nLab (infinity,1)-category of (infinity,1)-presheaves



(,1)(\infty,1)-Category theory

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos



For ∞Grpd the (∞,1)-category of ∞-groupoids, and for SS a (∞,1)-category (or in fact any simplicial set), an (,1)(\infty,1)-presheaf on SS is an (,1)(\infty,1)-functor

F:S opGrpd. F : S^{op} \to \infty Grpd \,.

The (,1)(\infty,1)-category of (,1)(\infty,1)-presheaves is the (∞,1)-category of (∞,1)-functors

PSh (,1)(S):=Func(S op,Grpd). PSh_{(\infty,1)}(S) := Func(S^{op}, \infinity Grpd) \,.



A model for an (,1)(\infty,1)-presheaf categories is the model structure on simplicial presheaves. See also the discussion at models for ∞-stack (∞,1)-toposes.


For CC a simplicially enriched category with Kan complexes as hom-objects, write [C op,sSet Quillen] proj[C^{op}, sSet_{Quillen}]_{proj} and [C op,sSet Quillen] inj[C^{op}, sSet_{Quillen}]_{inj} for the projective or injective, respectively, global model structure on simplicial presheaves. Write () (-)^\circ for the full sSet-enriched subcategory on fibrant-cofibrant objects, and N()N(-) for the homotopy coherent nerve that sends a Kan-complex enriched category to a quasi-category.

Then there is an equivalence of quasi-categories

PSh(N(C))N([C op,sSet Quillen] proj) . PSh(N(C)) \simeq N ([C^{op}, sSet_{Quillen}]_{proj})^\circ \,.

Similarly for the injective model structure.


This is a special case of the more general statement that the model structure on functors models an (∞,1)-category of (∞,1)-functors. See there for more details.

Notice that the result in particular means that any (,1)(\infty,1)-presheaf – an “\infty-pseudofunctor” – may be straightened or rectified to a genuine sSet-enriched functor, that respects horizontal compositions strictly.

Limits and colimits

In an ordinary category of presheaves, limits and colimits are computed objectwise, as described at limits and colimits by example. The analogous statement is true for (∞,1)-limits and colimits in an (,1)(\infty,1)-category of (,1)(\infty,1)-presheaves.

This is a special case of the general existence of limits and colimits in an (∞,1)-category of (∞,1)-functors. See there for more details.


For CC a small (,1)(\infty,1)-category, the (,1)(\infty,1)-category PSh(C)PSh(C) admits all small limits and colimits.

See around HTT, cor.

As the free completion under colimits

An ordinary category of presheaves on a small category CC is the free cocompletion of CC, the free completion under forming colimits.

The analogous result holds for (,1)(\infty,1)-category of (,1)(\infty,1)-presheaves.


Let CC be a small quasi-category and j:SPSh(C)j \colon S \to PSh(C) the (∞,1)-Yoneda embedding.

The identity (∞,1)-functor Id:PSh(C)PSh(C)Id \,\colon\, PSh(C) \to PSh(C) is the left (∞,1)-Kan extension of jj along itself.

This is HTT, lemma

For DD a quasi-category with all small colimits, write Func L(PSh(C),D)Func(PSh(C),D)Func^L(PSh(C),D) \subset Func(PSh(C),D) for the full sub-quasi-category of the (∞,1)-category of (∞,1)-functors on those that preserve small colimits.


Pre-composition with the Yoneda embedding j:CPSh(C)j \colon C \to PSh(C) induces an equivalence of quasi-categories

Func L(PSh(C),D)Func(C,D). Func^L(PSh(C),D) \to Func(C,D) \,.

This is HTT, theorem

In terms of the model given by the model structure on simplicial presheaves, this is statement made in Dugger (2001), which gives that article its name.


