$. For
every $i$, choose a representative $\widetilde{a}_{i}\in
R\langle\vec{X}\rangle[\vec{Y}]$ of $a_{i}\in A$. We note that
$\rho_{\widetilde{a}_{i}}$ is also saturated, and therefore
$A_{a_{i}}=R\langle\vec{X}\rangle[\vec{Y}]_{\widetilde{a}_{i}}/\rho_{\widetilde{a}_{i}}$.
Since both $A_{a_{i}}$ and
$R\langle\vec{X}\rangle[\vec{Y}]_{\widetilde{a}_{i}}$ are of finite PD
presentation over $R$, Lemma 5.6.4 tells us that
$\rho_{\widetilde{a}_{i}}=\overline{(r_{i,j}a_{i}^{-n_{i,j}})}=\overline{(r_{i,j})}$,
with finitely many $r_{i,j}\in\rho$. Let $c_{i}\in
R\langle\vec{X}\rangle[\vec{Y}]$ be elements with
$1-\sum_{i}c_{i}a_{i}\in\rho$. Then, setting
$\widetilde{\rho}\vcentcolon=\overline{(1-\sum_{i}c_{i}a_{i},r_{i,j})}\subseteq
R\langle\vec{X}\rangle[\vec{Y}]$, we have $\widetilde{\rho}\subseteq\rho$,
$(\widetilde{a}_{i})=(1)$ in
$R\langle\vec{X}\rangle[\vec{Y}]/\widetilde{\rho}$ and
$\widetilde{\rho}_{\widetilde{a}_{i}}=\rho_{\widetilde{a}_{i}}$ for all $i$.
This implies $\widetilde{\rho}=\rho$. ∎

######  Definition 5.6.8.

A morphism of PD schemes
$(T,\mathcal{I},\gamma)\to(S,\mathcal{I}_{S},\gamma_{S})$ is _locally of
finite PD type (presentation)_ if there are affine open coverings
$T=\bigcup_{i}\operatorname{Spec}A_{i}$ and
$S=\bigcup_{i}\operatorname{Spec}K_{j}$ such that every
$\operatorname{Spec}A_{i}$ maps to some $\operatorname{Spec}K_{j_{i}}$ where
$K_{j_{i}}\to A_{i}$ (with the induced structures of PD rings) is of finite PD
type (presentation).

The following good properties of morphisms locally of finite PD type and
morphisms locally of finite PD presentation now follow from Lemmas 5.6.5,
5.6.6 and 5.6.7 as in [9, Tag 01SQ]: If $T\to S$ is locally of finite PD type
(presentation), then for any affine open $\operatorname{Spec}A\subseteq T$
mapping into an affine open $\operatorname{Spec}K\subseteq S$, the
homomorphism of PD rings $K\to A$ is of finite PD type (presentation). In
particular, the morphisms locally of finite PD type (presentation) between
affine PD schemes correspond to the morphisms of finite PD type (presentation)
between PD rings. And morphisms locally of finite PD type (presentation) are
stable under composition and base change.

######  Definition 5.6.9.

We say that the PD ideal $I$ of a PD ring $(A,I,\gamma)$ is _PD-generated_ by
a subset $G\subseteq I$ if $\overline{(G)}=I$, that is, if it is generated (as
an ideal) by the elements $\gamma_{n}(g)$ for $g\in G$ and $n\geq 1$. We say
$I$ is _finitely PD-generated_ if it is PD-generated by a finite set $G$.

We also globalize the notion of a PD ideal being finitely PD-generated. Note
that by Lemma 5.6.4, if $(A,I,\gamma)$ is a PD ring and we equip $A/I$ with
the trivial PD structure, then $I$ is finitely PD-generated if and only if
$A\to A/I$ is of finite PD presentation.

######  Definition 5.6.10.

The PD ideal sheaf $\mathcal{I}$ of a PD scheme $(S,\mathcal{I},\gamma)$ is
_locally finitely PD-generated_ if for every affine open $U\subseteq S$, the
PD ideal $\mathcal{I}(U)\subseteq\mathcal{O}_{S}(U)$ is finitely PD-generated,
that is, if the closed embedding $V(\mathcal{I})\to S$ is locally of finite PD
presentation, where $V(\mathcal{I})$ is regarded as a PD scheme with trivial
PD structure.

###  5.7 Syntactic presentation of the big crystalline topos

We can now finish the definition of the precise variant of the crystalline
topos for which we will give a syntactic presentation. The objects we want to
allow in the site are those PD thickenings $(T,\mathcal{I},\gamma)$ over $S$
and $X$ where $T\to S$ is locally of finite PD presentation. An essentially
small dense subcategory is for example the one where $T$ is additionally
required to be an open subscheme of an affine scheme which is mapped into an
affine open of $S$. We also impose a finiteness assumption on $X$ without
which the connection to compact models motivating this definition would be
lost.

