derived smooth geometry
By analogy with the case of algebraic stacks one says that a morphism of 1-stacks in is a representable morphism of stacks if for any morphism of 1-stacks from a (stack associated to a) topological space to the pullback is (2-isomorphic to the stack associated to) a topological space.
Let be a property of a map of topological spaces. is said to be invariant under change of base if for all with property , if is any continuous map, the induced map also has property . is said to be invariant under restriction if this holds whenever is an embedding. A property which is invariant under restriction is said to be local on the target if any for which there exists an open cover such that the induced map has property , must also have property .
A representable map of stacks is said to have property if for any map from a topological space, the induced map has property
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 to a topological stack is representable (i.e. diagonal is always representable). For a topological stack , if is invariant under restriction and local on the target, a representable morphism of 1-stacks has this property if there exists an atlas such that the induced map has property .
If is an atlas for a topological stack, then is a topological groupoid, . The stackification of the presheaf of groupoids is (2-iso to) (where is considered as a topological groupoid with only identity arrows).
Conversely, given a topological groupoid , we can consider the stackification of . By direct inspection, one sees that is the groupoid of principal G-bundles over , . The canonical map yields a map . If is any map from a space, then is the total space of the principal -bundle over which corresponds to via Yoneda. If under the correspondence between principal bundles and generalized homomorphims corresponds to a map , then factors through the map . If instead corresponds to a map where is a cover, then factors through locally, hence, is an epimorphism. Therefore an alternative definition of a topological stack is:
By the Yoneda lemma, for all . Moreover, if is another topological groupoid, , where is the groupoid of principal -bundles over . 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.
Foundations are in
Metzler, Topological and smooth stacks (arXiv:math/0306176)