[000A] Étale morphisms

Étale morphisms play the central role in topos theory. They are defined in terms of slice categories.

Recall that slices of lex categories coclassify global sections.

[000B] Definition

Let $$C$$ be a category that has finite limits and let $$x:C$$ be an object. Let ${x}^{*}:C\rightarrow C/x$ denote the pullback functor along the morphism from $$x$$ to the final object. We define a morphism in $$C/x$$

\mathord {\textnormal {\textsf {d}}}_{x}:\mathord {\textnormal {\textsf {1}}}\rightarrow {x}^{*}(x)

to be the one represented by the diagonal morphism

x\rightarrow x\times x

\(C\)

[000C] Exercise

Let $$C$$ be a category that has finite limits and let $$x:C$$ be an object. Then the pair $({x}^{*},\mathord {\textnormal {\textsf {d}}}_{x})$ has the following universal property: For any category $$D$$ that has finite limits, the functor

G\mapsto (G\circ {x}^{*},G(\mathord {\textnormal {\textsf {d}}}_{x})):\mathord {\textnormal {\textsf {Lex}}}(C/x,D)\rightarrow (F:\mathord {\textnormal {\textsf {Lex}}}(C,D))\times (\mathord {\textnormal {\textsf {1}}}\rightarrow F(x))

is an equivalence. Moreover, the inverse of this functor at

\((F,u)\)

{u}^{*}\circ (F/x)

The same is true for lex cocomplete categories.

[000D] Proposition

Let $$U$$ be a universe, let $$C$$ be a lex $$U$$ -cocomplete category, and let $$x:C$$ be an object. Then the pair $({x}^{*},\mathord {\textnormal {\textsf {d}}}_{x})$ has the following universal property: For any lex $$U$$ -cocomplete category $$D$$ , the equivalence given in [000C] restricts to an equivalence

\mathord {\textnormal {\textsf {LexCocomp}}}_{U}(C/x,D)\simeq (F:\mathord {\textnormal {\textsf {LexCocomp}}}_{U}(C,D))\times (\mathord {\textnormal {\textsf {1}}}\rightarrow F(x))

Proof

This follows from the fact that, for any lex $$U$$ -cocomplete category $$C$$ and any morphism $u:x\rightarrow y$ in $$C$$ , the pullback functor ${u}^{*}:C/y\rightarrow C/x$ is a morphism of lex $$U$$ -cocomplete categories.

There is a generic slice category.

[000F] Definition

Let $$U$$ be a universe. We define ${\langle \mathord {\textnormal {\textsf {w}}}\rangle }_{\mathord {\textnormal {\textsf {LexCocomp}}}_{U}}$ to be the free lex $$U$$ -cocomplete category generated by one object $\mathord {\textnormal {\textsf {w}}}$ . That is, for any lex $$U$$ -cocomplete category $$C$$ , the functor

F\mapsto F(\mathord {\textnormal {\textsf {w}}}):\mathord {\textnormal {\textsf {LexCocomp}}}_{U}({\langle \mathord {\textnormal {\textsf {w}}}\rangle }_{\mathord {\textnormal {\textsf {LexCocomp}}}_{U}},C)\rightarrow C

is an equivalence.

[000G] Proposition

Let $$U$$ be a universe, let $$V$$ be a universe strictly greater than $$U$$ , let $$C$$ be a $$V$$ -small lex $$U$$ -cocomplete category, and let $$x:C$$ be an object. Let $F:{\langle \mathord {\textnormal {\textsf {w}}}\rangle }_{\mathord {\textnormal {\textsf {LexCocomp}}}_{U}}\rightarrow C$ denote the morphism of lex $$U$$ -cocomplete categories corresponding to $$x$$ . Then ${x}^{*}:C\rightarrow C/x$ is the pushout in $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ of ${\mathord {\textnormal {\textsf {w}}}}^{*}:{\langle \mathord {\textnormal {\textsf {w}}}\rangle }_{\mathord {\textnormal {\textsf {LexCocomp}}}_{U}}\rightarrow {\langle \mathord {\textnormal {\textsf {w}}}\rangle }_{\mathord {\textnormal {\textsf {LexCocomp}}}_{U}}/\mathord {\textnormal {\textsf {w}}}$ along $$F$$ .

Proof

One can check the universal property of the pushout using [000D].

[000H] Proposition

Let $$U$$ be a universe and let $$V$$ be a universe strictly greater than $$U$$ . Then the morphism ${\mathord {\textnormal {\textsf {w}}}}^{*}:{\langle \mathord {\textnormal {\textsf {w}}}\rangle }_{\mathord {\textnormal {\textsf {LexCocomp}}}_{U}}\rightarrow {\langle \mathord {\textnormal {\textsf {w}}}\rangle }_{\mathord {\textnormal {\textsf {LexCocomp}}}_{U}}/\mathord {\textnormal {\textsf {w}}}$ in $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ is coexponentiable. That is, the pushout functor along ${\mathord {\textnormal {\textsf {w}}}}^{*}$ has a left adjoint.

