Remarks On on constant$\mathrm{\infty }\mathrm{Top}$ \infty Top-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 $\bar const_X$ constant on $X$, meaning constant on the Kan complex that is the fundamental ∞-groupoid $Sing X = \Pi(X)$ 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 $S$

where of course

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 $S = Top$ itself, there are then two different ways to regard a topological space as an $\infty$-stack, and they have very different meaning.

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

This is because

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

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