nLab
(infinity,1)-site

Idea

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

$$\mathrm{Sh}(C)\hookrightarrow \mathrm{PSh}(C)$$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 $(\mathrm{\infty },1)$-sites parallels that of 1-categorical sites closely. In fact the structure of an $(\mathrm{\infty },1)$-site on an $(\mathrm{\infty },1)$-category is equivalent to that of a 1-categorical site on its homotopy category (see below).

Definition

($(\mathrm{\infty },1)$-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 $\mathrm{PSh}(C)$, with $j:C\to \mathrm{PSh}(C)$ 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 $(\mathrm{\infty },1)$-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 $\mathrm{Id}:j(c)\to j(c)$ 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

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

   in $\mathrm{PSh}(C)$ 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 $(\mathrm{\infty },1)$-category equipped with a Grothendieck topology is an $(\mathrm{\infty },1)$-site.

Properties

Of sieves

Lemma

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

Proof

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

Proposition

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

This is HTT, prop. 6.2.2.5.

Proof

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

An $(\mathrm{\infty },1)$-presheaf $F$ is $(-1)$-truncated if its value on any object is either the empty ∞-groupoid $\varnothing$ or a contractible $\mathrm{\infty }$-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 $\mathrm{\infty }$-groupoid). Conversely, given a sieve $S$ on $c$ we obtain a (-1)-truncated presheaf fixed by the demand that it takes the value $*=\Delta [0]\in \mathrm{\infty }\mathrm{Grpd}$ on those objects that are in $S$, and $\varnothing$ otherwise.

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

$$\mathrm{PSh}(C_{/c})\simeq \mathrm{PSh}(C{)}_{/j(c)}.$$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 $\mathrm{PSh}(C{)}_{/j(c)}$. But such a (-1)-truncated object is precisely a monomorphism $U\to j(c)$.

Of coverages

Observation

The set of Grothendieck topologies on an $(\mathrm{\infty },1)$-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-$(\mathrm{\infty },1)$-categories amounts to picking sub-sets/sub-classes of the set of equivalence classes of objects.

Corollary

If the $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-site on it is the same as the 1-categorical structure of a site on it.

Of sites

Proposition

Structures of $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-site appears as an

• sSet-site

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

• model site.

Examples

• The trivial Grothendieck-topology on an $(\mathrm{\infty },1)$-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 $\mathrm{PSh}(C)$ 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

• $(\mathrm{\infty },1)$-site

  • model site, simplicial site

  infinity-cohesive site

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

References

Section 6.2.2 of

• Jacob Lurie, Higher Topos Theory