Proof

By [000G], the pushout along ${\mathord {\textnormal {\textsf {w}}}}^{*}$ is computed by slicing. Since slicing commutes with limits and since the pushout functor commutes with colimits, the adjoint functor theorem applies.

[000X] Proposition

Proof

This follows from [000D] because, for any lex $$U$$ -cocomplete category $$C$$ , the projection $(F:\mathord {\textnormal {\textsf {LexCocomp}}}_{U}({\langle \mathord {\textnormal {\textsf {w}}}\rangle }_{\mathord {\textnormal {\textsf {LexCocomp}}}_{U}},C))\times (\mathord {\textnormal {\textsf {1}}}\rightarrow F(\mathord {\textnormal {\textsf {w}}}))\rightarrow \mathord {\textnormal {\textsf {LexCocomp}}}_{U}({\langle \mathord {\textnormal {\textsf {w}}}\rangle }_{\mathord {\textnormal {\textsf {LexCocomp}}}_{U}},C)$ is a left fibration.

A morphism of logoses is étale if it is equivalent to ${x}^{*}:C\rightarrow C/x$ for some $$x$$ . We see that $$x$$ is unique.

[0010] Definition

Let $$U$$ be a universe, let $$V$$ be a universe strictly greater than $$U$$ , and let $$C$$ be a $$V$$ -small lex $$U$$ -cocomplete category. We define a functor

\mathord {\textnormal {\textsf {E}}}_{C}:{C}^{\mathord {\textnormal {\textsf {op}}}}\rightarrow \mathord {\textnormal {\textsf {LexCocomp}}}(U,V)\backslash C

\mathord {\textnormal {\textsf {E}}}_{C}(x)=({x}^{*}:C\rightarrow C/x)

[000I] Proposition

Let $$U$$ be a universe, let $$V$$ be a universe strictly greater than $$U$$ , and let $$C$$ be a $$V$$ -small lex $$U$$ -cocomplete category. The functor $\mathord {\textnormal {\textsf {E}}}_{C}:{C}^{\mathord {\textnormal {\textsf {op}}}}\rightarrow \mathord {\textnormal {\textsf {LexCocomp}}}(U,V)\backslash C$ is fully faithful.

Proof

Let $$x,y:C$$ be objects. By [000D], the category of morphisms ${y}^{*}\rightarrow {x}^{*}$ in $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)\backslash C$ is equivalent to the discrete category of morphisms $\mathord {\textnormal {\textsf {1}}}\rightarrow {x}^{*}(y)$ in $$C/x$$ . But the latter is equivalent to the discrete category of morphisms $x\rightarrow y$ in $$C$$ .

[0011] Proposition

Let $$U$$ be a universe, let $$V$$ be a universe strictly greater than $$U$$ , and let $$C$$ be a $$V$$ -small lex $$U$$ -cocomplete category. Then the functor $\mathord {\textnormal {\textsf {E}}}_{C}:{C}^{\mathord {\textnormal {\textsf {op}}}}\rightarrow \mathord {\textnormal {\textsf {LexCocomp}}}(U,V)\backslash C$ preserves $$U$$ -small limits.

Proof

Observe that limits in $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ are computed in the category of $$V$$ -small categories. Then the claim is a consequence of descent.

[000V] Definition

Let $$U$$ be a universe and let $$V$$ be a universe strictly greater than $$U$$ . We say a morphism $f:C\rightarrow D$ in $\mathord {\textnormal {\textsf {LexCocomp}}}(U,V)$ is étale if it is in the image of the functor $\mathord {\textnormal {\textsf {E}}}_{C}:{C}^{\mathord {\textnormal {\textsf {op}}}}\rightarrow \mathord {\textnormal {\textsf {LexCocomp}}}(U,V)\backslash C$ .

Let us reverse morphisms.

[000K] Definition

Let $$U$$ be a universe. We define a morphism $\mathord {\textnormal {\textsf {p}}}:\mathbb {A}_{\bullet }\rightarrow \mathbb {A}$ in $\mathord {\textnormal {\textsf {Topos}}}(U)$ to be the one sent by $\mathord {\textnormal {\textsf {Sh}}}$ to ${\mathord {\textnormal {\textsf {w}}}}^{*}:{\langle \mathord {\textnormal {\textsf {w}}}\rangle }_{\mathord {\textnormal {\textsf {LexCocomp}}}_{U}}\rightarrow {\langle \mathord {\textnormal {\textsf {w}}}\rangle }_{\mathord {\textnormal {\textsf {LexCocomp}}}_{U}}/\mathord {\textnormal {\textsf {w}}}$ .

