[0000] Logoses and toposes

We review the $$2$$ -category of toposes. Following [Anel--Joyal--2021-0000], we first define the $$2$$ -category of logoses and then define the $$2$$ -category of toposes to be its opposite.

[004X] Definition

Let $$U$$ be a universe and let $$C$$ be a category that has finite limits and $$U$$ -small colimits. We say $$U$$ -small colimits in $$C$$ satisfy descent if, for any $$U$$ -small category $$I$$ , any functors $X,Y:{I}^{\triangleright }\rightarrow C$ , and any natural transformation $p:X\Rightarrow Y$ , if $$Y$$ is a colimit diagram and if the restriction of $$p$$ to $$I$$ is cartesian (that is, all naturality squares are pullbacks), then $$X$$ is a colimit diagram if and only if $$p$$ is cartesian.

[0001] Definition

Let $$U$$ be a universe. We say a category $$C$$ is lex $$U$$ -cocomplete if the following conditions are satisfied.

- $$C$$ has finite limits.
- $$C$$ has $$U$$ -small colimits.
- $$U$$ -small colimits in $$C$$ satisfy descent.

[0002] Definition

Let $$U$$ be a universe and let $$C$$ and $$D$$ be lex $$U$$ -cocomplete categories. We say a functor $F:C\rightarrow D$ is a morphism of lex $$U$$ -cocomplete categories if the following conditions are satisfied.

- $$F$$ commutes with finite limits.
- $$F$$ commutes with $$U$$ -small colimits.

[000E] Definition

Let $$U$$ be a universe and let $$C$$ and $$D$$ be lex $$U$$ -cocomplete universes. We define $\mathord {\textnormal {\textsf {LexCocomp}}}_{U}(C,D)$ to be the full subcategory of the category of functors from $$C$$ to $$D$$ spanned by the morphisms of lex $$U$$ -cocomplete categories.

[0003] Definition

Let $$U$$ be a universe and let $$V$$ be a universe strictly greater than $$U$$ . We define $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ to be the locally full subcategory of the $$2$$ -category of $$V$$ -small categories whose objects are the $$V$$ -small lex $$U$$ -cocomplete categories and morphisms are the morphisms of lex $$U$$ -cocomplete categories.

[004Y] Definition

Let $$U$$ be a universe.

-Let $\kappa$ be a $$U$$ -small sound doctrine. We say a category $$C$$ is $(U,\kappa )$ -presentable if there exist a $$U$$ -small category $$A$$ , a reflective full subcategory $$L$$ of the category of $$U$$ -small presheaves on $$A$$ closed under $$U$$ -small $\kappa$ -filtered colimits, and an equivalence $C\simeq L$ .
-We say a category $$C$$ is $$U$$ -presentable if it is $(U,\kappa )$ -presentable for some $$U$$ -small sound doctrine $\kappa$ .

[0006] Exercise

Let $$U$$ be a universe and let $$V$$ be a universe strictly greater than $$U$$ . Then $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ is $$(V,U)$$ -presentable.

Hint

The theory of lex $$U$$ -cocomplete categories is “algebraic”, so this should be straightforward.

Logoses are usually defined as lex cocomplete categories that have small presentations. Having small presentations can be replaced by compactness.

[004Z] Exercise

Let $$U$$ be a universe and let $$V$$ be a universe strictly greater than $$U$$ . Then a $$V$$ -small lex $$U$$ -cocomplete category $$C$$ is $$U$$ -presentable if and only if it is $$U$$ -compact in $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ .

Hint

The “only if” part follows from the following facts/observations.

-If $$C$$ is a $$U$$ -presentable lex $$U$$ -cocomplete category, then it is a lex localization of the category of $$U$$ -small presheaves on a $$U$$ -small lex category.
-A lex localization of a $$U$$ -presentable lex $$U$$ -cocomplete category is a coinverter in $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ .
-The category of $$U$$ -small presheaves on a $$U$$ -small lex category is $$U$$ -compact in $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ .

For the “if” part, observe that the $$U$$ -compact objects in $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ are constructible under $$U$$ -small colimits from the free lex $$U$$ -cocomplete category over one object. Then the proof proceeds by induction.

-The free lex $$U$$ -cocomplete category over one object is the category of $$U$$ -small presheaves on the free lex category over one object. Thus, it is $$U$$ -presentable.
- $$U$$ -presentable lex $$U$$ -cocomplete categories are closed under $$U$$ -small colimits in $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ . This is proved by tracking the construction of colimits of $$U$$ -presentable lex $$U$$ -cocomplete categories given in Section 6.3.4 of [Lurie--2009-0000]. The key observation is that colimits of $$U$$ -presentable lex $$U$$ -cocomplete categories are decomposed as colimits of $$U$$ -small lex categories and lex localizations both of which are colimits in $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ .

[0004] Definition

Let $$U$$ be a universe. The $$2$$ -category $\mathord {\textnormal {\textsf {Logos}}}(U)$ of $$U$$ -logoses is defined to be the smallest full subcategory of $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ spanned by the $$U$$ -compact objects, where $$V$$ is some universe strictly greater than $$U$$ . Note that the choice of $$V$$ does not matter.

[0005] Definition

Let $$U$$ be a universe. The $$2$$ -category $\mathord {\textnormal {\textsf {Topos}}}(U)$ of $$U$$ -toposes is defined to be ${(\mathord {\textnormal {\textsf {Logos}}}(U))}^{\mathord {\textnormal {\textsf {op}}}}$ .

[0018] Definition

Let $$U$$ be a universe. We define $\mathord {\textnormal {\textsf {Topos}}}^{(1)}(U)$ to be the $$1$$ -categorical core of $\mathord {\textnormal {\textsf {Topos}}}(U)$ .

[000J] Definition

Let $$U$$ be a universe. We refer to the canonical equivalence ${(\mathord {\textnormal {\textsf {Topos}}}(U))}^{\mathord {\textnormal {\textsf {op}}}}\simeq \mathord {\textnormal {\textsf {Logos}}}(U)$ as $\mathord {\textnormal {\textsf {Sh}}}$ .

[000O] Definition

Let $$U$$ be a universe. The action of $\mathord {\textnormal {\textsf {Sh}}}:{(\mathord {\textnormal {\textsf {Topos}}}(U))}^{\mathord {\textnormal {\textsf {op}}}}\simeq \mathord {\textnormal {\textsf {Logos}}}(U)$ on morphisms is also denoted by $f\mapsto {f}^{*}$ . We call ${f}^{*}$ the inverse image of a morphism $$f$$ of $$U$$ -toposes.

[0007] Lemma

Let $$U$$ be a universe and let $$V$$ be a universe strictly greater than $$U$$ . Then $\mathord {\textnormal {\textsf {Logos}}}(U)$ is closed in $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ under $$U$$ -small colimits.

Proof

This is because $$U$$ -compact objects in a $$(V,U)$$ -presentable category are closed under $$U$$ -small colimits.

[0008] Proposition

Let $$U$$ be a universe. Then $\mathord {\textnormal {\textsf {Topos}}}(U)$ has finite limits.

Proof

By [0007].

[0009] Proposition

Let $$U$$ be a universe. Then $\mathord {\textnormal {\textsf {Topos}}}(U)$ has $$U$$ -small products.

Proof

By [0007].