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Exponentials and Enrichment for Orbispaces

Banff International Research Station for Mathematical Innovation and Discovery

Orbifolds are defined like manifolds, by local charts. Where manifold charts are open subsets of Euclidean space, orbifold charts consist of an open subset of Euclidean space with an action by a finite group (thus allowing for local singularities). However, a more useful way to represent them is in terms of proper étale groupoids (which we will call orbispaces) and the maps between them are obtained as a bicategory of fractions of the 2-category of proper étale groupoids with respect to the class of essential equivalences. In recent work with Bustillo and Szyld we have shown that in any bicategory of fractions the hom-categories form a pseudo colimit of the hom categories of the original bicategory. We will show that this result can be extended to our topological context: for topological groupoids the hom-groupoids can again be topologized and under suitable conditions on the spaces these groupoids form both exponentials and enrichment. We will show that taking the appropriate pseudo colimit of these hom-groupoids within the 2-category of topological groupoids gives us a notion of hom-groupoids for the bicategory of orbispaces. When the domain orbispace is orbit compact, we see show that this groupoid is proper and satisfies the conditions to be an exponential. When we further cut back our morphisms between orbispaces to so-called admissible maps, we obtain a proper étale groupoid that is essentially equivalent to the pseudo colimit and hence is also the exponential. Furthermore, we show that the bicategory of orbit-compact orbispaces is enriched over orbispaces: the composition is given by a map of orbispaces rather than a continuous functor. This work rephrases the result from [Chen] in terms of groupoid representations for orbifolds and strengthens his result on enrichment: he expressed this in terms of a map between the quotient spaces of the mapping orbispaces, where we are able to give this in terms of a map between the orbispaces. I will end the talk with several examples of mapping spaces. This is joint work with Laura Scull and started out as a project of the first Women in Topology workshop. [Chen] Weimin Chen, On a notion of maps between orbifolds I: function spaces, Communications in Contemporary Mathematics 8 (2006), pp. 569-620.

Mathematics; Category Theory; Homological Algebra; Algebraic Topology; Algebraic Structures

video/mp4

Author affiliation: Dalhousie University

2023-10-29

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/86388

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/