Contents

category theory

topos theory

# Contents

## Idea

Let $\mathcal{C}$ be any small category.

### Over a representable

For $X \,\in\, \mathcal{C}$ an object, the category of presheaves $PSh\big( \mathcal{C}_{/X}\big)$ on the slice category $\mathcal{C}_{/X}$ is equivalent to the slice $PSh(\mathcal{C})_{/X}$ of the category of presheaves on $\mathcal{C}$ over the image of $X$ under the Yoneda embedding.

The former presheaf topos is manifestly a Grothendieck topos, whence this equivalence shows that also the slice $PSh(\mathcal{C})_{/X}$ is a Grothendieck topos. This is the archetypical special case of the fundamental theorem of topos theory which says that all slices of toposes are themselves toposes: slice toposes.

As shown in Prop. below, this equivalence is canonically an adjoint equivalence, where the right adjoint $R$ forms the hom-set in the slice over $y(X)$, hence is the functor which takes a bundle (in the broad sense) internal to presheaves to its system of sets $\Gamma_{(-)}(E)$ of local sections:

$\array{ PSh(\mathcal{C})_{/X} & \xrightarrow{\;\;\;\; \sim \;\;\;\;} & PSh \big( \mathcal{C}_{/X} \big) \\ \left( \array{ E \\ \downarrow \\ X } \right) &\mapsto& \big( (U \to X) \,\mapsto\, \Gamma_U(E) \big) \,. }$

If instead of presheaves of sets one considers simplicial presheaves then this adjoint equivalence becomes a Quillen equivalence with respect to the the projective model structure on simplicial presheaves and its slice model structure (Prop. below).

As such this Quillen equivalence models the analogous statement (Prop. below) for slice $\infty$-categories of $\infty$-categories of $\infty$-presheaves, which thus also are slice $\infty$-toposes. This is the archetypical case of the fundamental theorem of $\infty$-topos theory, see there for more.

### Over any presheaf

More generally, the analogous statement remains true when $X \,\in\, PSh(\mathcal{C})$ is any presheaf (not necessarily a representable in the image of the Yoneda embedding). In this more general case the equivalence reads just as before

$\array{ PSh(\mathcal{C})_{/X} & \xrightarrow{\;\;\;\; \sim \;\;\;\;} & PSh \big( \mathcal{C}_{/X} \big) \,, }$

only that now the site appearing on the right must be understood as a full subcategory of the slice category of the full category of presheaves, on those objects whose domain is a representable:

$\mathcal{C}_{/X} \;\xhookrightarrow{\;\;}\; PSh(\mathcal{C})_{/X} \,.$

This may equivalently be understood as the Grothendieck construction on the functor $X$.

## Preliminaries

### Presheaves

Let $\mathcal{C}$ be a small category, we write

$PSh(\mathcal{C}) \,\coloneqq\, Func(\mathcal{C}^{op} ,\, Set)$

for its category of presheaves and

(1)$y_{\mathcal{C}} \,\colon\, \mathcal{C} \xrightarrow{\;\;\;} PSh(\mathcal{C})$

for the Yoneda embedding.

Recall (from there) that every presheaf $F \,\in\, PSh(\mathcal{C})$ is a colimit of representables $y_{\mathcal{C}}(c)$ indexed by the comma category of morphisms $y_{\mathcal{C}}(c) \to F$. We will denote this “co-Yoneda lemma” by

(2)$F \;\; \simeq \;\; \underset {\underset{ y_{\mathcal{C}}(c) \to F }{\longrightarrow}} { lim } \; y_{\mathcal{C}}(c) \,.$

### Slice categories

For $\mathcal{D}$ any category and $B \in \mathcal{D}$, the hom-object in the slice category is given by the following fiber product (e.g, here):

(3)$\mathcal{D}_{/B} \big( (U,p_U) \,\, (U',p_{U'}) \big) \;\; \simeq \;\; \mathcal{D}(U, U') \underset{ \mathcal{D}(U, B) }{\times} \{p_{U}\} \,.$

