### Session S27 - Categories and Topology

Wednesday, July 21, 17:30 ~ 18:00 UTC-3

## Mapping Objects for Orbispaces

### Laura Scull

#### Fort Lewis College, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

Orbifolds, and more generally orbispaces, are spaces which have well-behaved singularities. They can be defined using atlases and charts analogously to manifolds, with charts consisting of open subsets of Euclidean space with an action of a finite group defining the local singularity structure. This approach is cumbersome and hard to work with, and instead, orbispaces are often modeled using topological groupoids. It is shown in [Moerdijk-Pronk] that orbispaces can be represented by topological groupoids with etale structure maps and proper diagonal. This representation is not unique, however, as two Morita equivalent groupoids represent the same orbispace.

Thus, to represent orbifolds and orbispaces, we turn to a bicategory of fractions where the Morita equivalences have been inverted. This gives a definition of a map between orbispaces $G \to H$ defined as a span of maps between groupoids $G \leftarrow K \to H$ where $K$ is Morita equivalent to $G$, giving an alternate representation of the domain orbispace.

The question I will address is how to create a topological mapping groupoid for orbispaces, $\mbox{OMap}(G, H)$, which encodes these spans and satisfies the properties of a mapping object. This question has been addressed in [Chen], but not in terms of orbigroupoids, and with only partial answers. Here, I will show how to define an etale proper groupoid $\mbox{OMap}(G, H)$ which is the exponential object for orbigroupoids and gives orbispaces the structure of an enriched bicategory, so that composition induces a map of orbispaces $\mbox{OMap}(G, H) \times \mbox{OMap}(H, K) \to \mbox{OMap}(G, K)$.

[Chen] Weimin Chen, On a notion of maps between orbifolds I: function spaces, Communications in Contemporary Mathematics 8 (2006), pp. 569-620.

[Moerdijk-Pronk] I. Moerdijk, D.A. Pronk, Orbifolds, sheaves and groupoids, K-Theory 12 (1997), pp. 3-21.

Joint work with Dorette Pronk (Dalhousie University).