######  Definition 5.7.1.

Let $(S,\mathcal{I}_{S},\gamma_{S})$ be a PD scheme and let $X$ be a scheme
locally of finite presentation over $S_{0}=V(\mathcal{I}_{S})$. Then we set

$(X/S)_{\mathrm{Cris}_{\mathrm{fp}}}\vcentcolon=(X/S)_{\mathrm{Cris}_{\mathcal{C}}}\hbox
to0.0pt{,\hss}$

where $\mathcal{C}$ is the class of PD thickenings $(T,\mathcal{I},\gamma)$
over $S$ and $X$ for which $T\to S$ is locally of finite PD presentation.

Now there is not much missing for the proof of the classification result. The
following lemma will be used to show that the axiom $(\mathrm{nil})$ is a
rigid-topology quotient.

######  Lemma 5.7.2.

Let $(A,\mathfrak{a},\gamma)$ be a PD ring. If $\mathfrak{a}$ is PD-generated
by nilpotent elements, then $\mathfrak{a}$ is a nil ideal.

###### Proof.

We need to show that $\mathfrak{a}$ is generated (as an ordinary ideal) by
nilpotent elements. So if $a\in\mathfrak{a}$ is nilpotent, say $a^{e}=0$, we
need to show that $\gamma_{n}(a)$ is still nilpotent for every $n\geq 1$. It
is most convenient to do the necessary calculation in $\mathbb{Q}[X]$ first
(with the unique PD structure $\gamma$ on any ideal):

$(\gamma_{n}(X))^{k}=\frac{1}{(n!)^{k}}X^{kn}=\frac{(kn-e)!}{(n!)^{k}}X^{e}\gamma_{kn-e}(X)\hbox
to0.0pt{.\hss}$

One can check that $c\vcentcolon=\frac{(kn-e)!}{(n!)^{k}}$ is an integer for
$k$ big enough ($k\geq\lceil\frac{e}{n}\rceil(n+1)$ suffices). But then we
have the equation $(\gamma_{n}(X))^{k}=cX^{e}\gamma_{kn-e}(X)$ also in the
sub-PD-ring $\mathbb{Z}\langle X\rangle\subseteq\mathbb{Q}[X]$, and then, via
the unique PD morphism $\mathbb{Z}\langle X\rangle\to A$ sending $X$ to $a$,
we obtain $(\gamma_{n}(a))^{k}=0$ for sufficiently large $k$. ∎

######  Proposition 5.7.3.

Let $(K,I_{K},\gamma_{K})$ be a PD ring with $I_{K}$ finitely PD-generated and
let $R$ be a finitely presented $K/I_{K}$-algebra. Then the theory
$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}_{\gamma_{K}}+(\mathrm{nil})$
is of presheaf type and its compact models are those $(\gamma\curvearrowright
I\triangleleft A\twoheadrightarrow B)$ where $(A,I,\gamma)$ is of finite PD
presentation over $(K,I_{K},\gamma_{K})$.

###### Proof.

We start with the cartesian theory
${K\text{-}\mathrm{Alg}}+\mathrm{Ideal}_{I_{K}}+\mathrm{PD}_{\gamma_{K}}$. By
Lemma 5.6.1, the compact models are the PD rings of finite PD presentation
over $(K,I_{K},\gamma_{K})$. It is an equivalence extension to add a sort $B$
with $K$-algebra structure for the quotient $A/I$, arriving at
$K\text{-}\mathrm{Alg}\text{-}(K/I_{K})\text{-}\mathrm{Quot}+\mathrm{PD}_{\gamma_{K}}$.
Since $R$ is a finitely presented $K/I_{K}$-algebra, finitely many constant
symbols and finitely many axioms in the empty context suffice to add an
$R$-algebra structure to $B$, so by Proposition 4.4.5 and Corollary 4.4.2,
$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}_{\gamma_{K}}$
is of presheaf type and a model $(\gamma\curvearrowright I\triangleleft
A\twoheadrightarrow B)$ is still compact if and only if $(A,I,\gamma)$ is of
finite PD presentation over $K$. All that remains is to show that
$(\mathrm{nil})$ is a rigid-topology quotient of this theory.