### Slices of presheaf categories

For any $X \,\in\, \mathcal{C}$ we denote the generic object of the slice category $\mathcal{C}_{/X}$ by

$c_X \,=\, \left( \array{ c \\ \big\downarrow{\mathrlap{{}^{c_X}}} \\ X } \;\; \right) \;\; \in \; \mathcal{C}_{/X} \,.$

Notice that the slice category $\mathcal{C}_{/X}$ has its own Yoneda embedding

$y_{\mathcal{C}_{/X}} \;\colon\; \mathcal{C}_{/X} \xrightarrow{\;\;\;\;} PSh \big( \mathcal{C}_{/X} \big)$

but that it is also the source of the slicing of the plain Yoneda embedding (1), which is still a fully faithful functor:

(4)$\array{ (y_{\mathcal{C}})_{/X} &\colon& \mathcal{C}_{/X} &\xhookrightarrow{\phantom{-----}}& PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)} \\ && c_X &\mapsto& \left( \array{ y_{\mathcal{C}}(c) \\ \downarrow^{\mathrlap{ y_{\mathcal{C}}(c_X) }} \\ y_{\mathcal{C}}(X) } \;\;\;\;\; \right) }$

## Statement

### In plain category theory

We discuss the statement in plain category theory with general abstract proofs that will work in any other context (notably in enriched category theory, see below, and $\infty$-category theory, see further below) that satisfies the basic theorems of category theory (e.g. the natural respect of hom-functors for (co-)limits etc.).

First we give a quick proof that (Prop. below) there is an equivalence of categories at all. Then we enhance this statement (in Prop. ) below to an adjoint equivalence whose right adjoint is concretely identified as forming sections.

###### Proposition

For $X \,\in\, PSh(\mathcal{C})$ any presheaf, we have an equivalence of categories

$PSh(\mathcal{C})_{/X} \;\simeq\; PSh \big( \mathcal{C}_{/X} \big) \,.$

###### Proof

Using that

1. every presheaf is a colimit of representables (the “co-Yoneda lemma”);

2. colimits in slice categories are computed in the underlying categories (see there)

$\left( \array{ U \\ \big\downarrow \mathrlap{{}^{p_{U}}} \\ X } \;\;\; \right) \;=\; \Big( U, \, p_U \Big) \;\simeq\; \Big( \underset{ \underset{i \in \mathcal{I}}{\longrightarrow} }{\lim} \, c_U(i) ,\, (p_{c_U(i)})_{i \in \mathcal{I}} \big) \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longrightarrow} }{\lim} \big( c_U(i) ,\, p_{c_U(i)} \big)$

we have an evident functor

$\array{ PSh(\mathcal{C})_{/X} &\longrightarrow& PSh \big( \mathcal{C}_{/X} \big) \\ (U,p_U) &\overset{\phantom{----}}{\mapsto}& \underset{\underset{ i \in \mathcal{I} }{\longrightarrow}}{\lim} \big( c_U(i) ,\, p_{c_U(i)} \big) }$

which is clearly essentially surjective. That it is also fully faithful is established by the following sequence of natural isomorphisms:

$\begin{array}{l} PSh(\mathcal{C})_{/X} \Big( \big( U, \, p_U \big) \,, \big( U ,\, p_{U'} \big) \Big) \\ \;\simeq\; PSh(\mathcal{C}) \big( U \,, U' \big) \underset{ PSh(\mathcal{C}) \big( U \,, X \big) }{\times} \{p_U\} \\ \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, PSh(\mathcal{C}) \big( c_U(i) \,, U' \big) \underset{ \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, PSh(\mathcal{C}) \big( c_U(i) \,, X \big) }{\times} \Big\{ \big( p_{c_U(i)} \big)_{i \in \mathcal{I}} \Big\} \\ \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, \bigg( PSh(\mathcal{C}) \big( c_U(i) \,, U' \big) \underset{ PSh(\mathcal{C}) \big( c_U(i) \,, X \big) }{\times} \big\{ p_{c_U(i)} \big\} \bigg) \\ \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, \bigg( \underset{ \underset{i' \in \mathcal{I}'}{\longrightarrow} }{\lim} PSh(\mathcal{C}) \big( c_U(i) \,, c_{U'}(i') \big) \underset{ PSh(\mathcal{C}) \big( c_U(i) \,, X \big) }{\times} \big\{ p_{c_U(i)} \big\} \bigg) \\ \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, \underset{ \underset{i' \in \mathcal{I}'}{\longrightarrow} }{\lim} \, \bigg( PSh(\mathcal{C}_{/X}) \Big( \big( c_U(i) ,\, p_{c_U(i)} \big) \,, \big( c_{U'}(i') ,\, p_{c_{U'}(i')} \big) \Big) \bigg) \\ \;\simeq\; PSh(\mathcal{C}_{/X}) \Big( \underset{ \underset{i \in \mathcal{I}}{\longrightarrow} }{\lim} \big( c_U(i) ,\, p_{c_U(i)} \big) \,, \underset{ \underset{i' \in \mathcal{I}'}{\longrightarrow} }{\lim} \big( c_{U'}(i') ,\, p_{c_{U'}(i')} \big) \Big) \bigg) \\ \;\simeq\; PSh(\mathcal{C}_{/X}) \Big( \big( U ,\, p_U \big) \,, \big( U' ,\, p_{U'} \big) \Big) \bigg) \mathrlap{\,.} \end{array}$

Here

• the first step is the definition (3) of the slice hom;

• the second step is the expression of $U$ as a colimit of representables (co-Yoneda lemma), together with the fact that any hom functor sends colimits in the first argument to limits;

• the third step is the the fiber product commutes with this limit (since limits commute with limits);

• the fourth step is the expression of $U'$ as a colimit of representables (co-Yoneda lemma), together with the Yoneda lemma and the fact that colimits of presheaves are computed objectwise (here);

• the fifth step is universal colimits in the ambient topos (of Sets);

• the sixths step recognizes that the slice hom (3) defining $\mathcal{C}_{/X}$ has appeared and moves the colimits back inside by the reverse of the previous arguments, now over the slice side.

The following statement enhances this equivalence of categories to an adjoint equivalence and identifying its right adjoint as the functor of forming sections. This stronger version further enhances to a simplicial Quillen equivalence below.

###### Proposition

For $X \,\in\, \mathcal{C} \xhookrightarrow{\;y\;} PSh(\mathcal{C})$. the following anti-parallel functors constitute an adjoint equivalence

Here:

1. the top functor $L$ is the colimit-preserving functor that makes the top triangle commute, hence which takes representables over the slice site to the slicing of the underlying representables on the plain site. These two conditions fix the functor completely, by the fact (2) that every presheaf is a colimit of representables.

2. the bottom functor is the hom-functor of the slice category, which means (3) that it is given by a pullback of the hom-functor in $PSh(\mathcal{C})$ itself:

(5)$PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)} \Big( \big( y_{\mathcal{C}})_{/X}(U \xrightarrow{\phi} X \big) ,\, \big( E \xrightarrow{p} y_{\mathcal{C}}(X) \big) \Big) \;=\; PSh(\mathcal{C}) \big( y_{\mathcal{C}}(U) ,\, E \big) \underset { PSh(\mathcal{C}) \big( y_{\mathcal{C}}(U) ,\, y_{\mathcal{C}}(X) \big) } {\times} \big\{ y_{\mathcal{C}}(\phi) \big\}$

###### Proof

First to see that the functors are adjoint, we check the required hom-isomorphism by observing the following sequence of natural bijections:

