Coend calculus in a compact closed virtual equipment
The talk is based on a project joint with Nathanael Arkor. We lay the foundations for a coend calculus in the framework of virtual double categories. To this end, we construct the concept of a compact closed virtual equipment \mathbb{X} and introduce the notion of a coend for each tight arrow f\colon\thinspace X\otimes A^\circ\otimes A\otimes Y\to C in \mathbb{X}. The goal is to generalize the coend calculus for locally internal categories due to Betti and Walters (R. Betti and R.F.C. Walters, 1989) to compact closed virtual equipments and recover some prominent results, such as the Fubini theorem for coends. To cite an example, we will consider the virtual equipment \mathbb{S}\textsf{pan}(\mathcal{E}) of spans in an ordinary category \mathcal{E} with pullbacks and describe a compact closure on it.
R. Betti and R.F.C. Walters (1989). The calculus of ends over a base topos, Journal of Pure and Applied Algebra.