# nLab concrete (infinity,1)-sheaf

### Context

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

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

## Definition

Let $H$ be a local (∞,1)-topos.

$H\stackrel{\stackrel{\stackrel{\mathrm{Disc}}{←}}{\underset{\Gamma }{\to }}}{\underset{\mathrm{Codisc}}{←}}\infty \mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{H} \stackrel{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}}{\underset{Codisc}{\leftarrow}} \infty Grpd \,.

Since $\mathrm{Codisc}$ is by definition a full and faithful (∞,1)-functor this means that

$\infty \mathrm{Grpd}\stackrel{\stackrel{\Gamma }{←}}{\underset{\mathrm{Codisc}}{↪}}H$\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 $\Gamma$ is the localization of an (∞,1)-category at a class $S\in \mathrm{Mor}H$ of morphisms. It factors therefore canonically through the (∞,1)-quasitopos of $S$-separated $\left(\infty ,1\right)$-sheaves

$\infty \mathrm{Grpd}\stackrel{\stackrel{\Gamma }{←}}{↪}\mathrm{Conc}\left(H\right)\stackrel{\stackrel{\mathrm{concretize}}{←}}{↪}H\phantom{\rule{thinmathspace}{0ex}}.$\infty Grpd \stackrel{\overset{\Gamma}{\leftarrow}}{\hookrightarrow} Conc(\mathbf{H}) \stackrel{\overset{concretize}{\leftarrow}}{\hookrightarrow} \mathbf{H} \,.

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

###### Definition

We say an object $X$ is $n$-concrete if the canonical morphism $X\to \mathrm{coDisc}\Gamma X$ is (n-1)-truncated.

If a 0-truncated object $X$ is $0$-concrete, we call it just concrete.

###### Proposition

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

###### Definition

For $X\in H$ and $n\in ℕ$, the $\left(n+1\right)$-concretification of $X$ is the morphism

$X\to {\mathrm{conc}}_{n+1}X$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 $\left(\Gamma ⊣\mathrm{coDisc}\right)$-unit

$\begin{array}{ccc}& & {\mathrm{conc}}_{n+1}X\\ & ↗& & ↘\\ X& & \to & & \mathrm{coDisc}\Gamma X\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 ${\mathrm{conc}}_{n+1}X$ is an $n+1$-concrete object.

## Examples

• concrete sheaf

• concrete $\left(\infty ,1\right)$-sheaf

## 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)