Let a compact model $M=(\gamma\curvearrowright I\triangleleft
A\twoheadrightarrow
B)\in(K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}_{\gamma_{K}})\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}$
be given. Since $I_{K}$ is finitely PD-generated and $A$ is of finite PD
presentation over $K$, the PD ideal $I$ is finitely PD-generated. Let $a\in I$
be one of the PD generators in a chosen finite family. The
$J_{(\mathrm{nil})}$-covering cosieve $S_{a}$ of Theorem 4.3.1 is the cosieve
of all arrows $M\to M^{\prime}$ that send $a$ to an nilpotent element; it is
generated by the countable family of the arrows $M\to
M^{n}=(\gamma\curvearrowright I/\overline{(a^{n})}\triangleleft
A/\overline{(a^{n})}\twoheadrightarrow B)$. In each of the (still compact)
models $M^{n}$, the PD ideal $I/\overline{(a^{n})}$ is PD generated by the
images of the PD generators of $I$, so we can apply this construction to the
other PD generators in turn to cover $M$ by models where the PD ideal is
generated by nilpotent elements and where thus, by Lemma 5.7.2, the axiom
$(\mathrm{nil})$ is fulfilled. ∎

######  Theorem 5.7.4.

Let $(K,I_{K},\gamma_{K})$ be a PD ring with $I_{K}$ finitely PD-generated and
let $R$ be a finitely presented $K/I_{K}$-algebra. Then the big crystalline
topos

$(\operatorname{Spec}R/\operatorname{Spec}K)_{\mathrm{Cris}_{\mathrm{fp}}}$

classifies the geometric theory

$K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}_{\gamma_{K}}+(\mathrm{nil})+(\mathrm{loc})\hbox
to0.0pt{.\hss}$

The universal model is the short exact sequence associated to the structure
sheaf

$\mathcal{J}\hookrightarrow\mathcal{O}\twoheadrightarrow\mathcal{O}^{\prime}\hbox
to0.0pt{,\hss}$

with the canonical $K$-algebra structure on $\mathcal{O}$ and $R$-algebra
structure on $\mathcal{O}^{\prime}$.

###### Proof.

Set

$\mathbb{T}_{0}\vcentcolon=K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}_{\gamma_{K}}+(\mathrm{nil})\hbox
to0.0pt{.\hss}$

The underlying category of the site of affine objects for
$(\operatorname{Spec}R/\operatorname{Spec}K)_{\mathrm{Cris}_{\mathrm{fp}}}$ is
dual to the subcategory of
$\mathbb{T}_{0}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})$ where
$(A,I,\gamma)$ is of finite PD presentation over $K$. By Proposition 5.7.3,
this is exactly
$\mathbb{T}_{0}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c}$, and the
associated presheaf topos

$[\mathbb{T}_{0}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c},\mathrm{Set}]$

classifies $\mathbb{T}_{0}$. The Zariski topology on this site, that is, the
restriction of the one in Lemma 5.4.5 to our compact models, coincides with
the topology induced by $(\mathrm{nil})$ via Theorem 4.3.1 (just as in
Corollary 5.5.5).

For the universal model, we note that the structure sheaf model
$\mathcal{J}\hookrightarrow\mathcal{O}\twoheadrightarrow\mathcal{O}^{\prime}$,
when restricted to the site of affine objects, becomes simply the tautological
model in
$[\mathbb{T}_{0}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})_{c},\mathrm{Set}]$,
which is the universal model of $\mathbb{T}_{0}$. In particular, the
presheaves of which the universal $\mathbb{T}_{0}$-model consists are sheaves
for the topology induced by $(\mathrm{loc})$, implying that the same model,
regarded as a model in the subtopos, is also the universal model of
$\mathbb{T}_{0}+(\mathrm{loc})$. ∎

######  Corollary 5.7.5.

Let $K$, $R$ be as in Theorem 5.7.4. Then the category of points of the big
crystalline topos
$(\operatorname{Spec}R/\operatorname{Spec}K)_{\mathrm{Cris}_{\mathrm{fp}}}$ is
equivalent to

$\Bigl{(}K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}_{\gamma_{K}}+(\mathrm{nil})+(\mathrm{loc})\Bigr{)}\text{-}\allowbreak\mathrm{mod}(\mathrm{Set})\hbox
to0.0pt{,\hss}$

the category of affine PD thickenings $(A,I,\gamma)$ over
$(K,I_{K},\gamma_{K})$ (implying that $I$ is a nil ideal) together with an
$R$-algebra structure on $A/I$, such that $A$ is a local ring.

###### Proof.

The points of a topos are by definition the geometric morphisms from
$\mathrm{Set}$ to it. So this follows immediately from Theorem 5.7.4 and the
definition of classifying topos. ∎

######  Remark 5.7.6.

If the ring $R$ has non-zero characteristic, then there is a different way of
dealing with the axiom $(\mathrm{nil})$. Namely, one can see that if the PD
ideal $I$ of $A$ contains a non-zero integer, then $(\mathrm{nil})$ is
actually equivalent to

$\top\;\vdash_{[]}\;\bigvee_{0\neq n\in\mathbb{Z}}(c_{n}=0)\hbox
to0.0pt{,\hss}$

which is an axiom in the empty context. So in Theorem 5.7.4, instead of
assuming that $I_{K}$ is finitely PD-generated, we can alternatively assume
$R$ to have non-zero characteristic.

