nLab concrete (infinity,1)-sheaf

Idea

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

Definition

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

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

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

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

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 $S \in Mor \mathbf{H}$ of morphisms. It factors therefore canonically through the (∞,1)-quasitopos of $S$-separated $(\infty,1)$-sheaves

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

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

Definition

We say an object $X$ is $n$-concrete if the canonical morphism $X \to coDisc \Gamma X$ 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 $X \in \mathbf{H}$ and $n \in \mathbb{N}$, the $(n+1)$-concretification of $X$ is the morphism

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

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

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

Remark

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

Examples

• concrete smooth ∞-groupoid

Related concepts

• concrete sheaf

• concrete $(\infty,1)$-sheaf

References

This entry goes back to some observations by David Carchedi.