nLab
topological stack

Idea

A topological stack is a geometric stack on the site Top: a topological groupoid regarded as presenting an object in the (2,1)-sheaf (2,1)-topos ${\mathrm{Sh}}_{(2,1)}(\mathrm{Top})$.

Definition

Let Top be the category of compactly generated spaces and continuous function. When equipped with a Grothendieck topology given by usual open covers this becomes a subcanonical large site.

Consider the (2,1)-topos (2,1)-sheaves=stacks

$$\mathrm{TopStack}:={\mathrm{Sh}}_{(2,1)}(\mathrm{Top})$$TopStack := Sh_{(2,1)}(Top)

of groupoids on $\mathrm{Top}$; by Yoneda $\mathrm{Top}$ is a full sub-(2,1)-category

$$\mathrm{Top}\hookrightarrow \mathrm{TopStack}.$$Top \hookrightarrow TopStack \,.

By analogy with the case of algebraic stacks one says that a morphism of 1-stacks $f:X\to Y$ in $\mathrm{TopStack}$ is a representable morphism of stacks if for any morphism of 1-stacks $T\to Y$ from a (stack associated to a) topological space $T$ to $Y$ the pullback $T{\times }_{Y}X$ is (2-isomorphic to the stack associated to) a topological space.

Let $P$ be a property of a map of topological spaces. $P$ is said to be invariant under change of base if for all $f:Y\to X$ with property $P$, if $g:Z\to X$ is any continuous map, the induced map $Z{\times }_{X}Y\to Z$ also has property $P$. $P$ is said to be invariant under restriction if this holds whenever $g$ is an embedding. A property $P$ which is invariant under restriction is said to be local on the target if any $f:Y\to X$ for which there exists an open cover $(U_{\alpha }\to X)$ such that the induced map ${\coprod }_{\alpha }{U}_{\alpha }{\times }_{X}Y\to {\coprod }_{\alpha }{U}_{\alpha }$ has property $P$, must also have property $P$.

Examples of such properties are being an open map, covering map, closed map, local homeomorphism etc.

A representable map $f:X\to Y$ of stacks is said to have property $P$ if for any map $T\to Y$ from a topological space, the induced map $T{\times }_{Y}X\to T$ has property $P$

Definition

A 1-stack $X$ of groupoids over $\mathrm{Top}$ having a representable epimorphism from a topological space $X_0\to X$ is a topological stack. Such an representable epimorphism is called an atlas (or chart).

This is what is called pretopological stack in Noohi . The terminology topological stack is reserved for those stacks whose atlas can be chosen to belong to a class of “local fibrations”; there are axioms which the class of local fibrations have to satisfy; there are several natural choices of this class which modify the variant of topological stacks considered.

Any map from a topological space $S$ to a topological stack $X$ is representable (i.e. diagonal $X\to X\times X$ is always representable). For a topological stack $Y$, if $P$ is invariant under restriction and local on the target, a representable morphism $f:X\to Y$ of 1-stacks has this property if there exists an atlas $T\to Y$ such that the induced map $X{\times }_{Y}T\to T$ has property $P$.

If $X_0\to X$ is an atlas for a topological stack, then $X_0{\times }_X{X}_0\rightrightarrows X_0$ is a topological groupoid, $X$. The stackification of the presheaf of groupoids $T\mapsto \mathrm{Hom}((T^{\mathrm{id}},X))$ is (2-iso to) $X$ (where $T^{\mathrm{id}}$ is $T$ considered as a topological groupoid with only identity arrows).

Conversely, given a topological groupoid $G$, we can consider the stackification of $\mathrm{Hom}(\mathrm{blank},G):=\left[G\right]$. By direct inspection, one sees that $\left[G\right](T)$ is the groupoid of principal G-bundles over $T$, ${\mathrm{Bun}}_G(T)$. The canonical map $(G_0{)}^{\mathrm{id}}\to G$ yields a map $a:G_0\to \left[G\right]$. If $p:T\to \left[G\right]$ is any map from a space, then $T{\times }_{\left[G\right]}{G}_0$ is the total space of the principal $G$-bundle over $T$ which $p$ corresponds to via Yoneda. If under the correspondence between principal bundles and generalized homomorphims $p$ corresponds to a map $T^{\mathrm{id}}\to G$, then $p$ factors through the map $a:G_0\to \left[G\right]$. If $p$ instead corresponds to a map $T_U\to G$ where $U\to T$ is a cover, then $p$ factors through $a$ locally, hence, $a$ is an epimorphism. Therefore an alternative definition of a topological stack is:

Definition

A 1-stack $X$ of groupoids over $\mathrm{Top}$ is a topological stack if it is [[]equivalence in an (infinity,1)-category|equivalent]] to the stack $G\mathrm{Bund}$ of groupoid-principal bundle for some topological groupoid $G$.

By the Yoneda lemma, $\mathrm{Hom}(T,{\mathrm{Bun}}_G)\cong {\mathrm{Bun}}_G(T)$ for all $T$. Moreover, if $H$ is another topological groupoid, $\mathrm{Hom}({\mathrm{Bun}}_H,{\mathrm{Bun}}_G)\cong {\mathrm{Bun}}_G(H)$, where ${\mathrm{Bun}}_G(H)$ is the groupoid of principal $G$-bundles over $H$. In fact, one can use this to show that the 2-category of topological stacks is equivalent to the bicategory of topological groupoids and principal bundles. One may also show that topological stacks are equivalent to the bicategory of fractions of topological groupoids with respect to formally inverting Morita-equivalences.

Related concept

• geometric stack

  • algebraic stack

  • topological stack, topological groupoid

  • differentiable stack, Lie groupoid

• geometric ∞-stack

• cohesive ∞-groupoid

  • discrete ∞-groupoid

  • Euclidean-topological ∞-groupoid

  • smooth ∞-groupoid

  • synthetic differential ∞-groupoid

References

• Behrang Noohi

  • Foundations of Topological Stacks I

  • Homotopy types of topological stacks

  • Mapping stacks of topological stacks

• Metzler, Topological and smooth stacks (arXiv:math/0306176)

• Kai Behrend, Gregory Ginot, Behrang Noohi, Ping Xu, String topology for stacks (arXiv)