###  5.8 The non-affine case

Now we apply the results of Section 3 to the big crystalline topos to obtain a
syntactic presentation of $(X/S)_{\mathrm{Cris}_{\mathrm{fp}}}$ in the case
where $S$ and $X$ are not affine. This will be an adaptation of the treatment
of the Zariski topos in Section 3 to the crystalline topos. and we will mostly
be concerned with the aspects which differ in these two situations and how to
handle them.

Since we only know a theory classified by
$(X/S)_{\mathrm{Cris}_{\mathrm{fp}}}$ in the case where both $S$ and $X$ are
affine, we will want to cover both $S$ and $X$ by affine opens $S_{i}$
respectively $X_{i}$ such that for each $i$, $X_{i}$ lies over $S_{i}$.

${S}$${X}$${S_{i}}$${X_{i}}$

This is always possible, as we can first choose an affine cover of $S$ by some
$S_{i}$ and then cover the $X\times_{S}S_{i}$ by affine opens. Also, an open
subscheme $S^{\prime}\subseteq S$ is canonically a PD scheme as
$(S^{\prime},\mathcal{I}_{S}|_{S^{\prime}},\gamma_{S}|_{S^{\prime}})$, and we
note that if $X^{\prime}\to S$ factors through $S^{\prime}$, then it
automatically also factors through
$S^{\prime}_{0}=V(\mathcal{I}_{S}|_{S^{\prime}})$.

When we turn to defining open subtoposes corresponding to such pairs of open
subschemes $(S^{\prime},X^{\prime})$, there are two things to note. Firstly,
it is no longer the case that $(S^{\prime},X^{\prime})$ is an object of the
site ${\mathrm{Cris}_{\mathrm{fp}}}(X/S)$, since $X^{\prime}$ is not at all
required to be isomorphic to $S^{\prime}_{0}$, and the PD scheme
$(S^{\prime},\mathcal{I}_{S}|_{S^{\prime}},\gamma_{S}|_{S^{\prime}})$ itself
does not have to be a PD thickening. But we can still consider the class of
all objects $(T,\mathcal{I},\gamma)\in{\mathrm{Cris}_{\mathrm{fp}}}(X/S)$ such
that $T\to S$ factors through $S^{\prime}$ and $V(\mathcal{I})\to X$ factors
through $X^{\prime}$, that is, the subterminal presheaf

$U_{(S^{\prime},X^{\prime})}\vcentcolon=\operatorname{Hom}_{X/S}(-,(S^{\prime},X^{\prime}))\;:\;{\mathrm{Cris}_{\mathrm{fp}}}(X/S)^{\mathrm{op}}\to\mathrm{Set}\hbox
to0.0pt{,\hss}$

and it is easy to see that this is in fact a sheaf for the Zariski topology on
${\mathrm{Cris}_{\mathrm{fp}}}(X/S)$.

Secondly, however, this subterminal object $U_{(S^{\prime},X^{\prime})}$
depends in fact only on $X^{\prime}$ and not on $S^{\prime}$, namely, we have
$U_{(S^{\prime},X^{\prime})}=U_{(S,X^{\prime})}$. This is because the closed
embedding $V(\mathcal{I})\hookrightarrow T$ is a homeomorphism, and whether or
not $T\to S$ factors through the open subscheme $S^{\prime}$ can be tested on
the level of points.

${S}$${X}$${S^{\prime}}$${X^{\prime}}$${T}$${V(\mathcal{I})}$

Thus, we only have induced open subtoposes
$U_{X^{\prime}}\vcentcolon=U_{(S,X^{\prime})}$ for all open subschemes
$X^{\prime}\subseteq X$.

This leads to the slightly subtle situation that a cover of $X$ by affine open
$X_{i}$ with the property that every $X_{i}\to S$ factors through some affine
open $S_{i}\subseteq S$ induces an open cover of
$(X/S)_{\mathrm{Cris}_{\mathrm{fp}}}$ by subtoposes for which we know
classified theories, but if $S_{i}=\operatorname{Spec}K_{i}$ and
$S^{\prime}_{i}=\operatorname{Spec}K^{\prime}_{i}$ are two different choices
for the same $X_{i}=\operatorname{Spec}R_{i}$, then we have no direct
algebraic relation between $K_{i}$ and $K^{\prime}_{i}$, so we have no way to
construct a diagonal extension between the classified theories. Therefore it
seems more reasonable to work with pairs $(S_{i},X_{i})$ of open subschemes
from the beginning and only drop the condition that the $S_{i}$ cover $S$.

Here is the analogue of Lemma 3.7.3 that we need.

######  Lemma 5.8.1.

  1. (i)

The mapping

$X^{\prime}\mapsto((X/S)_{\mathrm{Cris}_{\mathrm{fp}}})_{o(U_{X^{\prime}})}$

from open subschemes of $X$ to open subtoposes of
$(X/S)_{\mathrm{Zar}_{\mathrm{fp}}}$ is monotone and preserves finite
intersections and arbitrary unions.

