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.

One of the most fundamental theorems in algebra is the fundamental theorem of homomorphisms, which states that $\mathrm{Im}(f)\cong A/\mathrm{Ker}(f)$ for any homomorphism $f\colon A\to B$. In categorical terms, this states that strong epimorphisms and regular epimorphisms are equivalent in categories of algebraic structures such as groups, rings, and so on. Although this equivalence fails in general, it is known that in any locally presentable category, every strong epimorphism can be decomposed into a transfinite composite of regular epimorphisms. In this talk, I will discuss how many regular epimorphisms are needed in such a decomposition, which measures the extent to which the fundamental theorem of homomorphisms holds for general algebra-like structures such as $n$-categories.

This talk is based on joint work with Yuto Kawase.

Cartesian bicategories, introduced by Carboni and Walters, form a class of bicategories in which one can take “finite products.” Typical examples of those products include the direct products of sets in the bicategory Rel of sets and relations, and the product categories in the bicategory Prof of categories and profunctors. However, they do not coincide with the actual (bi)products in those bicategories, as those products do not possess the same universal property for relations or profunctors as for functions or functors. This discrepancy leads to the intricate formulation of cartesian bicategories.

In this talk, I will introduce cartesian equipments, a certain class of double categories introduced by Aleiferi, and explain in what sense they provide a more natural framework than cartesian bicategories and still cover most of the examples studied in terms of cartesian bicategories. I will then focus on a specific subclass of cartesian equipments that closely resemble Rel as a cartesian equipment.

This talk is based on my master’s thesis.