Seminario de Categorías

Resumen: 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). This affects the way that transitions 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, I.e., combined source-target map (s,t): G_1 -> G_0 x G_0. We have since generalized this notion further to orbispaces, represented by proper etale groupoids in the category of Hausdorff spaces. Two of these groupoids represent the same orbispace if they are Morita equivalent. However, Morita equivalences are generally not pseudo-invertible in this 2-category, so we consider the bicategory of fractions with respect to Morita equivalences.

For a pair of paracompact locally compact orbigroupoids G and H, with G orbit-compact, we want to study the mapping groupoid [G, H] of arrows and 2-cells in the bicategory of fractions. The question we want to address is how to define a topology on these mapping groupoids to obtain mapping objects for the bicategory of orbispaces. This question was addressed in [Chen], but not in terms of orbigroupoids, and with only partial answers.

We will present the following results:

1. When the orbifold G is compact, we define a topology on [G,H] to obtain a topological groupoid OMap(G, H), which is Morita equivalent to an orbigroupoid. To obtain a Morita equivalent orbigroupoid, we need to restrict ourselves to so-called admissible maps to form AMap(G,H), and Orbispaces(K × G, H) is equivalent to Orbispaces(K, AMap(G, H)).

So AMap(G,H) is an exponential object in the bicategory of orbispaces.

2. We will also show that AMap(G,H) thus defined provides the bicategory of orbit-compact orbispaces with bicategorical enrichment over the bicategory of orbispaces: composition can be given as a generalized map (an arrow in the bicategory of fractions) of orbispaces.

In this talk I will discuss how this work extends the work done by Chen and I will show several examples. 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.

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https://www.youtube.com/watch?v=7Ot9SCJLphU&feature=youtu.be