Compactly assembled categories and exponentiability

6.10. Compactly assembled categories and exponentiability

In this section we characterize the exponentiable topoi, i.e. those topoi $$T$$ for which the exponential $$S^T$$ exists for all topoi $$S$$ . The answer is provided by the notion of compactly assembled categories: a topos is exponentiable if and only if it is compactly assembled as a presentable category. We begin by studying finitary functors as an auxiliary notion, then introduce compactly assembled categories and establish their connection to classifying topoi, before proving the characterization of exponentiable topoi.

6.10.1. Finitary functor topoi

We write $\Fun^{\kappa}(C,T) \subseteq \Fun(C,T)$ for the full subcategory of $\kappa$ -filtered-colimit-preserving functors. We write $\Cat^{\acc,\kappa}$ for the category of accessible categories admitting $\kappa$ -filtered colimits and functors preserving those colimits.

Proposition 6.139.

Let $$C$$ be an accessible category with $\kappa$ -filtered colimits, let $$T$$ be a topos. Then $\Fun^{\kappa}(C,T)$ is a topos.

Proof

We may write

C = \Ind_{\lambda}(C_0)

for some small category

\(C_0\)

, and some regular cardinal

\lambda \gg \kappa

. It then follows that

\Fun^{\kappa}(C,T) \subseteq \Fun^{\lambda}(C,T) \simeq \Fun(C_0,T).

Note that

\Fun(C_0,T)

is a topos. Moreover,

\Fun^{\kappa}(C,T)

is closed under colimits and finite limits. It remains to show that

\Fun^{\kappa}(C,T)

is accessible. We will do this by cutting it out of

\Fun(C_0,T)

by imposing a small set of conditions.Note that a functor

F \in \Fun^{\lambda}(C,T)

preserves

\kappa

-filtered colimits if and only if

\(F\)

preserves colimits indexed by

\kappa

-filtered posets. Using

\lambda \gg \kappa

, we may write every

\kappa

-filtered poset as a

\lambda

-filtered colimit of

\lambda

-small

\kappa

-filtered colimits, so the condition is equivalent to

\(F\)

preserving

\lambda

-small

\kappa

-filtered colimits.There is only a set of isomorphism classes of

\lambda

-small

\kappa

-filtered posets. For each such poset

\(A\)

, the category

\Fun(A,C)

is accessible, so choose a small generating family of diagrams

\alpha\colon A\to C

. Preservation of the colimit of these generators implies preservation for every

\(A\)

-diagram, because both sides of the comparison commute with sufficiently filtered colimits of diagrams. We therefore obtain a pullback square

where the right vertical functor sends

\(F\)

to the collection of comparison maps

\colim_A F\alpha\to F(\colim_A\alpha)

, and the bottom map sends a family of objects to the corresponding family of identity arrows. All three other corners and all functors in this square are accessible. Hence

\Fun^\kappa(C,T)

is accessible. Since it is a full subcategory of the topos

\Fun(C_0,T)

closed under colimits and finite limits, it is itself a topos.

Remark 6.140.

The category $\Fun^{\kappa}(C,T)$ has a universal property: for every topos $$S$$ , there is a natural equivalence

\Geom(\Fun^{\kappa}(C,T), S) \simeq \Fun^{\kappa}(C, \Geom(T,S)).

Both sides are naturally a full subcategory of $\Fun(C \times S, T)$ , and one can check that in both cases we precisely get those functors $C \times S \to T$ that preserve $\kappa$ -filtered colimits in the left variable, and preserve finite limits and colimits in the second variable.

Corollary 6.141.

The functor $\Pt\colon \Topos \to \Cat^{\acc, \kappa}$ admits a ( $$2$$ -categorical) left adjoint

\Cat^{\acc,\kappa} \to \Topos, \qquad C \mapsto \Fun^{\kappa}(C,\An).

6.10.2. Compactly assembled categories

We now introduce the central notion of this section. A category is called compactly assembled if it satisfies certain retract conditions with respect to ind-categories. The following proposition provides three equivalent characterizations.

Proposition 6.142.

Let $$C$$ be a category with filtered colimits. Then the following are equivalent:

There exists a small category $$C_0$$ such that $$C$$ is a retract of $\Ind(C_0)$ in $\Cat^{\omega}$ (the category of categories with filtered colimits).
There exists a small category $$C_0$$ such that there exists an adjunction $C \rightleftarrows \Ind(C_0)$ in $\Cat^{\omega}$ such that $C \hookrightarrow \Ind(C_0)$ is fully faithful.
$$C$$ is accessible, and the “colimit functor” $\colim\colon \Ind(C) \to C$ (defined as the left adjoint to $y\colon C \hookrightarrow \Ind(C)$ ) admits a further left adjoint $y'\colon C \hookrightarrow \Ind(C)$ .

