Towards Tangent Structures for Orbifolds by Way of Tangent Categories

Orbifolds are geometric objects which are of interest in algebraic topology, differential topology, representation theory, certain flavours of differential geometry, and even in music theory (cf. [7] for an introduction and a discussion of the etymology of “orbifold”). An orbifold, broadly speaking, can be seen as a smooth manifold which is locally folded and glued along some local symmetries together with appropriate symmetry-preserving transition functions. As a result, orbifolds give ways to study smooth differential objects and the ways they interact with smooth symmetries and natural locations in which to study equivariant cohomology (of both Borel or Bredon flavour; cf. [3] for Borel cohomology for orbifolds and [4], [2] for Bredon cohomology for orbifolds). However, a difficulty with studying the tangential information encoded on an orbifold lies in the fact that defining the tangent tangent vectors which appropriately respect the symmetries along which the orbifold is glued require not mere tangent vectors, but instead require tangent vectors together with equivariant descent-theoretic information showing that these vectors are suitably symmetry-stable. In this talk we will propose how to define such tangent-theoretic information by using the language of tangent categories, a semantic tool for studying structural differential geometry discovered in [6] and [1], and pseudolimits of tangent categories.

In [4] we showed how to build pseudolimits in the $2$-category of tangent categories as well as how to use them to give descent-equivariant tangent structures for smooth $G$-spaces (for Lie groups $G$ acting smoothly on smooth manifolds). In this talk we will extend the work of [5] and illustrate how to use the tangent structures for smooth $G$-spaces to define tangent structures on orbifolds as follows. Because the charts in orbifold atlases are open subspaces of $U\subseteq {ℝ}^n$ equipped with a smooth action of a finite group $S$, we first equip each chart $U$ with its $S$-equivariant tangent structure and then take the pseudolimit of such tangent categories in order to arrive with a tangent structure for our given orbifold.