nLab
infinity-cohesive site

Context

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Contents

Idea

An (,1)(\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 CC is \infty-cohesive over ∞Grpd if it is

In detail this means that CC is

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 UU has an underlying set of points Hom C(*,U)Hom_C(*,U). We may think of each UU 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 C(U)\lim_\to C(U) is contractible means that UU itself is contractible, as seen by the Grothendieck topology on CC. 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 Hom C(*,C(U))Hom_C(*,C(U)) is contractible is the \infty-analog of the condition on a concrete site that Hom C(*, iU i)Hom C(*,U)Hom_C(*,\coprod_i U_i) \to Hom_C(*, U) is surjective. This expresses that the notion of topology on CC 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 {U iU}\{U_i \to U\} consist of only the identity morphism {UIdU}\{U \stackrel{Id}{\to} U\}. The Cech C{U}C\{U\} is then the simplicial object constant on UU 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 {U iU}\{U_i \hookrightarrow U\} by convex subsets U iU_i;

    we can take the morphisms k l\mathbb{R}^k \to \mathbb{R}^l in CartSpCartSp to be

  • the site ThCartSp 𝕃 \subset \mathbb{L} of smooth loci consisting of smooth loci of the form R n×D (k) lR^n \times D^l_{(k)} with the second factor infinitesimal, where covering families are those of the form {U i×D (k) lU×D (k) l}\{U_i \times D^l_{(k)} \to U \times D^l_{(k)}\} with {U iU}\{U_i \to U\} a covering family in CartSpCartSp 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 n\mathbb{R}^n is diffeomorphic to an open ball (see there for details) we have that the covers {U iU}\{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)C(U) are degreewise coproducts of representables.

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

Properties

\infty-Sheaves on \infty-Cohesive sites

Theorem

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

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

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

Corollary

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

To prove this, we need to show that

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

This follows with the discussion at ∞-connected site.

  1. Sh (,1)(C)Sh_{(\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 Π:Sh (,1)(C)Grpd\Pi : Sh_{(\infty,1)}(C) \to \infty Grpd preserves finite (∞,1)-products.

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

The last two conditions we demonstrate now.

Proposition

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

Π(A×B)Π(A)×Π(B). \Pi(A \times B) \simeq \Pi(A) \times \Pi(B) \,.
Proof

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

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

Proof

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

U 0U 1X 1U UX 0U, \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 iU_i as indicated.

The model for Γ\Gamma sends this to

U 0U 1X 0C(*,U 0) UX 0C(*,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 Π\Pi sends this to

U 0U 1X 0* UX 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)C(*,U) to the point. Since by assumption each C(*,U)C(*,U) is nonempty, this is componentwise an epi. Hence the whole morphism is an epi on π 0\pi_0.

Aufhebung

A cohesive (∞,1)-topos over an \infty-cohesive site satisfies Aufhebung of the moments of becoming. See at Aufhebung the section Aufhebung of becoming – Over cohesive sites.

and

Revised on November 26, 2014 21:48:10 by Urs Schreiber (82.224.164.72)