Coslice Colimits in Homotopy Type Theory

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.

💡 Research Summary

The paper develops a systematic theory of “coslice colimits” inside Homotopy Type Theory (HoTT). Given a universe 𝕌 and a type A : 𝕌, the authors consider the coslice category A/𝕌, whose objects are dependent maps A → T and whose morphisms are pairs of functions together with a homotopy witnessing compatibility. This construction is formalised as a “wild category”, a minimal categorical structure sufficient for HoTT’s higher‑dimensional reasoning.

The central technical contribution, called the “main connection” (Section 5.4), exhibits a left adjoint to the constant‑diagram functor from A/𝕌 to the category of diagrams in 𝕌. Concretely, the authors build the A‑colimit functor by a three‑stage process: (i) a pushout‑based construction of 1‑dimensional higher inductive types, (ii) a coproduct‑based assembly of many objects, and (iii) a combination of the two that yields a genuine colimit in the coslice. Each stage is accompanied by explicit path‑equalities and equivalences, guaranteeing that the resulting type satisfies the universal property of a colimit.

Using this connection, they prove that the forgetful functor U : A/𝕌 → 𝕌 creates colimits of diagrams indexed by contractible graphs, in particular trees (Corollary 5.4.6). This mirrors the classical result that a forgetful functor from a slice creates colimits of contractible diagrams, and it allows the transfer of many universal properties from ordinary colimits in the universe to coslice colimits.

Section 6 extends the universality result to broader classes of diagrams, showing that whenever ordinary colimits exist in 𝕌, the corresponding coslice colimits exist and enjoy the same universal property.

A major application concerns orthogonal factorisation systems (OFS). After developing OFS theory for wild categories (Section 3.3), the authors prove a general preservation theorem: if a functor L has a right adjoint G and G preserves the right class of an OFS, then L preserves the left class (Corollary 3.3.9). Applying this to the A‑colimit functor yields that coslice colimits preserve the left class of the (n‑connected, n‑truncated) OFS on 𝕌. Consequently, for A = 𝟙 (the unit type), the colimit of any diagram of pointed n‑connected types remains n‑connected. This implies that the higher groups defined by Buchholtz, van Doorn, and Rijke are closed under arbitrary colimits (Section 7.1), a non‑trivial closure property for higher algebraic structures.

In Section 8 the authors turn to cohomology. They show that, assuming the internal axiom of choice, any Eilenberg–Steenrod cohomology theory sends finite colimits of spaces to weak limits of sets. The proof leverages the pushout‑coproduct construction of coslice colimits together with the Mayer–Vietoris sequence, providing a type‑theoretic analogue of the classical Brown representability condition that a presheaf must turn homotopy colimits into weak limits.

All major results have been mechanised in Agda (see the accompanying repository). The formalisation includes the definition of the coslice colimit functor, the proof that the forgetful functor creates tree‑indexed colimits, the preservation of OFS left classes, and the weak‑limit property for cohomology. By providing a fully verified implementation, the paper supplies a solid computational foundation for further work on higher‑categorical constructions within HoTT.