Math and Stats - ATCAT seminar - "Orbispace Mapping Objects: Three Approaches, Two Results"
Speaker : Dorette Pronk (Dalhousie University)
Abstract : Orbispaces 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). This affects the way that transition between charts need to be described, and it is generally rather cumbersome to work with atlases. It has been shown in [Moerdijk-P] that one can represent orbifolds by groupoids internal to the category of manifolds, with etale structure maps and a proper diagonal. We have since generalized this notion further and we now consider orbispaces as represented by proper etale groupoids in the category of locally compact, paracompact topological spaces (they will also be called orbigroupoids). Two of these groupoids represent the same orbispace if they are Morita equivalent.
So we consider the bicategory of fractions with respect to Morita equivalences. For orbigroupoids G and H we can then consider the mapping groupoid [G, H] of maps and 2-cells in the bicategory of fractions. The question I want to address is how to define a topology on these mapping groupoids to obtain mapping objects for this bicategory. This question was addressed in [Chen], but not in terms of orbigroupoids, and with only partial answers.
I will approach this question from three different directions:
1. When the orbifold G is compact, we can define a topology on [G,H] to obtain a topological groupoid OMap(G, H) so that Orbispaces(K × G, H) is equivalent to Orbispaces(K, OMap(G, H)). We will also show that OMap(G,H) represents an orbispace.
2. For any pair of orbigroupoids G, H we can define a topology on [G,H] to obtain EMap(G,H) so that Orbispaces has the structure of an enriched bicategory: composition induces a continuous functor EMap(G,H) x EMap(H,K) --> EMap(G,K).
3. There is a fibration structure on the category of orbigroupoids with groupoid homomorphisms as defined in [P-Warren]. (This can be derived from unpublished work by Colman and Costoya.) This implies that when G and H are stack groupoids, we may restrict ourselves to ordinary groupoid homomorphisms and their usual 2-cells.
In this talk I will discuss the relationships between the topologies obtained in these ways, as well as the relationship with Chen's work. This is joint work with Laura Scull.
[Chen] Weimin Chen, On a notion of maps between orbifolds I: function spaces, Communications in Contemporary Mathematics 8 (2006), pp. 569-620.
[Moerdijk-P] I. Moerdijk, D.A. Pronk, Orbifolds, sheaves and groupoids, K-Theory 12 (1997), pp. 3-21.
[P-Warren] Dorette A. Pronk, Michael A. Warren, Bicategorical fibration structures and stacks, Theory and Applications of Categories, Vol. 29, 2014, No. 29, pp 836-873.
https://us02web.zoom.us/j/81872756284?pwd=VGVQcW8vRmFXK1hvTk5UZURGSXg5UT09 Meeting ID: 818 7275 6284 Passcode: ATCAT2020
Ellen LynchMath and Stats - ATCAT seminar - "Orbispace Mapping Objects: Three Approaches, Two Results"