nLab
constant infinity-stack

Definition

A constant ∞-stack or (∞,1)-sheaf is the ∞-stackification of a (∞,1)-presheaf which is constant as an (∞,1)-functor.

This is the categorification of the notion of constant sheaf

A section of the $\mathrm{\infty }$-stack constant on $\mathrm{Core}(\mathrm{Fin}\mathrm{\infty }\mathrm{Grpd})\in$ ∞Grpd is a locally constant ∞-stack.

Remarks on constant $\mathrm{\infty }$-stacks on Top

Notice that in the special case of ∞-stacks on Top, hence of topological ∞-groupoid, which may be thought of as Top-valued presheaves on Top(!), there are two different obvious ways to regard a topological space $X$ as an ∞-stack on Top:

• there is the ∞-stack ${\overline{\mathrm{const}}}_X$ constant on $X$, meaning constant on the Kan complex that is the fundamental ∞-groupoid $\mathrm{Sing}X=\Pi (X)$ of $X$;

• there is the Yoneda embedding $Y(X)$ of $X$ into ∞-stack.

The first regards $X$ really as an ∞-groupoid, forgetting its topology, the second regards $X$ as a locale, not caring about the homotopies that are inside.

For any (∞,1)-category $S$, there is the obvious embedding of ∞-groupoids into (∞,1)-presheaves on $S$

$$\mathrm{const}:\mathrm{\infty }\mathrm{Grpd}\to [S^{\mathrm{op}},\mathrm{\infty }\mathrm{Grpd}]$$const : \infty Grpd \to [S^{op}, \infty Grpd]

where of course

$${\mathrm{const}}_K:U\mapsto K$$const_K : U \mapsto K

for all $U$.

This is all very obvious, but deserves maybe a special remark in the case that ∞-groupoids are modeled as (compactly generated and weakly Hausdorff) topological spaces: in particular in the case that $S=\mathrm{Top}$ itself, there are then two different ways to regard a topological space as an $\mathrm{\infty }$-stack, and they have very different meaning.

In particular, with $X$ a topological space, the $\mathrm{\infty }$-stack constant on $X$ has the property that its loop space object $\Lambda X$ is indeed the $\mathrm{\infty }$-stack constant on the free loop space of $X$, while the loop space object of $X$ regarded as a representable $\mathrm{\infty }$-stack is just $X$ itself again.

This is because

• the $\mathrm{\infty }$-stack represented by $X$ regards $X$ as a categorically discrete topological groupoid;

• while the $\mathrm{\infty }$-stack constant on $X$ regards $X$ as a topologically discrete groupoid which however may have nontrivial morphisms.