Most categorical structures (categories, 2-categories, …) resist presentation as many-sorted algebras. Nevertheless, they are models of generalized algebraic theories, in which dependent sorts are allowed. Let us call a (generalized algebraic) theory simple when there are no non-trivial dependent sorts, as in simply typed lambda calculus. For instance, the theories of groups and rings are simple, while the theories of categories and 2-categories are not simple. However, the theory of 2-categories is even farther from being simple than that of categories, as it has a sort with iterated dependency. To measure how close two theories are with respect to dependency, we adopt a relative notion of simplicity. For example, we deem the theory of 2-categories simple relative to that of categories, since once a category $\mathcal{C}$ is fixed, the theory of 2-categories whose underlying category is $\mathcal{C}$ is simple.
We will discuss this topic of relative simplicity in terms of clans, introduced by Joyal. Just as models of an algebraic theory are seen as finite-product-preserving functors from its syntactic category into Set, models of a generalized algebraic theory are seen as clan morphisms from its syntactic clan into Set. We propose a definition of relative simplicity purely in terms of clans and their morphisms. As an application, we use this notion to prove that in the category of strict $n$-categories, every strong epimorphism is a composite of $(n+1)$ regular epimorphisms.
This talk is based on joint work with Yuto Kawase: Yuto Kawase, Hayato Nasu, “On the decomposition of a strong epimorphism into regular epimorphisms.” arXiv:2604.05744.