  2. (ii)

For open subschemes $S^{\prime}\subseteq S$ and $X^{\prime}\subseteq X$ with
$X^{\prime}\to S^{\prime}$, the open subtopos
$((X/S)_{\mathrm{Cris}_{\mathrm{fp}}})_{o(U_{X^{\prime}})}$ is equivalent to
$(X^{\prime}/S^{\prime})_{\mathrm{Cris}_{\mathrm{fp}}}$.

  3. (iii)

If $S=\operatorname{Spec}K$ and $X=\operatorname{Spec}R$ are affine, with
$I_{K}$ finitely PD-generated and $R$ finitely presented over $K/I_{K}$, so
that $(X/S)_{\mathrm{Cris}_{\mathrm{fp}}}$ classifies

$\mathbb{T}_{K,R}\;\vcentcolon=\;K\text{-}\mathrm{Alg}\text{-}R\text{-}\mathrm{Quot}+\mathrm{PD}_{\gamma_{K}}+(\mathrm{nil})+(\mathrm{loc})\hbox
to0.0pt{,\hss}$

and $h\in R$, then the open subtopos
$((X/S)_{\mathrm{Cris}_{\mathrm{fp}}})_{o(U_{D(h)})}$ is presented by the
closed geometric formula

$\operatorname{inv}(c_{h})\hbox to0.0pt{,\hss}$

which is also equivalent to

$\operatorname{inv}(c_{g}\,{:}\,A)\land\operatorname{inv}(c_{h}\,{:}\,B)$

for any $g\in K$ such that $g\mid h$ in $R$.

###### Proof.

  1. (i)

Monotonicity and finite intersections are clear. For unions, let
$X^{\prime}_{i}$ be a family of open subschemes of $X$ and let $F\subseteq
1_{(X/S)_{\mathrm{Cris}_{\mathrm{fp}}}}$ be a subterminal sheaf with
$U_{X^{\prime}_{i}}\leq F$ for all $i$. We want to show $U_{X^{\prime}}\leq F$
for $X^{\prime}\vcentcolon=\bigcup_{i}X^{\prime}_{i}$. So let
$T=(T,\mathcal{I},\gamma)\in{\mathrm{Cris}_{\mathrm{fp}}}(X/S)$ be given with
$V(\mathcal{I})\to X^{\prime}$. By pulling back the $X^{\prime}_{i}$, this
induced an open cover of $V(\mathcal{I})$, or equivalently an open cover
$T=\bigcup_{i}T_{i}$, such that $V(\mathcal{I}|_{T_{i}})\to X_{i}$. This means
$\lvert U_{X_{i}}(T_{i})\rvert=1$, which implies $\lvert F(T_{i})\rvert=1$,
and since $F$ is a sheaf for the Zariski topology on
${\mathrm{Cris}_{\mathrm{fp}}}(X/S)$, we obtain $\lvert F(T)\rvert=1$.

  2. (ii)

As explained above, $U_{X^{\prime}}$ selects exactly those objects of
${\mathrm{Cris}_{\mathrm{fp}}}(X/S)$ which belong to
${\mathrm{Cris}_{\mathrm{fp}}}(X^{\prime}/S^{\prime})$. The rest is exactly as
in the case of the Zariski topos.

  3. (iii)

We know that $(X/S)_{\mathrm{Cris}_{\mathrm{fp}}}$ classifies
$\mathbb{T}_{K,R}$ and

$((X/S)_{\mathrm{Cris}_{\mathrm{fp}}})_{o(U_{D(h)})}\simeq(D(h)/S)_{\mathrm{Cris}_{\mathrm{fp}}}$

classifies $\mathbb{T}_{K,R_{h}}$, which is equivalent to
$\mathbb{T}_{K,R}+\operatorname{inv}(c_{h})$. It follows that
$\operatorname{inv}(c_{h})$ presents this open subtopos, since one structure
sheaf pulls back to the other, including the extra structure that makes up the
universal models. For $g\in K$ with $g\mid h$ in $R$, the theory
$\mathbb{T}_{K,R}$ shows

$\operatorname{inv}(c_{h})\;\vdash_{[]}\;\operatorname{inv}(f(c_{g}))$

and also, by Remark 5.5.6,

$\operatorname{inv}(f(c_{g}))\;\dashv\vdash_{[]}\;\operatorname{inv}(c_{g})\hbox
to0.0pt{.\hss}$

∎

As the final prerequisite, we show that intersections of pairs of open
subschemes can be covered by appropriate pairs of standard opens.

