nLab constant infinity-stack

Definition

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

With the global section (∞,1)-functor the constant $\infty$-stack functor $LConst$ forms the terminal (∞,1)-geometric morphism

$$(LConst \dashv \Gamma) : Sh_{(\infty,1)}(C) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd \,.$$

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 $S$

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

where of course

$$const_K : U \mapsto K$$

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 $S = Top$ 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.

Pattern

• A locally constant function is a section of a constant sheaf;

• a locally constant sheaf is a section of a constant stack;

• a locally constant stack is a section of (… and so on…)

• a locally constant ∞-stack is a section of a constant $\infty$-stack.

A locally constant sheaf / $\infty$-stack is also called a local system.