# nLab infinity-cohesive site

### Context

#### Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

## Structures in a cohesive $\left(\infty ,1\right)$-topos

structures in a cohesive (∞,1)-topos

## Structures with infinitesimal cohesion

infinitesimal cohesion

# Contents

## Idea

An $\left(\infty ,1\right)$-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 $\infty$-cohesive over ∞Grpd if it is

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 $\left\{{U}_{i}\to U\right\}$ in $C$

• the Cech nerve $C\left(U\right)\in \left[{C}^{\mathrm{op}},\mathrm{sSet}\right]$ 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)

$\underset{\to }{\mathrm{lim}}C\left(U\right)\stackrel{\simeq }{\to }*$\lim_\to C(U) \stackrel{\simeq}{\to} *
• the simplicial set of points in $C\left(U\right)$ is weakly equivalent to the set of points of $U$:

${\mathrm{Hom}}_{C}\left(*,C\left(U\right)\right)\stackrel{\simeq }{\to }{\mathrm{Hom}}_{C}\left(*,U\right)\phantom{\rule{thinmathspace}{0ex}}.$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 ${\mathrm{Hom}}_{C}\left(*,U\right)$. 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 ${\mathrm{lim}}_{\to }C\left(U\right)$ 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 ${\mathrm{Hom}}_{C}\left(*,C\left(U\right)\right)$ is contractible is the $\infty$-analog of the condition on a concrete site that ${\mathrm{Hom}}_{C}\left(*,{\coprod }_{i}{U}_{i}\right)\to {\mathrm{Hom}}_{C}\left(*,U\right)$ 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 $\infty$-cohesive if it has finite products.

###### Proof

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

###### Examples/Proposition

The following sites are $\infty$-cohesive:

• the category CartSp with covering families given by open covers $\left\{{U}_{i}↪U\right\}$ by convex subsets ${U}_{i}$;

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

• the site ThCartSp $\subset 𝕃$ of smooth loci consisting of smooth loci of the form ${R}^{n}×{D}_{\left(k\right)}^{l}$ with the second factor infinitesimal, where covering families are those of the form $\left\{{U}_{i}×{D}_{\left(k\right)}^{l}\to U×{D}_{\left(k\right)}^{l}\right\}$ with $\left\{{U}_{i}\to U\right\}$ 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 $\left\{{U}_{i}\to U\right\}$ 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 $C\left(U\right)$ are degreewise coproducts of representables.

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

## Properties

###### Theorem

Let $C$ be an $\infty$-cohesive site. Then the (∞,1)-sheaf (∞,1)-topos ${\mathrm{Sh}}_{\left(\infty ,1\right)}\left(C\right)$ 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 $C\left(*,U\right)$ is inhabited, then the axiom “pieces have points” also holds.

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

###### Corollary

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

To prove this, we need to show that

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

This follows with the discussion at ∞-connected site.

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

This follows with the discussion at ∞-local site.

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

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

The last two conditions we demonstrate now.

###### Proposition

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

$\Pi \left(A×B\right)\simeq \Pi \left(A\right)×\Pi \left(B\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 ${\mathrm{lim}}_{\to }:{\mathrm{PSh}}_{\left(\infty ,1\right)}\left(C\right)\to \infty \mathrm{Grpd}$. 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 ${\mathrm{Sh}}_{\left(\infty ,1\right)}\left(C\right)$ 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 $\left[{C}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{proj},\mathrm{loc}}$. 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

$\cdots \stackrel{\to }{\stackrel{\to }{\to }}\coprod _{{U}_{0}\to {U}_{1}\to {X}_{1}}U\stackrel{\to }{\to }\coprod _{U\to {X}_{0}}U\phantom{\rule{thinmathspace}{0ex}},$\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

$\cdots \stackrel{\to }{\stackrel{\to }{\to }}\coprod _{{U}_{0}\to {U}_{1}\to {X}_{0}}C\left(*,{U}_{0}\right)\stackrel{\to }{\to }\coprod _{U\to {X}_{0}}C\left(*,U\right)\phantom{\rule{thinmathspace}{0ex}},$\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

$\cdots \stackrel{\to }{\stackrel{\to }{\to }}\coprod _{{U}_{0}\to {U}_{1}\to {X}_{0}}*\stackrel{\to }{\to }\coprod _{U\to {X}_{0}}*\phantom{\rule{thinmathspace}{0ex}}.$\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 $C\left(*,U\right)$ to the point. Since by assumption each $C\left(*,U\right)$ is nonempty, this is componentwise an epi. Hence the whole morphism is an epi on ${\pi }_{0}$.

and

Revised on April 24, 2012 09:34:39 by David Corfield (129.12.18.29)