nLab
topological stack

Context

Higher geometry

Topology

Contents

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 Sh (2,1)(Top)Sh_{(2,1)}(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

TopStack:=Sh (2,1)(Top) TopStack := Sh_{(2,1)}(Top)

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

TopTopStack. Top \hookrightarrow TopStack \,.

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

Let PP be a property of a map of topological spaces. PP is said to be invariant under change of base if for all f:YXf: Y \to X with property PP, if g:ZXg:Z \to X is any continuous map, the induced map Z× XYZZ \times_X Y \to Z also has property PP. PP is said to be invariant under restriction if this holds whenever gg is an embedding. A property PP which is invariant under restriction is said to be local on the target if any f:YXf: Y \to X for which there exists an open cover (U αX)\left(U_\alpha \to X\right) such that the induced map αU α× XY αU α\coprod_{\alpha} {U_\alpha } \times_{X} Y \to \coprod_\alpha {U_\alpha } has property PP, must also have property PP.

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

A representable map f:XYf:\X \to \Y of stacks is said to have property PP if for any map TYT \to \Y from a topological space, the induced map T× YXTT \times_{\Y} \X \to T has property PP

Definition

A 1-stack XX of groupoids over Top\mathrm{Top} having a representable epimorphism from a topological space X 0XX_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 SS to a topological stack XX is representable (i.e. diagonal XX×XX\to X\times X is always representable). For a topological stack YY, if PP is invariant under restriction and local on the target, a representable morphism f:XYf : X \to Y of 1-stacks has this property if there exists an atlas TYT\to Y such that the induced map X× YTTX\times_Y T\to T has property PP.

If X 0XX_0 \to X is an atlas for a topological stack, then X 0× XX 0X 0X_0 \times_{X} X_0 \rightrightarrows X_0 is a topological groupoid, X\mathbf{X}. The stackification of the presheaf of groupoids THom((T id,X))T \mapsto Hom((T^{id},\mathbf{X})) is (2-iso to) XX (where T idT^{id} is TT considered as a topological groupoid with only identity arrows).

Conversely, given a topological groupoid GG, we can consider the stackification of Hom(blank,G):=[G]Hom(blank,G):= \left[ G\right]. By direct inspection, one sees that [G](T)\left[ G\right](T) is the groupoid of principal G-bundles over TT, Bun G(T)Bun_G(T). The canonical map (G 0) idG(G_0)^id \to G yields a map a:G 0[G]a:G_0 \to \left[ G\right]. If p:T[G]p:T \to \left[ G\right] is any map from a space, then T× [G]G 0T \times_{\left[ G\right]} G_0 is the total space of the principal GG-bundle over TT which pp corresponds to via Yoneda. If under the correspondence between principal bundles and generalized homomorphims pp corresponds to a map T idGT^{id} \to G, then pp factors through the map a:G 0[G]a:G_0 \to \left[ G\right]. If pp instead corresponds to a map T UGT_U \to G where UTU \to T is a cover, then pp factors through aa locally, hence, aa is an epimorphism. Therefore an alternative definition of a topological stack is:

Definition

A 1-stack XX of groupoids over Top\mathrm{Top} is a topological stack if it is equivalent to the stack GBundG Bund of groupoid-principal bundle for some topological groupoid GG.

By the Yoneda lemma, Hom(T,Bun G)Bun G(T)Hom(T,Bun_G) \cong Bun_G(T) for all TT. Moreover, if HH is another topological groupoid, Hom(Bun H,Bun G)Bun G(H)Hom(Bun_H,Bun_G) \cong Bun_G(H), where Bun G(H)Bun_G(H) is the groupoid of principal GG-bundles over HH. 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.

References

Foundations are in

The mapping stacks of topological stacks are discussed in

See also

Revised on August 12, 2013 13:24:20 by Urs Schreiber (89.204.130.218)