[000Z] Proposition

Let $$U$$ be a universe. Then $\mathbb {A}$ represents the functor $\mathord {\textnormal {\textsf {Sh}}}:{(\mathord {\textnormal {\textsf {Topos}}}(U))}^{\mathord {\textnormal {\textsf {op}}}}\rightarrow \mathord {\textnormal {\textsf {Logos}}}(U)$ . That is, we have a natural equivalence $\mathord {\textnormal {\textsf {Hom}}}_{\mathord {\textnormal {\textsf {Topos}}}(U)}(X,\mathbb {A})\simeq \mathord {\textnormal {\textsf {Sh}}}(X)$ .

Proof

This is by definition.

[000L] Proposition

Let $$U$$ be a universe. Then the morphism $\mathord {\textnormal {\textsf {p}}}:\mathbb {A}_{\bullet }\rightarrow \mathbb {A}$ in $\mathord {\textnormal {\textsf {Topos}}}(U)$ is exponentiable.

Proof

By [000H].

[000Y] Proposition

Let $$U$$ be a universe. Then the morphism $\mathord {\textnormal {\textsf {p}}}:\mathbb {A}_{\bullet }\rightarrow \mathbb {A}$ in $\mathord {\textnormal {\textsf {Topos}}}(U)$ is a left fibration.

Proof

By [000X].

[000M] Proposition

Let $$U$$ be a universe. Then the morphism $\mathord {\textnormal {\textsf {p}}}:\mathbb {A}_{\bullet }\rightarrow \mathbb {A}$ in $\mathord {\textnormal {\textsf {Topos}}}(U)$ satisfies directed univalence. That is, for any object $X:\mathord {\textnormal {\textsf {Topos}}}(U)$ , the functor

\mathord {\textnormal {\textsf {Hom}}}_{\mathord {\textnormal {\textsf {Topos}}}(U)}(X,\mathbb {A})\rightarrow \mathord {\textnormal {\textsf {Topos}}}(U)/X

defined by the pullback of the left fibration

\mathord {\textnormal {\textsf {p}}}:\mathbb {A}_{\bullet }\rightarrow \mathbb {A}

is fully faithful.

Proof

By [000I].

[000N] Definition

Let $$U$$ be a universe. We say a morphism $f:Y\rightarrow X$ in $\mathord {\textnormal {\textsf {Topos}}}(U)$ is étale if there exists a (necessarily unique by [000M]) pullback square from $$f$$ to $\mathord {\textnormal {\textsf {p}}}:\mathbb {A}_{\bullet }\rightarrow \mathbb {A}$ .

[000S] Proposition

Let $$U$$ be a universe. Then étale morphisms in $\mathord {\textnormal {\textsf {Topos}}}(U)$ are closed under pullback.

Proof

This is by definition.

[000P] Proposition

Let $$U$$ be a universe. Then a morphism $f:Y\rightarrow X$ in $\mathord {\textnormal {\textsf {Topos}}}(U)$ is étale if and only if the morphism of lex $$U$$ -cocomplete categories ${f}^{*}:\mathord {\textnormal {\textsf {Sh}}}(X)\rightarrow \mathord {\textnormal {\textsf {Sh}}}(Y)$ is étale.

Proof

By [000G].

[000Q] Proposition

Let $$U$$ be a universe. Then all the identity morphisms in $\mathord {\textnormal {\textsf {Topos}}}(U)$ are étale.

Proof

Let $$X$$ be a $$U$$ -topos. We have the equivalence $\mathord {\textnormal {\textsf {Sh}}}(X)\simeq \mathord {\textnormal {\textsf {Sh}}}(X)/\mathord {\textnormal {\textsf {1}}}$ . Then [000P] applies.

[000R] Proposition

Let $$U$$ be a universe. Then étale morphisms in $\mathord {\textnormal {\textsf {Topos}}}(U)$ are closed under composition.

Proof

Let $$X$$ be a $$U$$ -topos, let $a:\mathord {\textnormal {\textsf {Sh}}}(X)$ be an object, and let $s:\mathord {\textnormal {\textsf {Sh}}}(X)/a$ be an object. Let $u:b\rightarrow a$ denote the morphism that represents $$s$$ . We have the equivalence $(\mathord {\textnormal {\textsf {Sh}}}(X)/a)/s\simeq \mathord {\textnormal {\textsf {Sh}}}(X)/b$ . Then [000P] applies.

