We examine how the standard proof that left adjoints preserve colimits behaves in the setting of wild categories, a natural setting for synthetic homotopy theory inside homotopy type theory. We prove that the proof may fail for adjunctions between wild categories. Our core contribution, however, is a sufficient condition on the left adjoint for the proof to go through. The condition, called 2-coherence, expresses that the naturality structure of the hom-isomorphism commutes with composition of morphisms. We present two useful examples of this condition in action. First, we use it, along with a new version of a known trick for homogeneous types, to show that the suspension functor preserves graph-indexed colimits. Second, we show that every modality, viewed as a functor on coslices of a type universe, is 2-coherent as a left adjoint to the forgetful functor from the subcategory of modal types, thereby proving this subcategory is cocomplete. We have formalized our main results in Agda.

Under the homotopy hypothesis, higher dimensional groups are defined as pointed homotopy types whose homotopy groups vanish outside a certain range. In particular, a 2-group is a pointed connected homotopy 2-type. Classically, 2-groups have two equivalent algebraic descriptions: one in terms of weak monoidal categories and the other in terms of group cohomology. We present these two classifications of pointed connected 2-types in homotopy type theory, thereby providing internal, constructive counterparts to the traditional classifications of 2-groups. Our first classification (in terms of monoidal categories) takes the form of a bicategorical equivalence, while our second is a type equivalence that extends to n-groups for all n ≥ 2. We have mechanized our results in Agda.

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.