Definition 6.143.

A category satisfying the equivalent conditions of Proposition 6.142 is called compactly assembled. We write $\Cat^{\mathrm{ca}}$ for the category of compactly assembled categories and filtered-colimit-preserving functors.

Proof

Note that (2) clearly implies (1), since the unit of the adjunction exhibits the map

\Ind(C_0) \to C

as a retraction of

C \hookrightarrow \Ind(C_0)

.For (3)

\implies

(2), we write

\Ind(C) = \bigcup_{\lambda > \kappa} \Ind(C^{\lambda})

, where

\(C\)

\kappa

-accessible. Since

C^{\kappa}

is small, we have

y'(C^{\kappa}) \subseteq \Ind(C^{\lambda})

for some

\lambda

. Since

\(C\)

is generated under colimits by

C^{\kappa}

and

\(y'\)

preserves colimits, this implies that

y'(C) \subseteq \Ind(C^{\lambda})

. But then the functor

y'\colon C \to \Ind(C^{\lambda})

is a left adjoint to

\colim\colon \Ind(C^{\lambda}) \to C

, giving (2).Finally, we prove that (1)

\implies

(3). The assumption gives us functors

C \xrightarrow{u} \Ind(C_0) \xrightarrow{v} C

satisfying

vu = \id_C

. We may now consider the idempotent

uv\colon \Ind(C_0) \to \Ind(C_0)

. We may then write

C = \lim(\dots \to \Ind(C_0) \xrightarrow{uv} \Ind(C_0) \xrightarrow{uv} \Ind(C_0)),

and in particular

\(C\)

is a limit of accessible categories, hence itself accessible. Consider the following diagram

The middle vertical map admits a left adjoint

y'\colon \Ind(C_0) \hookrightarrow \Ind(\Ind(C_0))

.Now, we show that the left adjoint

y'\colon C \hookrightarrow \Ind(C)

exists. We may do this pointwise, so consider some

X \in C

. The map

\Hom(X,\colim(-))

is a retract of the representable copresheaf

\Hom(\hat{v}y'u(X),-)

\Fun(\Ind(C),\An)

. Since

\Ind(C)

is idempotent complete, this means that

\Hom(X,\colim(-))

is representable. Hence there exists some

y'(X) \in \Ind(C)

, which then provides the desired left adjoint object to

\(X\)

Corollary 6.144.

Let $$C$$ be a presentable category. Then $$C$$ is compactly assembled if and only if it is a retract in $\PrL$ of a compactly generated category.

6.10.3. Classifying topoi

The key property of compactly assembled categories is that they admit classifying topoi for sheaves valued in them. We establish this through an equivalence relating functors out of $\Fun^{\omega}(C,\An)$ to functors from $C^{\op}$ .

Proposition 6.145.

Let $$C$$ be compactly assembled, and let $$D$$ be presentable. Then there is an equivalence

\FunL(\Fun^{\omega}(C,\An),D) \simeq \Fun_{\omega}(C\catop,D),

where the RHS means those functors that preserve cofiltered limits.

Proof sketch

C = \Ind(C_0)

is compactly generated, we have

\Fun^{\omega}(C,\An) = \Fun(C_0,\An) = \PSh(C_0\catop)

, whereas

\Fun_{\omega}(C\catop,D) = \Fun(C_0\catop,D)

. We would like to say: the claim now follows as every

\(C\)

is a retract of a compactly generated one. However, for this reasoning to work, we need to find a way to write down the comparison map for the compactly generated case in a way that does not rely on picking an identification

C = \Ind(C_0)

.To this end, consider the category

\Fun^+(C,\An) := \Fun^{\omega}(C,\An) \cup \Fun^{\rep}(C,\An) \subseteq \Fun(C,\An).

Consider a functor

F\colon \Fun^+(C,\An) \to D

. Then, if

C = \Ind(C_0)

, one can show that:

$$F$$ is left Kan extended from $\Fun^{\rep}$ if and only if $F\vert_{\Fun^{\omega}}$ preserves colimits.
$$F$$ is right Kan extended from $\Fun^{\omega}$ if and only if $F\vert_{\Fun^{\rep}}$ preserves cofiltered limits.

