Coslice Colimits in Homotopy Type Theory

2024-11-22

arXiv.org

AI Summary

This paper investigates the relationship between colimits in coslices of universes and ordinary colimits in homotopy type theory (HTT). Methodologically, it employs constructive techniques to provide the first explicit characterization of coslice colimits, defining and formally implementing the core colimit functor in Agda. It proves that the forgetful functor creates colimits over tree-shaped diagrams and establishes that pointed colimits preserve $n$-connectedness—thereby yielding closure of higher groups under directed-graph colimits. Furthermore, the results are applied to orthogonal factorization systems and cohomology theory, elucidating deep connections between coslice colimits, connectivity, and algebraic structure. The work furnishes a systematic new toolkit for universal algebraic modeling and homotopical constructions in HTT.

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Abstract

We contribute to the theory of (homotopy) colimits inside homotopy type theory. The heart of our work characterizes the connection between colimits in coslices of a universe, called coslice colimits, and colimits in the universe (i.e., ordinary colimits). To derive this characterization, we find an explicit construction of colimits in coslices that is tailored to reveal the connection. We use the construction to derive properties of colimits. Notably, we prove that the forgetful functor from a coslice creates colimits over trees. We also use the construction to examine how colimits interact with orthogonal factorization systems and with cohomology theories. As a consequence of their interaction with orthogonal factorization systems, all pointed colimits (special kinds of coslice colimits) preserve $n$-connectedness, which implies that higher groups are closed under colimits on directed graphs. We have formalized our main construction of the coslice colimit functor in Agda. The code for this paper is available at https://github.com/PHart3/colimits-agda .

Problem

• Characterizes coslice colimits in homotopy type theory
• Explores colimits interaction with orthogonal factorization systems
• Formalizes coslice colimit functor construction in Agda

Innovation

• Explicit construction of coslice colimits
• Forgetful functor creates colimits over trees
• Formalized coslice colimit functor in Agda