\begin{aligned} PSh(\mathcal{C}_{/X}) \big( A ,\, R(B) \big) & \;=\; PSh(\mathcal{C}_{/X}) \Big( A ,\, PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)} \big( (y_{\mathcal{C}})_{/X}(-) ,\, B \big) \Big) \\ & \;\simeq\; PSh \big( \mathcal{C}_{/X} \big) \Big( \underset {\underset{c_X \to A}{\longrightarrow}} {\mathrm{lim}} y_{(\mathcal{C}_{/X})}(c_{X}) ,\, PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)} \big( (y_{\mathcal{C}})_{/X}(-) ,\, B \big) \Big) \\ & \;\simeq\; \underset{ \underset{c_X \to A}{\longleftarrow} }{\mathrm{lim}} PSh(\mathcal{C}_{/X}) \Big( y_{(\mathcal{C}_{/X})}(c_{X}) ,\, PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)} \big( (y_{\mathcal{C}})_{/X}(-) ,\, B \big) \Big) \\ & \; \simeq \; \underset{ \underset{c_X \to A}{\longleftarrow} }{\mathrm{lim}} PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)} \big( (y_{\mathcal{C}})_{/X}(c_X) ,\, B \big) \\ & \; \simeq \; PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)} \Big( \underset{ \underset{c_X \to A}{\longrightarrow} }{\mathrm{lim}} (y_{\mathcal{C}})_{/X}(c_X) ,\, B \Big) \\ & \;=\; PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)} \big( L(A) ,\, B \big) \end{aligned}

Here:

• the first step is the above definition of the right adjoint,

• the second step is (2),

• the third is that any hom-functor sends colimits in its first argument into limits (here),

• the fourth step is the Yoneda lemma over the slice site,

• the fifth step takes the limit back into the hom-functor, but now that of the other category,

• the sixth step is the above definition of the would-be left adjoint, using again (2).

Now to see that these two functors are weak inverses of each other.

In one direction we have the following sequence of natural bijections for $A \,\in\, PSh\big( \mathcal{C}_{/X}\big)$:

\begin{aligned} R \circ L (A) & \;\simeq\; PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)} \big( (y_{\mathcal{C}})_{/X}(-) ,\, L(A) \big) \\ & \;\simeq\; PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)} \Big( (y_{\mathcal{C}})_{/X}(-) ,\, \underset{\underset{c_X \to A}{\longrightarrow}}{\lim} (y_{\mathcal{C}})_{/X}(c_X) \Big) \\ & \;\simeq\; \underset{\underset{c_X \to A}{\longrightarrow}}{\lim} y_{(\mathcal{C}_{/X})}(c_X) \\ & \;\simeq\; A \end{aligned}

Here:

• the first two steps just unwind again the above definitions of the functors;

• the third step follows by the Yoneda lemma over $\mathcal{C}$, to which applies by observing that that:

1. colimits in slices are reflected as colimits in the underlying category (by this Prop),

2. the slice hom is a pullback of the plain hom (5),

3. colimits in a topos such as $PSh(\mathcal{C})$ are pullback-stable;

• the last step re-assembles the argument, by (2).

In the other direction we have the following sequence of natural bijections, for $B \,\in\, PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)}$:

\begin{aligned} L \circ R(B) & \;\simeq\; L \Big( \underset{ \underset { y_{(\mathcal{C}_{/X})}(c_X) \to R(B) } {\longrightarrow} } {\lim} y_{(\mathcal{C}_{/X})}(c_X) \Big) \\ & \;\simeq\; \underset{ \underset { y_{(\mathcal{C}_{/X})}(c_X) \to R(B) } {\longrightarrow} } {\lim} (y_{\mathcal{C}})_{/X}(c_X) \\ & \;\simeq\; \underset{ \underset { L\big( y_{(\mathcal{C}_{/X})}(c_X) \big) \to B } {\longrightarrow} } {\lim} (y_{\mathcal{C}})_{/X}(c_X) \\ & \;\simeq\; \underset{ \underset { (y_{\mathcal{C}})_{/X}(c_X) \to B } {\longrightarrow} } {\lim} (y_{\mathcal{C}})_{/X}(c_X) \\ & \;\simeq\; B \end{aligned}

Here:

• the first step is the co-Yoneda lemma (2) for $R(B)$,

• the second step unwinds the definition of $L$ from above,

• the third step uses the adjunction $L \dashv R$ established above on the indexing category of the colimit;

• the fourth step applies again the definition of $L$ from above,

• the last step is again the co-Yoneda lemma (2), now for $B$ itself.