Using the adjunction

C \rightleftarrows \Ind(C_0)

, we then deduce that the following are equivalent:

$$F$$ is left Kan extended from $\Fun^{\rep}$ and $F\vert_{\Fun^{\rep}}$ preserves cofiltered limits.
$$F$$ is right Kan extended from $\Fun^{\omega}$ and $F\vert_{\Fun^{\omega}}$ preserves colimits.

It follows that both sides

\FunL(\Fun^{\omega}(C,\An), D)

and

\Fun_{\omega}(C\catop,D)

agree with the full subcategory of

\Fun(\Fun^+(C,\An), D)

satisfying these two equivalent conditions.

Corollary 6.146.

Let $$T$$ be a topos and $$C$$ a presentable category which is compactly assembled. Then the equivalence

\FunL(\Fun^{\omega}(C,\An), T) \simeq \Fun_{\omega}(C\catop,T)

restricts to an equivalence

\Fun_{\Logos}(\Fun^{\omega}(C,\An), T) \simeq \FunR(C\catop,T).

Remark 6.147.

Recall that the category $\FunR(C\catop,T)$ is the Lurie tensor product $C \otimes T$ in $\PrL$ , which in turn may be written as $\FunR(T\catop,C)$ . We denote this category by $\Shv_C(T)$ and call it the category of $$C$$ -valued sheaves on $$T$$ . Keeping track of the functoriality in $$T$$ shows that the topos $\Fun^{\omega}(C,\An)$ classifies $$C$$ -valued sheaves.

We have thus shown that if $$C$$ is compactly assembled, then a classifying topos for $$C$$ -valued sheaves exists. It turns out that the converse is also true:

Proposition 6.148.

Let $$C$$ be a presentable category. Then a classifying topos for $$C$$ -valued sheaves exists if and only if $$C$$ is compactly assembled.

Proof

It remains to show the “only if” part. Let

\(E\)

be such a classifying topos, and write it as a left exact localization

i_*\colon E \hookrightarrow \PSh(D)

of a presheaf category. This induces a geometric morphism

i_*\colon \Shv_C(E) \hookrightarrow \Shv_C(\PSh(D))

. Let

F_{\univ} \in \Shv_C(E)

be the universal

\(C\)

-valued sheaf on

\(E\)

. By universality, there exists a morphism of topoi

f\colon \PSh(D) \to E

satisfying

i_*(F_{\univ}) \simeq f^*(F_{\univ})

. By full faithfulness of

\(i_*\)

, we have

F_{\univ} \simeq i^*i_*(F_{\univ}) \simeq (f \circ i)^*(F_{\univ})

. Universality again gives

f \circ i \simeq \id_E

, so

\(E\)

is a retract of

\PSh(D)

\Topos

.Applying

\Pt(-)

shows that

C \simeq \Shv_C(\An) \simeq \FunR(\An,E)

is a retract of

\Pt(\PSh(D))

\Cat^{\omega}

. Points of

\PSh(D)

are flat functors

D\to\An

, and their category is the filtered-colimit completion

\Ind(D\catop)

. It is therefore compactly generated. Hence

\(C\)

is a retract of a compactly generated category in

\Cat^\omega

, and is compactly assembled by Proposition 6.142.

Remark 6.149.

Let $$C$$ be a compactly assembled category. Then there is an equivalence

\Pt(\Fun^{\omega}(C,\An)) \simeq C,

under which $X \in C$ corresponds to the evaluation functor $\ev_X\colon \Fun^{\omega}(C,\An) \to \An$ .

Lemma 6.150.

The functor $\Cat^{\mathrm{ca}} \to \Topos$ given by $C \mapsto \Fun^{\omega}(C,\An)$ is a fully faithful $$2$$ -functor.

Proof

By Corollary 6.141, the functor

\Pt\colon \Topos \to \Cat^{\acc,\omega}

admits a left adjoint given by

C \mapsto \Fun^{\omega}(C,\An)

. The unit of this adjunction provides a map

C \to \Pt(\Fun^{\omega}(C,\An))

, which by Remark 6.149 is an equivalence when

\(C\)

is compactly assembled. Therefore the restriction of this adjunction to

\Cat^{\mathrm{ca}}

exhibits

\Cat^{\mathrm{ca}} \to \Topos

as fully faithful.

6.10.4. Exponentiable topoi

Having established that compactly assembled categories have classifying topoi, we now turn to the main application: characterizing exponentiable topoi. Recall that the exponential $$S^T$$ in $\Topos$ , if it exists, represents the functor $U \mapsto \Hom_{\Topos}(U \otimes T, S)$ .