######  Lemma 5.8.2.

Let $X\to S$ be a morphism of schemes, let $S_{1},S_{2}\subseteq S$ and let
$X_{1},X_{2}\subseteq X$ be affine open subschemes,
$S_{i}=\operatorname{Spec}K_{i}$, $X_{i}=\operatorname{Spec}R_{i}$, such that
$X_{i}\to S_{i}$. Then there are families of elements $g_{i}^{j}\in K_{i}$,
$h_{i}^{j}\in R_{i}$, $j\in J$, $i\in\\{1,2\\}$, such that for every $j\in J$
we have $g_{i}^{j}\mid h_{i}^{j}$ in $R_{i}$ and

$\operatorname{Spec}K_{1}\supseteq
D(g_{1}^{j})=D(g_{2}^{j})\subseteq\operatorname{Spec}K_{2}\hbox
to0.0pt{,\hss}$ $\operatorname{Spec}R_{1}\supseteq
D(h_{1}^{j})=D(h_{2}^{j})\subseteq\operatorname{Spec}R_{2}$

as open subschemes of $S$ respectively $X$, and

$X_{1}\cap X_{2}=\bigcup_{j\in J}D(g_{1}^{j})=\bigcup_{j\in
J}D(g_{2}^{j})\hbox to0.0pt{.\hss}$

###### Proof.

As mentioned before, we can cover $S_{1}\cap S_{2}$ by open subschemes
simultaneously standard open in $S_{1}$ and $S_{2}$. The same is true for
$X_{1}\cap X_{2}$, and since the simultaneously standard opens even form a
basis of opens of $X_{1}\cap X_{2}$, we can also arrange that $D(h_{1}^{j})$
maps to $D(g_{1}^{j})$, and, equivalently, $D(h_{2}^{j})$ maps to
$D(g_{2}^{j})$. This means that for each $i$ and $j$, we have the dashed arrow
in

${R_{i}}$${(R_{i})_{h_{i}^{j}}}$${K_{i}}$${(K_{i})_{g_{i}^{j}}\hbox
to0.0pt{,\hss}}$

so $g_{i}^{j}$ is invertible in $(R_{i})_{h_{i}^{j}}$, so $g_{i}^{j}$ divides
some power of $h_{i}^{j}$, and replacing $h_{i}^{j}$ with this power, we
obtain $g_{i}^{j}\mid h_{i}^{j}$ in $R_{i}$. ∎

Recall the notation ${R\text{-}\mathrm{AlgStr}}_{K}(B)$ for the extension
adding to a sort $B$ an $R$-algebra structure compatible with a previously
given $K$-algebra structure, from Definition 5.3.2. For a PD ring
$(K,I_{K},\gamma_{K})$, we also denote $\gamma/\gamma_{K}$ the axioms
distinguishing the extension $\mathrm{PD}_{\gamma_{K}}$ of
${K\text{-}\mathrm{Alg}}+\mathrm{Ideal}_{I_{K}}$ from $\mathrm{PD}$.

######  Theorem 5.8.3.

Let $(S,\mathcal{I}_{S},\gamma_{S})$ be a PD scheme with $\mathcal{I}_{S}$
locally finitely PD-generated, and let $X$ be a scheme locally of finite
presentation over $S_{0}=V(\mathcal{I}_{S})$. Let

$\operatorname{Spec}K_{i}=S_{i}\subseteq
S,\qquad\operatorname{Spec}R_{i}=X_{i}\subseteq X$

be open subschemes such that $X_{i}\to S_{i}$ and $X=\bigcup_{i\in I}X_{i}$.
For every $i\neq i^{\prime}\in I$, let

$g_{i,i^{\prime}}^{j}\in K_{i},\quad g_{i^{\prime},i}^{j}\in
K_{i^{\prime}},\quad h_{i,i^{\prime}}^{j}\in R_{i},\quad
h_{i^{\prime},i}^{j}\in R_{i^{\prime}},$

$j\in J_{\\{i,i^{\prime}\\}}$, be families of elements as in Lemma 5.8.2, with
corresponding ring isomorphisms

$\varphi_{i,i^{\prime}}^{j}=(\varphi_{i^{\prime},i}^{j})^{-1}:(K_{i})_{g_{i,i^{\prime}}^{j}}\to(K_{i^{\prime}})_{g_{i^{\prime},i}^{j}},\qquad\widetilde{\varphi}_{i,i^{\prime}}^{j}=(\widetilde{\varphi}_{i^{\prime},i}^{j})^{-1}:(R_{i})_{h_{i,i^{\prime}}^{j}}\to(R_{i^{\prime}})_{h_{i^{\prime},i}^{j}}\hbox
to0.0pt{.\hss}$

