nLab (infinity,1)-site

Idea

The structure of an $(\infty,1)$-site on an (∞,1)-category $C$ is precisely the data encoding an (∞,1)-category of (∞,1)-sheaves

$$Sh(C) \hookrightarrow PSh(C)$$

inside the (∞,1)-category of (∞,1)-presheaves on $C$.

The notion is the analog in (∞,1)-category theory of the notion of a site in 1-category theory.

Definition

The definition of $(\infty,1)$-sites parallels that of 1-categorical sites closely. In fact the structure of an $(\infty,1)$-site on an $(\infty,1)$-category is equivalent to that of a 1-categorical site on its homotopy category (see below).

Definition

($(\infty,1)$-Grothendieck topology)

A sieve in an (∞,1)-category $C$ is a full sub-(∞,1)-category $D \subset C$ which is closed under precomposition with morphisms in $C$.

A sieve on an object $c \in C$ is a sieve in the overcategory $C_{/c}$.

Equivalently, a sieve on $c$ is an equivalence class of monomorphisms $U \to j(c)$ in the (∞,1)-category of (∞,1)-presheaves $PSh(C)$, with $j : C \to PSh(C)$ the (∞,1)-Yoneda embedding. (See below for the proof of this equivalence).

For $S$ a sieve on $c$ and $f : d \to c$ a morphism into $c$, we take the pullback sieve $f^* S$ on $d$ to be that spanned by all those morphisms into $d$ that become equivalent to a morphism in $S$ after postcomposition with $f$.

A Grothendieck topology on the $(\infty,1)$-category $C$ is the specification of a collection of sieves on each object of $C$ – called the covering sieves , subject to the following conditions:

1. the trivial sieve covers – For each object $c \in C$ the overcategory $C_{/c}$ regarded as a maximal subcategory of itself is a covering sieve on $c$. Equivalently: the monomorphism $Id : j(c) \to j(c)$ covers.

2. the pullback of a sieve covers – If $S$ is a covering sieve on $c$ and $f : d \to c$ a morphism, then the pullback sieve $f^* S$ is a covering sieve on $d$. Equivalently, the pullback

   $$\array{ f^* U &\to& U \\ \downarrow && \downarrow \\ d &\stackrel{f}{\to}& c }$$

   in $PSh(C)$ is covering.

3. a sieve covers if its pullbacks cover – For $S$ a covering sieve on $c$ and $T$ any sieve on $c$, if the pullback sieve $f^* T$ for every $f \in S$ is covering, then $T$ itself is covering.

An $(\infty,1)$-category equipped with a Grothendieck topology is an $(\infty,1)$-site.

Properties

Of sieves

Lemma

A sieve $S'$ on $c$ that contains a covering sieve $S \subset S'$ is itself covering.

Proof

For every $f : d \to c$ an object of $S \subset C_{/c}$, the pullback sieve $f^* S'$ equals the pullback sieve $f^* S$. So it covers $d$ by the second axiom on sieves. So by the third axiom $S'$ itself is covering.

Proposition

There is a natural bijection between sieves on $c$ in $C$ and equivalence class of monomorphisms $U \to j(C)$ in $PSh(C)$.

This is HTT, prop. 6.2.2.5.

Proof

First observe that equivalence classes of $(-1)$-truncated object of $PSh(C_{/c})$ are in bijection with sieves on $c$:

An $(\infty,1)$-presheaf $F$ is $(-1)$-truncated if its value on any object is either the empty ∞-groupoid $\emptyset$ or a contractible $\infty$-groupoid. The full subcategory of $C_{/c}$ on those objects on which $F$ takes a contractible value is evidently a sieve (because there is no morphism from a contractible to the empty $\infty$-groupoid). Conversely, given a sieve $S$ on $c$ we obtain a (-1)-truncated presheaf fixed by the demand that it takes the value $* = \Delta[0] \in \infty Grpd$ on those objects that are in $S$, and $\emptyset$ otherwise.

Now, as described at Interaction of presheaves and overcategories we have an equivalence

$$PSh(C_{/c}) \simeq PSh(C)_{/j(c)} \,.$$

Under this equivalence our bijection above maps to the statement that there is a bijection between sieves on $c$ and equivalence class of $(-1)$-truncated objects in $PSh(C)_{/j (c)}$. But such a (-1)-truncated object is precisely a monomorphism $U \to j(c)$.

Of coverages

Lemma

The set of Grothendieck topologies on an $(\infty,1)$-category $C$ is in natural bijection with the set of Grothendieck topologies on its homotopy category.

This is HTT, remark 6.2.2.3.

Proof

Because picking full sub-1-categories as well as full sub-$(\infty,1)$-categories amounts to picking sub-sets/sub-classes of the set of equivalence classes of objects.

Corollary

If the $(\infty,1)$-category $C$ happens to be an ordinary category (for instance in its incarnation as a quasi-category it is the nerve of an ordinary category), then the structure of an $(\infty,1)$-site on it is the same as the 1-categorical structure of a site on it.

Of sites

Proposition

Structures of $(\infty,1)$-sites on an (∞,1)-category $C$ correspond bijectively to topological localizations of the (∞,1)-category of (∞,1)-presheaves to a (∞,1)-category of (∞,1)-sheaves. See there for more details.

Incarnations and models

If (∞,1)-categories are incarnated as simplicially enriched categories, then an $(\infty,1)$-site appears as an

• sSet-site

If $(\infty,1)$-categories are presented by model categories, then the notion of $(\infty,1)$-site appears as that of

• model site.

Examples

• The trivial Grothendieck-topology on an $(\infty,1)$-category is that where the only covering sieve on each object $c$ is $C_{/c}$ itself. Equivalently, where the only covering monomorphisms $U \to j(c)$ in $PSh(C)$ are the equivalences.

  The (∞,1)-category of (∞,1)-sheaves on this site is just the (∞,1)-category of (∞,1)-presheaves itself. The localization is an equivalence.

• étale (∞,1)-site

Related concepts

• site

• 2-site, (2,1)-site

• $(\infty,1)$-site

  • model site, simplicial site

  infinity-cohesive site

• internal site / internal (infinity,1)-site?

References

• Jacob Lurie, Section 6.2.2 of: Higher Topos Theory (2009)

• Raffael Stenzel, Notions of $(\infty,1)$-sites and related formal structures [arXiv:2306.06619]

• Raffael Stenzel, Higher sites and their higher categorical logic, talk at HoTT Electronic Seminar (18 November 2021) [video:YT, slides:pdf]