Let AA and BB be model categories, DD a plain category and

D r A γ B \array{ D &\overset{r}{\longrightarrow}& A \\ & \mathllap{{}_{\gamma}}\searrow \\ && B }

two plain functors. Say that a model-category theoretic factorization of γ\gamma through AA is

  1. a Quillen adjunction (LR):ARLB(L \dashv R) : A \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}} B

  2. a natural weak equivalence η:Lrγ\eta : L \circ r \to \gamma

    D r A γ η L B. \array{ D &&\stackrel{r}{\to}&& A \\ & \mathllap{{}_\gamma}\searrow &{}^\eta\swArrow & \swarrow_{\mathrlap{L}} \\ && B } \,.

Let the category of such factorizations have morphisms ((LR),η)((LR),η)\big((L \dashv R), \eta \big) \to \big((L' \dashv R'\big), \eta' ) given by natural transformations LLL \to L' such that for all all objects dDd \in D the diagrams

Lr(d) Lr(d) η d η d γ() \array{ L\circ r(d) &&\longrightarrow&& L'\circ r(d) \\ & {}_{\eta_{d}}\searrow && \swarrow_{\eta'_{d}} \\ && \gamma() }


Notice that the (∞,1)-category presented by a model category – at least by a combinatorial model category – has all (∞,1)-categorical colimits, and that the Quillen left adjoint functor LL presents, via its derived functor, a left adjoint (∞,1)-functor that preserves (,1)(\infty,1)-categorical colimits. So the notion of factorization as above is really about factorizations through colimit-preserving (,1)(\infty,1)-functors into (,1)(\infty,1)-categories that have all colimits.


(model category presentation of free (,1)(\infty,1)-cocompletion)

For CC a small category, the projective global model structure on simplicial presheaves [C op,sSet] proj[C^{op}, sSet]_{proj} on CC is universal with respect to such factorizations of functors out of CC:

every functor CBC \to B to any model category BB has a factorization through [C op,sSet] proj[C^{op}, sSet]_{proj} as above, and the category of such factorizations is contractible.

This is Dugger (2001), theorem 1.1, where the proof appears on page 30.


To produce the factorization [C op,sSet]B[C^{op},sSet] \to B given the functor γ\gamma, first notice that the ordinary Yoneda extension [C op,Set]B[C^{op},Set] \to B would be given by the left Kan extension given by the coend formula

F cCγ(c)F(c), F \mapsto \int^{c \in C} \gamma(c) \cdot F(c) \,,

where the dot in the integrand is the tensoring of cocomplete category BB over Set. To refine this to a left Quillen functor L:[C op,sSet]BL : [C^{op},sSet] \to B, choose a cosimplicial resolution

Γ:C[Δ,B] \Gamma \;\colon\; C \to [\Delta,B]

of γ\gamma. Then set

L:F cC [n]ΔΓ n(c)F n(c). L \;\colon\; F \mapsto \int^{c \in C} \int^{[n] \in \Delta} \Gamma^n(c) \cdot F_n(c) \,.

The right adjoint R:B[C op,sSet]R \colon B \to [C^{op},sSet] of this functor is given by

R(X):cHom B(Γ (c),X). R(X) \colon c \mapsto Hom_B(\Gamma^\bullet(c), X) \,.

For (LR):[C op,sSet] projB(L \dashv R) \colon [C^{op}, sSet]_{proj} \stackrel{\to}{\leftarrow} B to be a Quillen adjunction, it is sufficient to check that RR preserves fibrations and acyclic fibrations. By definition of the projective model structure this means that for every (acyclic) fibration b 1b 2b_1 \to b_2 in BB we have for every object cCc \in C that that

Hom C(Γ (c),b 1b 2) Hom_C\big(\Gamma^\bullet(c), b_1 \to b_2\big)

is an (acyclic) fibration of simplicial sets. But this is one of the standard properties of cosimplicial [[resolutions.

Finally, to find the natural weak equivalence η:Ljγ\eta \colon L \circ j \simeq \gamma, write j:C[C op,sSet]j : C \to [C^{op},sSet] for the Yoneda embedding and notice that by Yoneda reduction it follows that for xCx \in C we have

L(j(x))= cC [n]ΔΓ n(c)C(c,x)=Γ 0(x) L(j(x)) \;=\; \int^{c \in C} \int^{[n] \in \Delta} \Gamma^n(c) \cdot C(c,x) \;=\; \Gamma^0(x)

(where equality signs denote isomorphisms).

