nLab
concrete (infinity,1)-sheaf

Context

Discrete and concrete objects

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

Inside a local (∞,1)-topos H\mathbf{H} there are objects that may be thought of as ∞-groupoids equipped with extra structure (“cohesive structure” if H\mathbf{H} is even a cohesive (∞,1)-topos). These are the concrete objects in H\mathbf{H}.

Definition

Let H\mathbf{H} be a local (∞,1)-topos.

HCodiscΓDiscGrpd. \mathbf{H} \stackrel{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}}{\underset{Codisc}{\leftarrow}} \infty Grpd \,.

Since CodiscCodisc is by definition a full and faithful (∞,1)-functor this means that

GrpdCodiscΓH \infty Grpd \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}} \mathbf{H}

is a geometric embedding. By the discussion at reflective sub-(∞,1)-category this means that Γ\Gamma is the localization of an (∞,1)-category at a class SMorHS \in Mor \mathbf{H} of morphisms. It factors therefore canonically through the (∞,1)-quasitopos of SS-separated (,1)(\infty,1)-sheaves

GrpdΓConc(H)concretizeH. \infty Grpd \stackrel{\overset{\Gamma}{\leftarrow}}{\hookrightarrow} Conc(\mathbf{H}) \stackrel{\overset{concretize}{\leftarrow}}{\hookrightarrow} \mathbf{H} \,.

We call Conc(H)Conc(\mathbf{H}) the (∞,1)-quasitopos of concrete objects of the local (,1)(\infty,1)-topos H\mathbf{H}.

Definition

We say an object XX is nn-concrete if the canonical morphism XcoDiscΓXX \to coDisc \Gamma X is (n-1)-truncated.

If a 0-truncated object XX is 00-concrete, we call it just concrete.

Proposition

For CC an ∞-cohesive site, a 0-truncated object in the (∞,1)-topos over CC is concrete precisely if it is a concrete sheaf in the traditional sense.

Definition

For XHX \in \mathbf{H} and nn \in \mathbb{N}, the (n+1)(n+1)-concretification of XX is the morphism

Xconc n+1X X \to conc_{n+1} X

that is the left factor in the decomposition with respect to the n-connected/n-truncated factorization system of the (ΓcoDisc)(\Gamma \dashv coDisc)-unit

conc n+1X X coDiscΓX. \array{ && conc_{n+1} X \\ & \nearrow && \searrow \\ X &&\to&& coDisc \Gamma X } \,.
Remark

By that very n-connected/n-truncated factorization system we have that conc n+1Xconc_{n+1} X is an n+1{n+1}-concrete object.

Examples

References

This entry goes back to some observations by David Carchedi.

Revised on November 23, 2011 17:50:00 by Urs Schreiber (131.174.40.49)