Soichiro Fujii, RIMS, Kyoto University
Algebraic theories, bimodels and Gabriel-Ulmer duality
In this talk, I will revisit classical topics in categorical algebra, centring around the notion of bimodels of algebraic theories. An algebraic theory T is a small category with finite products, and its model in a category C is a finite-product-preserving functor from T to C. Dually, a comodel of an algebraic theory S in C is a finite-coproduct-preserving functor from the opposite of S to C. Combining these, we define an (S, T)-bimodel in C to be an S-comodel in the category of T-models in C. Bimodels valued in Set can be regarded as morphisms between algebraic theories. Indeed, we have the following chain of notions of morphisms between algebraic theories: profunctors preserving finite products in their second (covariant) variable, Set-valued bimodels, and finite-product-preserving functors. Modulo a caveat on Cauchy completeness, these correspond to the following classes of functors between their respective categories of Set-valued models: functors preserving sifted colimits, left adjoint functors, and left adjoint functors whose right adjoints preserve sifted colimits. Similar statements hold for essentially algebraic theories, i.e., small categories with finite limits, yielding Gabriel-Ulmer duality and its extensions.