Building the bicategory Span$_2(\mathcal {C})$

Abstract

Given any category $\mathcal{C}$ with pullbacks and a terminal object, we show that the data consisting of the objects of $\mathcal{C}$, the spans of $\mathcal{C}$, and the isomorphism classes of spans of spans of $\mathcal{C}$, forms a bicategory. We denote this bicategory Span$_2(\mathcal{C})$ and show that this construction can be applied to give a bicategory of $n$-manifolds, cobordisms of those manifolds, and cobordisms of cobordisms.