nLab
infinity-cohesive site

Idea

An $(\mathrm{\infty },1)$-cohesive site is a site such that the (∞,1)-category of (∞,1)-sheaves over it is a cohesive (∞,1)-topos.

Definition

Definition

A site $C$ is $\mathrm{\infty }$-cohesive over ∞Grpd if it is

• a strongly ∞-connected site

• and an ∞-local site.

In detail this means that $C$ is

• a site – a small category $C$ equipped with a coverage;

• with the property that

  • it has a terminal object $*$;

  • it is a cosifted category (for instance in that it has all finite products);

  • for every covering family $\{U_i\to U\}$ in $C$

    • the Cech nerve $C(U)\in [C^{\mathrm{op}},\mathrm{sSet}]$ is degreewise a coproduct of representables;

    • the simplicial set obtained by replacing each copy of a representable by a point is contractible (weakly equivalent to the point in the standard model structure on simplicial sets)

      $$\underrightarrow{\lim }C(U)\stackrel{\simeq }{\to }*$$\lim_\to C(U) \stackrel{\simeq}{\to} *

    • the simplicial set of points in $C(U)$ is weakly equivalent to the set of points of $U$:

      $${\mathrm{Hom}}_C(*,C(U))\stackrel{\simeq }{\to }{\mathrm{Hom}}_C(*,U).$$Hom_C(*, C(U)) \stackrel{\simeq}{\to} Hom_C(*,U) \,.

Remark

These conditions are stronger than for a cohesive site, which only guarantees cohesiveness of the 1-topos over it.

This definition is supposed to model the following ideas:

• every object $U$ has an underlying set of points ${\mathrm{Hom}}_C(*,U)$. We may think of each $U$ as specifying one way in which there can be cohesion on this underlying set of points;

• in view of the nerve theorem the condition that ${\lim }_{\to }C(U)$ is contractible means that $U$ itself is contractible, as seen by the Grothendieck topology on $C$. This reflects the local aspect of cohesion: we only specify cohesive structure on contractible lumps of points;

• in view of this, the remaining condition that ${\mathrm{Hom}}_C(*,C(U))$ is contractible is the $\mathrm{\infty }$-analog of the condition on a concrete site that ${\mathrm{Hom}}_C(*,{\coprod }_i{U}_i)\to {\mathrm{Hom}}_C(*,U)$ is surjective. This expresses that the notion of topology on $C$ and its concreteness over Set are consistent.

Examples

Example

The site for a presheaf topos, hence with trivial topology, is $\mathrm{\infty }$-cohesive if it has finite products.

Proof

All covers $\{U_i\to U\}$ consist of only the identity morphism $\{U\stackrel{\mathrm{Id}}{\to }U\}$. The Cech $C\{U\}$ is then the simplicial object constant on $U$ and hence satisfies its two conditions above trivially.

Examples/Proposition

The following sites are $\mathrm{\infty }$-cohesive:

• the category CartSp with covering families given by open covers $\{U_i\hookrightarrow U\}$ by convex subsets $U_i$;

  we can take the morphisms ${ℝ}^k\to {ℝ}^l$ in $\mathrm{CartSp}$ to be

  • continuous maps

    – in which case the sheaf topos over it models generalized topological spaces, the 2-sheaf 2-topos contains for instance topological stacks;

  • or smooth maps

    – in which case the sheaf topos models generalized smooth spaces such as diffeological spaces, the (∞,1)-sheaf (∞,1)-topos is that of ∞-Lie groupoids;

• the site ThCartSp $\subset 𝕃$ of smooth loci consisting of smooth loci of the form $R^n\times D_{(k)}^l$ with the second factor infinitesimal, where covering families are those of the form $\{U_i\times D_{(k)}^l\to U\times D_{(k)}^l\}$ with $\{U_i\to U\}$ a covering family in $\mathrm{CartSp}$ as above.

  This is a site of definition for the Cahiers topos.

More discussion of these two examples is at ∞-Lie groupoid and ∞-Lie algebroid.

Proof

Since every star-shaped region in ${ℝ}^n$ is diffeomorphic to an open ball (see there for details) we have that the covers $\{U_i\to U\}$ on CartSp by convex subsets are good open covers in the strong sense that any finite non-empty intersection is diffeomorphic to an open ball and hence diffeomorphic to a Cartesian space. Therefore these are good open covers in the strong sense of the term and their Cech nerves $C(U)$ are degreewise coproducts of representables.

The fact that ${\lim }_{\to }C(U)\simeq *$ follows from the nerve theorem, using that a Cartesian space regarded as a topological space is contractible.