By the very definition of cosimplicial resolutions, there is a natural weak equivalence Γ(x)\Gamma(x) \stackrel{\simeq}{\to}. We can take this to be the component of η\eta.


The (∞,1)-Yoneda embedding j:CPSh(C)j \colon C \to PSh(C) generates PSh(C)PSh(C) under small colimits:

a full (∞,1)-subcategory of PSh(C)PSh(C) that contains all representables and is closed under forming (,1)(\infty,1)-colimits is already equivalent to PSh(C)PSh(C).


This is HTT, corollary


The two descriptions of PSh(S)PSh(S) as the functor category Func(S op,Grpd)Func(S^{op}, \infty Grpd) and as the free cocompletion SPSh(S)S \to PSh(S) (above) both extend to contravariant and covariant functors on ( , 1 ) Cat (\infty, 1)Cat .

These two constructions are related.

Func(,Grpd):(,1)Cat op(,1)Cat^Func(-, \infty Grpd) \colon (\infty, 1)Cat^{op} \to (\infty, 1)\widehat{Cat} takes values in the subcategory of presentable (∞,1)-categories and functors preserving small limits and colimits, and thus having left and right adjoints.

Let Pr L(,1)CatPr^L(\infty,1)Cat be the (∞,1)category of presentable (∞,1)-categories and functors that are left adjoints, and similarly for Pr R(,1)CatPr^R(\infty,1)Cat.


The local left adjoint to Func(,Grpd):(,1)Cat opPr R(,1)CatFunc(-, \infty Grpd) \colon (\infty, 1)Cat^{op} \to Pr^R(\infty,1)Cat is naturally equivalent to the free cocompletion functor P:(,1)CatPr L(,1)CatP \colon (\infty, 1)Cat \to Pr^L(\infty,1)Cat by a natural equivalence whose components at a small (∞,1)-category SS are homotopic to the identity functor on PSh(S)PSh(S).


Consider the (∞,1)-category of possibly large (∞,1)-categories that have small colimits, and functors preserving small colimits. By HTT,, the inclusion into the full (∞,1)-category of (∞,1)-categories has a left adjoint PP, which is characterized by the natural equivalence

Func L(P(S),)Func(S,) Func^L\big(P(S), -\big) \simeq Func(S, -)

given by pre-composition with the Yoneda embedding.

Let Func(,Grpd) ladjFunc(-, \infty Grpd)^{ladj} denote the local left adjoint to Func(,Grpd)Func(-, \infty Grpd). Then for CPr L(,1)CatC \in Pr^L(\infty,1)Cat and S(,1)CatS \in (\infty,1)Cat, there are natural equivalences

Func L(Func(S op,Grpd) ladj,C) Func R(C radj,Func(S op,Grpd)) op Func(S op,Func R(C radj,Grpd)) op Func(S,Func L(Grpd,C)) Func(S,C) \begin{aligned} Func^L\big(Func(S^{op}, \infty Grpd)^{ladj}, C\big) &\simeq Func^R\big(C^{radj}, Func(S^{op}, \infty Grpd)\big)^{op} \\ &\simeq Func\big(S^{op}, Func^R(C^{radj}, \infty Grpd)\big)^{op} \\&\simeq Func\big(S, Func^L(\infty Grpd, C)\big) \\&\simeq Func(S, C) \end{aligned}

The second equivalence holds because, for presentable CC, the property of being in Func RFunc^R is characterized by being accessible and preserving small limits, which can be determined pointwise.

The first and third equivalences are general properties of local left adjoints and opposite functor categories.

The last equivalence is because Func L(Grpd,C)Func(1,C)Func^L(\infty Grpd, C) \simeq Func(1, C).

Since both constructions corepresent the same functor Func(S,)Func(S, -), the yoneda lemma ensures there is a natural equivalence P(S)Func(S op,Grpd) ladj P(S) \simeq Func(S^{op}, \infty Grpd)^{ladj} .

