nLab
slice of presheaves is presheaves on slice

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Category theory

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topos theory

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In higher category theory

Theorems

Contents

Idea

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

The former presheaf topos is manifestly a Grothendieck topos, whence this equivalence shows that also the slice PSh(𝒞) /XPSh(\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 RR forms the hom-set in the slice over y(X)y(X), hence is the functor which takes a bundle (in the broad sense) internal to presheaves to its system of sets Γ ()(E)\Gamma_{(-)}(E) of local sections:

PSh(𝒞) /X PSh(𝒞 /X) (E X) ((UX)Γ U(E)). \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.


Preliminaries

Presheaves

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

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

for its category of presheaves and

(1)y 𝒞:𝒞PSh(𝒞) y_{\mathcal{C}} \,\colon\, \mathcal{C} \xrightarrow{\;\;\;} PSh(\mathcal{C})

for the Yoneda embedding.

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

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

Slices

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

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

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

y 𝒞 /X:𝒞 /XPSh(𝒞 /X) 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:

(3)(y 𝒞) /X : 𝒞 /X PSh(𝒞) /y 𝒞(X) c X (y 𝒞(c) y 𝒞(c X) y 𝒞(X)) \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

Proposition

The following anti-parallel functors constitute an adjoint equivalence

Here:

  1. the top functor LL 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 that it is given by a pullback of the hom-functor in PSh(𝒞)PSh(\mathcal{C}) itself:

    (4)PSh(𝒞) /y 𝒞(X)((y 𝒞) /X(UϕX),(Epy 𝒞(X)))=PSh(𝒞)(y 𝒞(U),E)×PSh(𝒞)(y 𝒞(U),y 𝒞(X)){y 𝒞(ϕ)} PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)} \big( (y_{\mathcal{C}})_{/X}(U \xrightarrow{\phi} X) ,\, (E \xrightarrow{p} y_{\mathcal{C}}(X) ) \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:

PSh(𝒞 /X)(A,R(B)) =PSh(𝒞 /X)(A,PSh(𝒞) /y 𝒞(X)((y 𝒞) /X(),B)) PSh(𝒞 /X)(limc XAy (𝒞 /X)(c X),PSh(𝒞) /y 𝒞(X)((y 𝒞) /X(),B)) limc XAPSh(𝒞 /X)(y (𝒞 /X)(c X),PSh(𝒞) /y 𝒞(X)((y 𝒞) /X(),B)) limc XAPSh(𝒞) /y 𝒞(X)((y 𝒞) /X(c X),B) PSh(𝒞) /y 𝒞(X)(limc XA(y 𝒞) /X(c X),B) =PSh(𝒞) /y 𝒞(X)(L(A),B) \begin{aligned} PSh(\mathcal{C}_{/X}) \big( A ,\, R(B) \big) & \;=\; PSh(\mathcal{C}_{/X}) \Big( A ,\, PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)} ( (y_{\mathcal{C}})_{/X}(-) ,\, B ) \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)} ( (y_{\mathcal{C}})_{/X}(-) ,\, B ) \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)} ( (y_{\mathcal{C}})_{/X}(-) ,\, B ) \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 APSh(𝒞 /X)A \,\in\, PSh\big( \mathcal{C}_{/X}\big):

RL(A) PSh(𝒞) /y 𝒞(X)((y 𝒞) /X(),L(A)) PSh(𝒞) /y 𝒞(X)((y 𝒞) /X(),limc XA(y 𝒞) /X(c X)) limc XAy (𝒞 /X)(c X) A \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 𝒞\mathcal{C}, to which applies by observing that that:

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

    2. colimits of presheaves are computed objectwise,

    3. the slice hom is a pullback of the plain hom (4),

    4. colimits in a topos such as PSh(𝒞)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 BPSh(𝒞) /y 𝒞(X)B \,\in\, PSh(\mathcal{C})_{/y_{\mathcal{C}}(X)}:

LR(B) L(limy (𝒞 /X)(c X)R(B)y (𝒞 /X)(c X)) limy (𝒞 /X)(c X)R(B)(y 𝒞) /X(c X) limL(y (𝒞 /X)(c X))B(y 𝒞) /X(c X) lim(y 𝒞) /X(c X)B(y 𝒞) /X(c X) B \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)R(B),

  • the second step unwinds the definition of LL from above,

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

  • the fourth step applies again the definition of LL from above,

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

In enriched category theory

For 𝒱\mathcal{V} any Bénabou cosmos for enriched category theory, the statement and proof of Prop. holds and applies verbatim also in 𝒱\mathcal{V}-enriched category theory for enriched presheaves and enriched slice categories (with the colimit-of-representables-expression for enriched presheaves now being the corresponding coend, as discussed at co-Yoneda lemma).

Example

(for simplicial presheaves)
With Bénabou cosmos 𝒱=\mathcal{V} \,=\, sSet being the category of simplicial sets with its cartesian monoidal category-structure (see at products of simplicial sets), the enriched presheaves are simplicial presheaves over simplicial sites 𝒞\mathcal{C}. With categories of simplicial presheaves denoted sPSh()sPSh(-), Prop. reads:

In simplicial model category theory

For

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 QusSet_{Qu}-enriched model category sSh(𝒞)sSh(\mathcal{C}).

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

Together this implies that their composite (4) 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 fsPsh(𝒞) /y 𝒞(X)f \,\in\, sPsh(\mathcal{C})_{/y_{\mathcal{C}}(X)} between fibrations such that for all UϕXU \xrightarrow{\phi} X in 𝒞 /X\mathcal{C}_{/X} the base change (4) of its values on U𝒞U \,\in\, \mathcal{C} is a weak equivalence:

Here the right vertical morphisms are Kan fibrations by the fact that sPSh(𝒞)(y 𝒞(U),)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 ϕ\phi, by assumption – this Prop. implies that f(U)PSh(𝒞)(y 𝒞(U),f) f(U) \,\simeq\, \mathrm{PSh}(\mathcal{C}) \big( y_{\mathcal{C}}(U) ,\, f \big) is a weak equivalence. And since this holds for all U𝒞U \,\in\, \mathcal{C}, this means that ff is a weak equivalence in the slice of the projective model structure.

In \infty-category theory

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:

Proposition

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

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

Example

(cohesion of global- over G-equivariant homotopy theory)

In the case that C=SnglrtGrpd 1,1 fin\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 GG-orbi-singularity, for some equivariance group GG. Together with the adjoint quadruple that is induced (see here) via \infty -Kan extension from the reflection onto the GG-orbit category, this implies the cohesion of global- over G-equivariant homotopy theory. See there for more.

References

Textbook accounts for the statement in plain category theory:

Discussion in \infty -category theory via quasi-categories:

Last revised on October 11, 2021 at 12:21:18. See the history of this page for a list of all contributions to it.