Properties

Theorem

Let $C$ be an $\mathrm{\infty }$-cohesive site. Then the (∞,1)-sheaf (∞,1)-topos ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)$ over $C$ is a cohesive (∞,1)-topos that satisfies the axiom “discrete objects are concrete” .

If moreover for all objects $U$ of $C$ we have that $C(*,U)$ is inhabited, then the axiom “pieces have points” also holds.

Since the (n,1)-topos over a site for any $n\in \mathbb{N}$ arises as the full sub-(∞,1)-category of the $(\mathrm{\infty },1)$-topos on the $n$-truncated objects and since the definition of cohesive $(n,1)$-topos is compatible with such truncation, it follows that

Corollary

Let $C$ be an $\mathrm{\infty }$-cohesive site. Then for all $n\in \mathbb{N}$ the (n,1)-topos ${\mathrm{Sh}}_{(n,1)}(C)$ is cohesive.

To prove this, we need to show that

1. ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)$ is a locally ∞-connected (∞,1)-topos and a ∞-connected (∞,1)-topos.

This follows with the discussion at ∞-connected site.

1. ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)$ is a local (∞,1)-topos.

   This follows with the discussion at ∞-local site.

2. The fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos $\Pi :{\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)\to \mathrm{\infty }\mathrm{Grpd}$ preserves finite (∞,1)-products.

3. If $\Gamma (U)$ is not empty for all $U\in C$, then pieces have points in ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)$.

The last two conditions we demonstrate now.

Proposition

The functor $\mathrm{Pi}:{\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)\to \mathrm{\infty }\mathrm{Grpd}$ whose existence is guaranteed by the above proposition preserves products:

$$\Pi (A\times B)\simeq \Pi (A)\times \Pi (B).$$\Pi(A \times B) \simeq \Pi(A) \times \Pi(B) \,.

Proof

By the discussion at ∞-connected site we have that $\Pi$ is given by the (∞,1)-colimit ${\lim }_{\to }:{\mathrm{PSh}}_{(\mathrm{\infty },1)}(C)\to \mathrm{\infty }\mathrm{Grpd}$. By the assumption that $C$ is a cosifted (∞,1)-category, it follows that this operation preserves finite products.

Finally we prove that pieces have points in ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)$ if all objects of $C$ have points.

Proof

By the above discussion both $\Gamma$ and $\Pi$ are presented by left Quillen functors on the projective model structure $[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}$. By Dugger’s cofibrant replacement theorem (see model structure on simplicial presheaves) we have for $X$ any simplicial presheaf that a cofibrant replacement is given by an object that in the lowest two degrees is

$$\cdots \overrightarrow{\overrightarrow{\to }}\coprod _{U_0\to U_1\to X_1}U\overrightarrow{\to }\coprod _{U\to X_0}U,$$\cdots \stackrel{\to}{\stackrel{\to}{\to}} \coprod_{U_0 \to U_1 \to X_1} U \stackrel{\to}{\to} \coprod_{U \to X_0} U \,,

where the coproduct is over all morphisms out of representable presheaves $U_i$ as indicated.

The model for $\Gamma$ sends this to

$$\cdots \overrightarrow{\overrightarrow{\to }}\coprod _{U_0\to U_1\to X_0}C(*,U_0)\overrightarrow{\to }\coprod _{U\to X_0}C(*,U),$$\cdots \stackrel{\to}{\stackrel{\to}{\to}}\coprod_{U_0 \to U_1 \to X_0} C(*,U_0) \stackrel{\to}{\to} \coprod_{U \to X_0} C(*,U) \,,

whereas the model for $\Pi$ sends this to

$$\cdots \overrightarrow{\overrightarrow{\to }}\coprod _{U_0\to U_1\to X_0}*\overrightarrow{\to }\coprod _{U\to X_0}*.$$\cdots \stackrel{\to}{\stackrel{\to}{\to}}\coprod_{U_0 \to U_1 \to X_0} * \stackrel{\to}{\to} \coprod_{U \to X_0} * \,.

The morphism from the first to the latter is the evident one that componentwise sends $C(*,U)$ to the point. Since by assumption each $C(*,U)$ is nonempty, this is componentwise an epi. Hence the whole morphism is an epi on ${\pi }_0$.

Related concepts

• locally connected topos / locally ∞-connected (∞,1)-topos

  • connected topos / ∞-connected (∞,1)-topos

  • strongly connected topos / strongly ∞-connected (∞,1)-topos

  • totally connected topos / totally ∞-connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos

and

• locally connected site / locally ∞-connected site

  • connected site / ∞-connected site

  • strongly connected site / strongly ∞-connected site

  • totally connected site / totally ∞-connected site

• local site / ∞-local site

• cohesive site, ∞-cohesive site