Coslice Colimits in Homotopy Type Theory

TL;DR

This paper advances the understanding of homotopy colimits in homotopy type theory by characterizing coslice colimits, constructing them explicitly, and exploring their interactions with various structures, formalized in Agda.

Contribution

It provides a new construction of coslice colimits in homotopy type theory and analyzes their properties and interactions with other categorical structures.

Findings

Coslice colimits can be constructed to reveal their connection to colimits in a universe.

The forgetful functor from a coslice creates colimits over trees.

Colimits of pointed types preserve n-connectedness, affecting higher group structures.

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.