[000T] Lemma

Let $$U$$ be a universe. Then étale morphisms in $\mathord {\textnormal {\textsf {Topos}}}(U)$ has the left cancellation property. That is, for morphisms $f:Y\rightarrow X$ and $g:Z\rightarrow Y$ in $\mathord {\textnormal {\textsf {Topos}}}(U)$ , if $$f$$ and $f\circ g$ are étale, then so is $$g$$ .

Proof

Let $$X$$ be a $$U$$ -topos, let $a,b:\mathord {\textnormal {\textsf {Sh}}}(X)$ be objects, and let $F:\mathord {\textnormal {\textsf {Sh}}}(X)/a\rightarrow \mathord {\textnormal {\textsf {Sh}}}(X)/b$ be a morphism in $\mathord {\textnormal {\textsf {Logos}}}(U)\backslash \mathord {\textnormal {\textsf {Sh}}}(X)$ . By [000I], $$F$$ is equivalent to the pullback functor ${u}^{*}$ for some morphism $u:b\rightarrow a$ . Let $s:\mathord {\textnormal {\textsf {Sh}}}(X)/a$ be the object represented by $$u$$ . We have the equivalence $\mathord {\textnormal {\textsf {Sh}}}(X)/b\simeq (\mathord {\textnormal {\textsf {Sh}}}(X)/a)/s$ . Then [000P] applies.

[000U] Proposition

Let $$U$$ be a universe. Then étale morphisms in $\mathord {\textnormal {\textsf {Topos}}}(U)$ are closed under diagonal. That is, if a morphism $f:Y\rightarrow X$ in $\mathord {\textnormal {\textsf {Topos}}}(U)$ is étale, then so is the diagonal morphism $Y\rightarrow Y\mathbin {{}_{f}\mathord {\times _{f}}}Y$ .

Proof

The diagonal morphism in question is a section of the first (or second) projection $Y\mathbin {{}_{f}\mathord {\times _{f}}}Y\rightarrow Y$ . By [000T] and [000Q], it suffices to show that the first projection is étale, but this follows from [000S].

[000W] Definition

Let $$U$$ be a universe and let $$X$$ be a $$U$$ -topos. We define $\mathord {\textnormal {\textsf {Etale}}}(X)$ to be the full subcategory of $\mathord {\textnormal {\textsf {Topos}}}(U)/X$ spanned by the étale morphisms with codomain $$X$$ .

[0012] Proposition

Let $$U$$ be a universe and let $$X$$ be a $$U$$ -topos. Then the functor $\mathord {\textnormal {\textsf {E}}}_{\mathord {\textnormal {\textsf {Sh}}}(X)}:{(\mathord {\textnormal {\textsf {Sh}}}(X))}^{\mathord {\textnormal {\textsf {op}}}}\rightarrow \mathord {\textnormal {\textsf {Logos}}}(U)\backslash \mathord {\textnormal {\textsf {Sh}}}(X)$ restricts to an equivalence $\mathord {\textnormal {\textsf {Sh}}}(X)\simeq \mathord {\textnormal {\textsf {Etale}}}(X)$ .

Proof

By definition.

[0013] Proposition

Let $$U$$ be a universe. Then, for any $$U$$ -topos $$X$$ , the category $\mathord {\textnormal {\textsf {Etale}}}(X)$ has $$U$$ -small colimits. Moreover, for any morphism $f:X\rightarrow Y$ in $\mathord {\textnormal {\textsf {Topos}}}(U)$ , the pullback functor ${f}^{*}:\mathord {\textnormal {\textsf {Etale}}}(Y)\rightarrow \mathord {\textnormal {\textsf {Etale}}}(X)$ commutes with $$U$$ -small colimits.

Proof

By [0012].

[0014] Proposition

Let $$U$$ be a universe and let $$X$$ be a $$U$$ -topos. Then the inclusion $\mathord {\textnormal {\textsf {Etale}}}(X)\rightarrow \mathord {\textnormal {\textsf {Topos}}}(U)/X$ preserves $$U$$ -small colimits.

Proof

By [0011].

[0017] Definition

Let $$U$$ be a universe, let $$I$$ be a $$U$$ -small category, and let $X:I\rightarrow \mathord {\textnormal {\textsf {Topos}}}(U)$ be a functor. Suppose that $$X$$ factors through the domain projection $\mathord {\textnormal {\textsf {Etale}}}(Y)\rightarrow \mathord {\textnormal {\textsf {Topos}}}(U)$ for some $$U$$ -topos $$Y$$ . By [0013] and [0014], the colimit of $$X$$ exists. We call colimits obtained in this way étale colimits. Limits in $\mathord {\textnormal {\textsf {Logos}}}(U)$ that are étale colimits in $\mathord {\textnormal {\textsf {Topos}}}(U)$ are called étale limits.