Then $(X/S)_{\mathrm{Cris}_{\mathrm{fp}}}$ classifies the geometric theory

$\displaystyle\mathbb{T}_{X/S}\;\vcentcolon=\;{}$
$\displaystyle\mathbb{Z}\text{-}\mathrm{Alg}\text{-}\mathbb{Z}\text{-}\mathrm{Quot}+\mathrm{PD}+(\mathrm{nil})+(\mathrm{loc})+\langle
p_{i}\rangle_{i\in I}+(\bigvee_{i\in I}p_{i})$
$\displaystyle+\Bigl{(}\Bigl{(}{K_{i}\text{-}\mathrm{AlgStr}}(A)+{R_{i}\text{-}\mathrm{AlgStr}}_{K_{i}}(B)+\gamma/\gamma_{K_{i}}\Bigr{)}/p_{i}\Bigr{)}_{i\in
I}$
$\displaystyle+\Bigl{(}x\in\widetilde{c}_{h_{i,i^{\prime}}^{j}}\land\operatorname{inv}(x)\;\vdash_{x\,{:}\,B}\;p_{i^{\prime}}\Bigr{)}_{i\neq
i^{\prime}\in I,j\in J_{\\{i,i^{\prime}\\}}}$
$\displaystyle+\Bigl{(}p_{i}\land p_{i^{\prime}}\;\vdash_{[]}\;\bigvee_{j\in
J_{\\{i,i^{\prime}\\}}}\exists
x{\,:\,}B.\;(x\in\widetilde{c}_{h_{i,i^{\prime}}^{j}}\land\operatorname{inv}(x))\Bigr{)}_{i\neq
i^{\prime}\in I,j\in J_{\\{i,i^{\prime}\\}}}$
$\displaystyle+\Bigl{(}x\in\widetilde{c}_{g_{i^{\prime},i}^{j}}\land\operatorname{inv}(x)\land
y\in\widetilde{c}_{\lambda}\land
z\in\widetilde{c}_{\lambda^{\prime}}\;\vdash_{x,y,z\,{:}\,A}\;x^{n}y=z\Bigr{)}_{i\neq
i^{\prime}\in I,j\in J_{\\{i,i^{\prime}\\}},\lambda\in K_{i}}$
$\displaystyle+\Bigl{(}x\in\widetilde{c}_{h_{i^{\prime},i}^{j}}\land\operatorname{inv}(x)\land
y\in\widetilde{c}_{\mu}\land
z\in\widetilde{c}_{\mu^{\prime}}\;\vdash_{x,y,z\,{:}\,B}\;x^{n}y=z\Bigr{)}_{i\neq
i^{\prime}\in I,j\in J_{\\{i,i^{\prime}\\}},\mu\in R_{i}}\hbox to0.0pt{,\hss}$

where in the last two families of axioms, $\lambda^{\prime}\in K_{i^{\prime}}$
and $n\in\mathbb{N}$ are chosen for each $\lambda\in K_{i}$ such that
$\varphi_{i,i^{\prime}}^{j}(\lambda)=(g_{i^{\prime},i}^{j})^{-n}\lambda^{\prime}$,
and $\mu^{\prime}\in R_{i^{\prime}}$ and $n\in\mathbb{N}$ are chosen for each
$\mu\in R_{i}$ such that
$\widetilde{\varphi}_{i,i^{\prime}}^{j}(\mu)=(h_{i^{\prime},i}^{j})^{-n}\mu^{\prime}$,

###### Proof.

Since $\mathcal{I}_{S}$ is locally finitely PD-generated, the PD ideals
$I_{K_{i}}\vcentcolon=\mathcal{I}_{S}(S_{i})\subseteq K_{i}$ with PD structure
$\gamma_{K_{i}}\vcentcolon=\gamma_{S}(S_{i})$ are finitely PD-generated, and
since $X$ is locally of finite presentation over $S_{0}$, each $R_{i}$ is a
finitely presented $K_{i}/I_{K_{i}}$-algebra. So
$(X_{i}/S_{i})_{\mathrm{Cris}_{\mathrm{fp}}}$ classifies

$\mathbb{T}_{K_{i},R_{i}}=K_{i}\text{-}\mathrm{Alg}\text{-}R_{i}\text{-}\mathrm{Quot}+\mathrm{PD}_{\gamma_{K_{i}}}+(\mathrm{nil})+(\mathrm{loc})\hbox
to0.0pt{.\hss}$

By Lemma 5.8.1, these toposes

$\mathcal{E}_{i}\vcentcolon=((X/S)_{\mathrm{Cris}_{\mathrm{fp}}})_{o(U_{X_{i}})}\simeq(X_{i}/S_{i})_{\mathrm{Cris}_{\mathrm{fp}}}$

form an open cover of $(X/S)_{\mathrm{Cris}_{\mathrm{fp}}}$, and the universal
models are extensions of the restrictions of the short exact sequence
$\mathcal{J}\hookrightarrow\mathcal{O}\twoheadrightarrow\mathcal{O}^{\prime}$
surrounding the structure sheaf. So we can use the base theory and base model