The above proof of Prop. does not actually depend on the assumption that the base object is representable ($y(X)$).

### For simplicial presheaves on plain categories

We discuss generalizations of the above situation from presheaves to the homotopy theory of simplicial presheaves.

#### Over 0-truncated objects

For

• $\mathcal{C}$ an category,

• $X \,\in\, \mathcal{C}$ an object,

• $\mathcal{C}_{/X}$ the sSet-enriched slice category

###### Lemma

With categories of simplicial presheaves denoted $sPSh(-)$, Prop. generalizes to:

##### Globally

Write:

###### Proposition

Relative to these projective (slice) model structures, the comparison functor from Exp. is a right Quillen functor, hence the right adjoint in a simplicial Quillen adjunction, which is a Quillen equivalence:

###### Proof

Observe that:

1. Since representables are cofibrant (evidently so in the projective model structure, since acyclic Kan fibrations are surjective), the unsliced simplicial hom out of a representable is a right Quillen functor by the pullback-power axiom in the $sSet_{Qu}$-enriched model category $sSh(\mathcal{C})$.

2. The base change-functor by pullback is a right Quillen functor on slice model categories of $sSet_{Qu}$ (by this Prop.).

Together this implies that their composite (5) is a right Quillen functor.

By Ken Brown's lemma (here) it follows that the right adjoint preserves weak equivalences between fibrant objects. We claim that it also reflects weak equivalences between fibrant objects, in that a morphism between fibrant objects on the left is a weak equivalence if and only if its image under the right adjoint functor is a weak equivalences. Since the functor is also an equivalence of categories, by Prop. , this immediately implies that the derived adjunction is an equivalence of homotopy categories, and hence that we have a Quillen adjunction.

To see this remaining claim (that the right adjoint reflects weak equivalences between fibrant objects) consider a morphism $f \,\in\, sPsh(\mathcal{C})_{/y_{\mathcal{C}}(X)}$ between fibrations such that for all $U \xrightarrow{\phi} X$ in $\mathcal{C}_{/X}$ the base change (5) of its values on $U \,\in\, \mathcal{C}$ is a weak equivalence:

Here the right vertical morphisms are Kan fibrations by the fact that $sPSh(\mathcal{C})\big( y_{\mathcal{C}}(U),\, - \big)$ is a right Quillen functor as in the first item above. Therefore – since this holds for all $\phi$, by assumption – this Prop. implies that $f(U) \,\simeq\, \mathrm{PSh}(\mathcal{C}) \big( y_{\mathcal{C}}(U) ,\, f \big)$ is a weak equivalence. And since this holds for all $U \,\in\, \mathcal{C}$, this means that $f$ is a weak equivalence in the slice of the projective model structure.

##### Locally

We now promote the Quillen equivalence in the previous section to the case of Čech-local model structures on simplicial presheaves.

Recall that these are obtained as a left Bousfield localization of the (say) projective model structure on simplicial presheaves with respect to Čech nerves of covering families.

We reuse the notation of the previous section.

###### Proposition

The Quillen equivalence of descends to a Quillen equivalence of the corresponding Čech-local projective model structures.

###### Proof

Both model categories are left proper and combinatorial. Therefore we can take left Bousfield localizations with respect to arbitrary sets of morphisms.

We localize both sides with respect to Čech nerves of respective covering families. Observe that Čech nerves of covering families in $\mathcal{C}_{/X}$ are mapped to Čech nerves of covering families in $sPSh(\mathcal{C})$ and therefore also in the slice category $sPSh(\mathcal{C})_{y_{\mathcal{C}}(X)}$. Thus, we have an induced Quillen adjunction between localized model categories.

It remains to show that this Quillen adjunction is a Quillen equivalence.

It suffices to show that the right adjoint reflects weak equivalences between fibrant objects. Here fibrant objects are objectwise Kan complexes that satisfy the appropriate variant of the homotopy descent property?. Local weak equivalences between locally fibrant objects coincide with objectwise weak equivalences. As established in the previous section, the right adjoint functor reflects objectwise weak equivalences between objectwise fibrant presheaves, which completes the proof.

