Coslice Colimits in Homotopy Type Theory
This work advances foundational mathematics in homotopy type theory, with implications for higher groups and cohomology theories, though it is incremental within this specialized domain.
The paper tackles the characterization of coslice colimits in homotopy type theory, revealing their connection to colimits in type universes and proving that the forgetful functor from a coslice creates colimits over trees, with results including that all colimits of pointed types preserve n-connectedness.
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.