$\displaystyle\mathbb{T}_{0}$
$\displaystyle\vcentcolon=\mathbb{Z}\text{-}\mathrm{Alg}\text{-}\mathbb{Z}\text{-}\mathrm{Quot}+\mathrm{PD}+(\mathrm{nil})+(\mathrm{loc})\hbox
to0.0pt{,\hss}$ $\displaystyle M_{0}$
$\displaystyle\vcentcolon=(\mathcal{J}\hookrightarrow\mathcal{O}\twoheadrightarrow\mathcal{O}^{\prime})\hbox
to0.0pt{,\hss}$

where $\mathbb{T}_{0}$ is of course equivalent to
$\mathrm{Ring}+\mathrm{Ideal}+\mathrm{PD}+(\mathrm{nil})+(\mathrm{loc})$, but
we want to have the sort $B=A/I$ in the base theory. And we have presentations
of the $\mathcal{E}_{i}$ over $(\mathbb{T}_{0},M_{0})$ with

$\mathbb{E}_{i}\vcentcolon={K_{i}\text{-}\mathrm{AlgStr}}(A)+{R_{i}\text{-}\mathrm{AlgStr}}_{K_{i}}(B)+\gamma/\gamma_{K_{i}}\hbox
to0.0pt{.\hss}$

Closed geometric formulas for the intersections
$\mathcal{E}_{i}\cap\mathcal{E}_{i^{\prime}}\subseteq\mathcal{E}_{i}$ are
given by

$\phi_{i,i^{\prime}}\;\vcentcolon=\;\bigvee_{j\in
J_{\\{i,i^{\prime}\\}}}\operatorname{inv}(c_{h_{i,i^{\prime}}^{j}})\hbox
to0.0pt{.\hss}$

The diagonal quotient extensions $\mathbb{Q}_{\\{i,i^{\prime}\\}}$ have to
make both the $K_{i^{\prime}}$-algebra structure on $A$ and the
$R_{i^{\prime}}$-algebra structure on $B$ definable in terms of
$\mathbb{T}_{0}+\mathbb{E}_{i}+\phi_{i,i^{\prime}}$. Since the formula
$\phi_{i,i^{\prime}}$ also implies

$\bigvee_{j\in
J_{\\{i,i^{\prime}\\}}}\operatorname{inv}(c_{g_{i,i^{\prime}}^{j}})\hbox
to0.0pt{,\hss}$

we can proceed as in Theorem 3.7.6 for both $K_{i}$ and $R_{i}$. That is, for
every $\lambda\in K_{i}$ and $\mu\in R_{i}$, we write

$\varphi_{i,i^{\prime}}^{j}(\lambda)=(g_{i^{\prime},i}^{j})^{-n}\lambda^{\prime}$

for some $\lambda^{\prime}\in K_{i^{\prime}}$, $n\in\mathbb{N}$, respectively

$\widetilde{\varphi}_{i,i^{\prime}}^{j}(\mu)=(h_{i^{\prime},i}^{j})^{-n}\mu^{\prime}$

for some $\mu^{\prime}\in R_{i^{\prime}}$, $n\in\mathbb{N}$. Then
$\mathcal{O}_{((X_{i}\cap X_{i^{\prime}})/(S_{i}\cap
S_{i^{\prime}}))_{\mathrm{Cris}_{\mathrm{fp}}}}$ satisfies

$\displaystyle\mathbb{Q}_{i,i^{\prime}}\quad$
$\displaystyle\vcentcolon=\quad\Bigl{\\{}\quad\operatorname{inv}(c_{g_{i^{\prime},i}^{j}})\;\vdash_{[]}\;(c_{g_{i^{\prime},i}^{j}})^{n}c_{\lambda}=c_{\lambda^{\prime}}\quad\mid\quad
j\in J_{\\{i,i^{\prime}\\}},\lambda\in K_{i}\quad\Bigr{\\}}\hbox
to0.0pt{,\hss}$ $\displaystyle\widetilde{\mathbb{Q}}_{i,i^{\prime}}\quad$
$\displaystyle\vcentcolon=\quad\Bigl{\\{}\quad\operatorname{inv}(c_{h_{i^{\prime},i}^{j}})\;\vdash_{[]}\;(c_{h_{i^{\prime},i}^{j}})^{n}c_{\mu}=c_{\mu^{\prime}}\quad\mid\quad
j\in J_{\\{i,i^{\prime}\\}},\mu\in R_{i}\quad\Bigr{\\}}\hbox to0.0pt{,\hss}$

and

$\mathbb{Q}_{\\{i,i^{\prime}\\}}\vcentcolon=\mathbb{Q}_{i,i^{\prime}}+\mathbb{Q}_{i^{\prime},i}+\widetilde{\mathbb{Q}}_{i,i^{\prime}}+\widetilde{\mathbb{Q}}_{i^{\prime},i}$

is a diagonal quotient extension of $\mathbb{E}_{i}+\phi_{i,i^{\prime}}$ and
$\mathbb{E}_{i^{\prime}}+\phi_{i^{\prime},i}$ over $\mathbb{T}_{0}$.

Applying Corollary 3.6.1 to these data and simplifying the axioms slightly
yields the theory $\mathbb{T}_{X/S}$ in the statement. ∎

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