Coslice Colimits in Homotopy Type Theory

Abstract

We contribute to the theory of (homotopy) colimits inside homotopy type theory. The heart of our work characterizes the connection between (graph-indexed) colimits in a type universe and colimits in coslices of the universe, called coslice colimits. To derive this characterization, we give a construction of coslice colimits that is tailored to reveal the connection. We use the construction to prove that the forgetful functor from a coslice creates colimits over trees. We also use it to study how coslice colimits interact with orthogonal factorization systems and with cohomology theories. As a result of their interaction with orthogonal factorization systems, all colimits of pointed types preserve $n$-connectedness, which implies that higher groups, in the sense of Buchholtz, van Doorn, and Rijke, are closed under colimits. We have formalized major portions of this work (see https://github.com/PHart3/colimits-agda for the Agda code), including our main construction of the coslice colimit functor.