Cell-sets and decomposition spaces

This project, termed Cell Sets and Decomposition Spaces, is an introduction to the connections between homotopy and algebraic combinatorics. The main goal is to understand the category Cell of cell-sets and cell-maps, introduced by N. Ray and W. Schmitt, in the more modern language of 2-Segal spaces, also known as decomposition spaces. Recent work on decomposition spaces has shown them to provide a powerful language for combinatorial structures and their symmetries, and it is illuminating to see the examples and constructions of Ray-Schmitt from this perspective. In order to do so we construct a functor between the category of (integer graded, discrete) decomposition spaces and Cell, which factors through the bicategory of spans of (integer graded, discrete) groupoids by identifying the cell-sets and cell-maps of Ray--Schmitt as discrete groupoids over the integers and spans or correspondences between them, respectively. We use it, together with the free abelian group functor defined by Ray-Schmitt in their work, to connect decomposition spaces with graded coalgebras and, more generally, graded Hopf algebras