nLab
concrete (infinity,1)-sheaf

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.

$$H\stackrel{\stackrel{\stackrel{\mathrm{Disc}}{\leftarrow }}{\underset{\Gamma }{\to }}}{\underset{\mathrm{Codisc}}{\leftarrow }}\mathrm{\infty }\mathrm{Grpd}.$$\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

$$\mathrm{\infty }\mathrm{Grpd}\stackrel{\stackrel{\Gamma }{\leftarrow }}{\underset{\mathrm{Codisc}}{\hookrightarrow }}H$$\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 $(\mathrm{\infty },1)$-sheaves

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

We call $\mathrm{Conc}(H)$ the (∞,1)-quasitopos of concrete objects of the local $(\mathrm{\infty },1)$-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 $n\in \mathbb{N}$, the $(n+1)$-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 $(\Gamma ⊣\mathrm{coDisc})$-unit

$$\begin{array}{ccc}& & {\mathrm{conc}}_{n+1}X\\ & \nearrow & & \searrow \\ X& & \to & & \mathrm{coDisc}\Gamma X\end{array}.$$\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 smooth ∞-groupoid

Related concepts

• concrete sheaf

• concrete $(\mathrm{\infty },1)$-sheaf

References

This entry goes back to some observations by David Carchedi.