Now consider the chain of equivalences

Func(S,C)Func L(Func(S op,Grpd) ladj,C)Func L(P(S),C)Func(S,C), Func(S, C) \leftarrow Func^L\big(Func(S^{op}, \infty Grpd)^{ladj}, C\big) \to Func^L\big(P(S), C\big) \to Func(S, C) \mathrlap{\,,}

where the first map is inverse to composition with the Yoneda embedding, the second map is the equivalence constructed previously, and the third map is composition with the Yoneda embedding.

Be careful to note that the first equivalence has not yet been shown to be natural in SS.

The overall composite is essentially uniquely determined by an automorphism of α S\alpha_S of SS. It remains to be shown that α S\alpha_S is an identity.

When S=Δ nS = \Delta^n, SS has no nontrivial automorphisms, so α Δ n\alpha_{\Delta^n} is the identity.

For a functor ϕ:Δ nS\phi : \Delta^n \to S. The naturality of the above equivalence gives a commutative square

HTT, asserts P(ϕ)P(\phi) is left adjoint to Func(ϕ,Grpd)\Func(\phi, \infty Grpd), so the left and right arrows are homotopic.

We’ve already seen the top arrow is homotopic to the identity, and so we infer α Sϕϕ\alpha_S \phi \simeq \phi. Since the Δ n\Delta^n generate (,1)Cat(\infty,1)Cat, α S\alpha_S is thus homotopic to the identity.

Note that a natural automorphism of either functor extending PShPSh would induce a natural automorphism of the identity functor on (,1)Cat(\infty,1)Cat. By HTT,, the space of such automorphisms is contractible.

Relation to slicing

The following analog of the corresponding result for 1-categories of presheaves holds for (,1)(\infty,1)-presheaves. See functors and comma categories.


(slicing commutes with passing to presheaves)

Let 𝒞\mathcal{C} be a small (∞,1)-category and p:𝒦𝒞p \colon \mathcal{K} \to \mathcal{C} a diagram.

Write 𝒞 /p\mathcal{C}_{/p} and PSh (𝒞)/ ypPSh_\infty(\mathcal{C})/_{y p} for the corresponding over categories, where y:𝒞PSh (𝒞)y \colon \mathcal{C} \to PSh_\infty(\mathcal{C}) is the (∞,1)-Yoneda embedding.

Then we have an equivalence of (∞,1)-categories

PSh (𝒞 /p)PSh (𝒞) /yp. PSh_\infty(\mathcal{C}_{/p}) \stackrel{\simeq}{\to} PSh_\infty(\mathcal{C})_{/y p} \,.

This appears as HTT,

(,1)(\infty,1)-subcategories of ()(\infty)-presheaf categories

Locally presentable (,1)(\infty,1)-categories

A reflective (∞,1)-subcategory of an (,1)(\infty,1)-category of (,1)(\infty,1)-presheaves is called a presentable (∞,1)-category.

(,1)(\infty,1)-Sheaf (,1)(\infty,1)-categories

If that left adjoint (∞,1)-functor to the embedding of the reflective (∞,1)-subcategory furthermore preserves finite limits, then the subcategory is an (∞,1)-category of (∞,1)-sheaves: an (∞,1)-topos

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

A\phantom{A}(n,r)-categoriesA\phantom{A}A\phantom{A}toposesA\phantom{A}locally presentableloc finitely preslocalization theoremfree cocompletionaccessible
(0,1)-category theorylocalessuplatticealgebraic latticesPorst’s theorempowersetposet
category theorytoposeslocally presentable categorieslocally finitely presentable categoriesAdámek-Rosický‘s theorempresheaf categoryaccessible categories
model category theorymodel toposescombinatorial model categoriesDugger's theoremglobal model structures on simplicial presheavesn/a
(∞,1)-category theory(∞,1)-toposeslocally presentable (∞,1)-categoriesSimpson’s theorem(∞,1)-presheaf (∞,1)-categoriesaccessible (∞,1)-categories


Last revised on June 18, 2023 at 09:59:00. See the history of this page for a list of all contributions to it.