Freely generated $n$-categories, coinserters and presentations of low dimensional categories

Abstract

Composing with the inclusion $\mathsf{Set}\to\mathsf{Cat} $, a graph $G$ internal to $\mathsf{Set} $ becomes a graph of discrete categories, the coinserter of which is the category freely generated by $G$. Introducing a suitable definition of $n$-computad, we show that a similar approach gives the $n$-category freely generated by an $n$-computad. Suitable $n$-categories with relations on $n$-cells are presented by these $(n+1)$-computads, which allows us to prove results on presentations of thin groupoids and thin categories. So motivated, we introduce a notion of deficiency of (a presentation of) a groupoid via computads and prove that every small connected thin groupoid has deficiency $0$. We compare the resulting notions of deficiency and presentation with those induced by monads. In particular, we find our notion of group deficiency to coincide with the classical one. Finally, we study presentations of $2$-categories via $3$-computads, focusing on locally thin groupoidal $2$-categories. Under suitable hypotheses, we give efficient presentations of some locally thin and groupoidal $2$-categories. A fundamental tool is a $2$-dimensional analogue of the association of a "topological graph" to every graph internal to $\mathsf{Set} $. Concretely, we construct a left adjoint $\mathcal{F}_ {\mathsf{Top} _ 2} : 2\textrm{-}\mathsf{cmp}\to \mathsf{Top} $ associating a $2$-dimensional CW-complex to each small $2$-computad. Given a $2$-computad $\mathfrak{g} $, the groupoid it presents is equivalent to the fundamental groupoid of $\mathcal{F} _ {\mathsf{Top} _2}(\mathfrak{g}) $. Finally, we sketch the $3$-dimensional version $\mathcal{F}_ {\mathsf{Top} _ 3}$.