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

Definition

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

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

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

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

Properties

Models

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

Proposition

For $C$ a simplicially enriched category with Kan complexes as hom-objects, write $[C^{op}, sSet_{Quillen}]_{proj}$ and $[C^{op}, sSet_{Quillen}]_{inj}$ 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 $N(-)$ for the homotopy coherent nerve that sends a Kan-complex enriched category to a quasi-category.

Then there is an equivalence of quasi-categories

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

Similarly for the injective model structure.

Proof

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 $(\infty,1)$-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 $(\infty,1)$-category of $(\infty,1)$-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.

Corollary

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

See around HTT, cor. 5.1.2.4.

As the free completion under colimits

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

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

Lemma

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

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

This is HTT, lemma 5.1.5.3.

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

Lemma

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

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

This is HTT, theorem 5.1.5.6.

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.

Definition

Let $A$ and $B$ be model categories, $D$ a plain category and

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

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

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

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

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

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

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

commutes.

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 $L$ presents, via its derived functor, a left adjoint (∞,1)-functor that preserves $(\infty,1)$-categorical colimits. So the notion of factorization as above is really about factorizations through colimit-preserving $(\infty,1)$-functors into $(\infty,1)$-categories that have all colimits.

Theorem

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

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

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

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

Proof

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

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

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

$$\Gamma \;\colon\; C \to [\Delta,B]$$

of $\gamma$. Then set

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

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

$$R(X) \colon c \mapsto Hom_B(\Gamma^\bullet(c), X) \,.$$

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

$$Hom_C\big(\Gamma^\bullet(c), b_1 \to b_2\big)$$

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 $\eta \colon L \circ j \simeq \gamma$, write $j : C \to [C^{op},sSet]$ for the Yoneda embedding and notice that by Yoneda reduction it follows that for $x \in C$ we have

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

(where equality signs denote isomorphisms).

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

Corollary

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

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

Proof

This is HTT, corollary 5.1.5.8.

Functorality

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

These two constructions are related.

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

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

Theorem

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

Proof

Consider the (∞,1)-category of possibly large (∞,1)-categories that have small colimits, and functors preserving small colimits. By HTT, 5.3.6.10, the inclusion into the full (∞,1)-category of (∞,1)-categories has a left adjoint $P$, which is characterized by the natural equivalence

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

given by pre-composition with the Yoneda embedding.

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

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

The second equivalence holds because, for presentable $C$, the property of being in $Func^R$ 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 $Func^L(\infty Grpd, C) \simeq Func(1, C)$.

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

Now consider the chain of equivalences

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

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 $S$.

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

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

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

HTT, 5.2.6.3 asserts $P(\phi)$ is left adjoint to $\Func(\phi, \infty Grpd)$, so the left and right arrows are homotopic.

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

Note that a natural automorphism of either functor extending $PSh$ would induce a natural automorphism of the identity functor on $(\infty,1)Cat$. By HTT, 5.2.9.1, the space of such automorphisms is contractible.

Relation to slicing

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

Proposition

(slicing commutes with passing to presheaves)

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

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

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

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

This appears as HTT, 5.1.6.12.

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

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

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

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

Related concepts

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.

$\phantom{A}$(n,r)-categories$\phantom{A}$	$\phantom{A}$toposes$\phantom{A}$	locally presentable	loc finitely pres	localization theorem	free cocompletion	accessible
(0,1)-category theory	locales	suplattice	algebraic lattices	Porst’s theorem	powerset	poset
category theory	toposes	locally presentable categories	locally finitely presentable categories	Gabriel–Ulmer’s theorem	presheaf category	accessible categories
model category theory	model toposes	combinatorial model categories		Dugger's theorem	global model structures on simplicial presheaves	n/a
(∞,1)-category theory	(∞,1)-toposes	locally presentable (∞,1)-categories		Simpson’s theorem	(∞,1)-presheaf (∞,1)-categories	accessible (∞,1)-categories

References

• Dan Dugger, Universal homotopy theories, Advances in Mathematics 164 (2001) 144-176 [arXiv:math/0007070, doi:10.1006/aima.2001.2014]

• Jacob Lurie, section 5.1 in: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press (2009) [pup:8957]