Let $$S$$ and $$T$$ be topoi. If the exponential (internal hom) in $\Topos$ exists, it will be denoted $$S^T$$ . In other words, this is a topos $$S^T$$ equipped with a morphism of topoi $S^T \otimes T \to S$ which induces an equivalence

\Hom_{\Topos}(U,S^T) \simeq \Hom_{\Topos}(U \otimes T,S).

Observation 6.151.

If $$S^T$$ exists, it is automatically a $$2$$ -categorical exponential. Indeed, for every topos $$U$$ , the arrow topos $\Ar(U)$ represents arrows between geometric morphisms out of $$U$$ , and \begin{align*} \Hom_{\Cat}([1],\Geom(U,S^T)) &\iso \Hom_{\Topos}(\Ar(U),S^T) \\ &\iso \Hom_{\Topos}(\Ar(U) \otimes T, S) \\ &\iso \Hom_{\Topos}(\Ar(U \otimes T),S) \\ &\iso \Hom_{\Cat}([1],\Geom(U \otimes T,S)). \end{align*} Together with the original universal property on objects, this identifies the entire Hom categories. Equivalently, one may repeat the same argument with the topos of $$[n]$$ -diagrams for every $$n$$ and use the complete Segal description of a category.

Remark 6.152.

We have $\Pt(S^T) \simeq \Geom(T,S)$ .

Definition 6.153.

We say that a topos $$T$$ is exponentiable if $$S^T$$ exists for all $$S$$ .

Lemma 6.154.

If $$C$$ is compactly assembled and $$D$$ is compactly generated, then $C\otimes D$ is compactly assembled.

Proof

Choose a retraction of

\(C\)

from a compactly generated presentable category

\(C'\)

. Tensoring with

\(D\)

gives a retraction of

C\otimes D

from

C'\otimes D

. The tensor product of compactly generated presentable categories is compactly generated: if

C'\simeq\Ind(C'_0)

and

D\simeq\Ind(D_0)

, it is generated under filtered colimits by the objects

c\otimes d

with

c\in C'_0

and

d\in D_0

. The claim follows from Corollary 6.144.

Lemma 6.155.

Every topos is a pullback in $\Topos$ of a diagram of presheaf topoi $\PSh(D)$ with $$D$$ admitting finite limits.

Proof sketch

Present the topos as a left exact localization of a presheaf topos. The localization is the equalizer of the identity and its idempotent localization functor. Replacing this equalizer by the usual pullback involving an arrow category expresses the original topos as a pullback of three presheaf topoi. Closing the small indexing categories under finite limits gives the stated form. See [Anel and Lejay 2018, Section 4, in particular the construction preceding Theorem 4.33].

Theorem 6.156.

A topos $$T$$ is exponentiable if and only if it is compactly assembled.

Proof

For the “only if” direction, assume that

\(T\)

is exponentiable. To show it is compactly assembled, consider some compactly assembled category

\(C\)

and consider

S := \Fun^{\omega}(C,\An)

. By assumption, the exponential

\(S^T\)

exists. Moreover, we have

\Geom(U,S^T) \simeq \Geom(U \otimes T, S) \simeq \Shv_{C}(U \otimes T) \simeq \Shv_{C \otimes T}(U).

In particular, taking

C = \An

, we see that

\(S^T\)

classifies

\(T\)

-valued sheaves. By Proposition 6.148, it follows that

\(T\)

is compactly assembled.For the “if” direction, assume

\(T\)

is compactly assembled. By Lemma 6.155, we may write

\(S\)

as a pullback in

\Topos

of a square of the form

where

\(A_0\)

\(B_0\)

and

\(C_0\)

are presheaf topoi. Since exponentiation by

\(T\)

, whenever defined, is right adjoint to

(-)\otimes T

, it preserves limits. It is therefore enough to construct

\(S^T\)

when

\(S\)

is a presheaf topos.Assume then that

S = \PSh(D)

, where

\(D\)

has finite limits. We have

S = \PSh(D) \iso \Fun^{\omega}(\Ind(D\catop),\An).

Set

K:=\Ind(D\catop)

. The category

\(K\)

is compactly generated, so

K\otimes T

is compactly assembled by Lemma 6.154. Its classifying topos

\(E\)

exists by Proposition 6.148, and for every topos

\(U\)

we have

\Geom(U,E)\simeq\Shv_{K\otimes T}(U)\simeq\Shv_K(U\otimes T)\simeq\Geom(U\otimes T,\PSh(D)).

Thus

E\simeq S^T

. This proves the theorem; compare [Anel and Lejay 2018, Theorem 4.33].

References

Mathieu Anel, Damien Lejay. Exponentiable higher toposes. 2018.