#### Over 1-truncated objects

Over a 1-truncated object (hence over a simplicial presheaf with values in the simplicial nerve of a groupoid):

This happens in Hollander 2008, Def. 5.1.

The resulting Quillen equivalences for the global model structure on simplicial presheaves is Hollander 2008, Them. 5.2, and that for any local model structure is Hollander 2008, Thm. 6.2.

### In $\infty$-category theory

All the natural equivalences used in a category-theoretic proof such as of Prop. above also hold in $\infty$-category theory.

More explicitly, since simplicial localization at the Quillen equivalences identifies the homotopy theory ($\infty$-category) of combinatorial model categories (such as model categories of simplicial presheaves and their slice model structures) with that of presentable $\infty$-categories, Prop. immediately implies simplicial Quillen equivalences which present the equivalence of $\infty$-categories.

Either way, we obtain the following conclusion (and again this hold verbatim also over non-representable base presheaves):

###### Proposition

(fundamental theorem of $\infty$-presheaf $\infty$-topos theory)
For $\mathbf{C}$ a small $\infty$-category and $X \,\in\, \mathbf{S}$ an object, the operation of forming systems of local sections of bundles of $\infty$-presheaves over $y(X)$ is an equivalence of $\infty$-categories:

from the slice $\infty$-category of the $\infty$-category of $\infty$-presheaves over $\mathbf{C}$ to the $\infty$-category of $\infty$-presheaves over the slice $\infty$-category of $\mathbf{C}$.

An alternative proof of this statement in terms of quasi-categories is in Lurie 2009, Prop. 5.1.6.12. (See also here at slice $\infty$-topos.)

A(nother) model category-theory argument for the statement over any 1-truncated simplicial presheaf is in (Hollander 2008).

A quasi-category-proof of this statement over representables is in Lurie 2009, Cor. 5.1.6.12. (This states a more general theorem which superficially looks like it may cover the case of non-representable base presheaves, but at least not directly so.)

###### Example

($\mathcal{G}$-$\infty$-actions as slice over $B\mathcal{G}$)
Consider

$\mathcal{G} \,\simeq\, \Omega B \mathcal{G} \,\in\, Grp\big(Grpd_\infty\big)$

an $\infty$-group with delooping/classifying space $B \mathcal{G} \,\in\, Grpd_\infty^{\geq 1}$. Observe that the external slice of the point by any $\mathcal{X} \,\in\, Grpd_\infty$ is that $\infty$-groupoid:

$\ast_{/\mathcal{X}} \;\simeq\; \mathcal{X} \,,$

so that Prop. here gives the following sequence of equivalences:

$\array{ \big(Grpd_\infty\big)_{/\mathbf{B}\mathcal{G}} \,\simeq\, \big( PSh_\infty(\ast) \big)_{/\mathbf{B}\mathcal{G}} & \underoverset {\phantom{--}\sim\phantom{--}} {\Gamma} {\longrightarrow} & PSh_\infty \big( \ast_{/\mathbf{B}\mathcal{G}} \big) \;\simeq\; PSh_\infty( \mathbf{B}\mathcal{G} \big) \;\simeq\; \mathcal{G} Act \big( Grpd_\infty \big) \,. }$

This is the identification of the slice over $B \mathcal{G}$ with $\infty$-actions of $\mathcal{G}$. For a model category-presentation see at Borel model structure – Relation to the slice over the simplicial classifying space.

###### Example

In the case that $\mathbf{C} \,=\, Snglrt \,\coloneqq\, Grpd^{fin}_{1,\geq 1}$ is the global orbit category (a (2,1)-category) the equivalence of Prop. extracts the system of fixed loci of an object in global equivariant homotopy theory sliced over the archetypical $G$-orbi-singularity, for some equivariance group $G$. Together with the adjoint quadruple that is induced (see here) via $\infty$-Kan extension from the reflection onto the $G$-orbit category, this implies the cohesion of global- over G-equivariant homotopy theory. See there for more.

Textbook accounts for the statement in plain category theory:

via model categories:

and via quasi-categories: