Gabriel Goren-Roig

Categorical logic can be understood as the perspective that logics, understood as inductively defined syntax for well-formed formulas and well-formed proofs, constitute syntactic presentations for categorical structures. Although the field has a long history and a deep connection to type theory and the theory of programming languages, traditionally it has been disconnected from other areas of logic more connected to combinatorics and computational complexity, as is the case of finite model theory and the theory of databases. An intersection between these two traditions, Structure and Power, has begun to take form owing greatly to so-called comonadic semantics, a research program based on game comonads and arboreal categories. Currently, I am particularly interested in using this framework to understand what kinds of logics are possible near the lower end of the expressive power spectrum as exemplified by Basic Modal Logic. In this context I have worked on a variation of Basic Modal Logic with (non-unary) relations instead of propositional variables and developed its fundamental model theory using the framework of game comonads and arboreal categories.

More recently, I have been studying the conceptual neighborhood of classical Kripke semantics using arboreal categories. I have approached this from two sides, which correspond to what I call arboreal coreflections and arboreal categories of shape trees, respectively (the latter in collaboration with Tomáš Jakl and Luca Reggio). None of this work is published yet, but I should write some more about it and it constitutes a big part of my PhD thesis.

Computational category theory means two things, essentially: from the applied side, it means computer algebra for category theory. From the theoretical side, it means a theory of presentations for category theory (as in “group presentations”). I am interested in it for both applied and theoretical reasons.

On the applied side, being able to compute answers to categorical questions algorithmically seems fundamental to making category theory more applicable in the sciences. Categorical database theory is an excellent example of application where it is crucial—if the idea is to make any sense at all-to not only develop a mathematical theory at a semantic level but also explicitly develop its syntactic and algorithmic counterpart.

On the theoretical side, I am interested in how syntax refines semantics, in the sense that new phenomena arise when we become stricter about our notions of mathematical equivalence. For instance, once we have focused on syntactic descriptions we can narrow the scope even further to the finite ones and analyze the classes of finitely presentable objects that arise.

In my work, an interesting example of this phenomenon came up when trying to understand the syntactic representations of profunctors, which is an important point for categorical database theory. In particular, the fundamental equivalence between functors \(\mathcal{C}^\text{op} \times \mathcal{D} \to \mathsf{Set}\) and functors \(\mathcal{C}^\text{op} \to \mathsf{Set}^\mathcal{D}\) does not carry over to the world of finite syntactic descriptions. This is the main observation that led to the paper Presenting Profunctors.

I am interested in developing a mathematical yet qualitative understanding of emergence (“the whole being more than the sum of the parts”), in contrast with-but also complementary to-predominant quantitative approaches based on information theory. I believe category theory is key in achieving this since it gives the appropriate language to talk about the composition algebra of interconnected systems. From this perspective, emergence stands tightly connected to the concept of compositionality.

I am also interested in qualitative notions of information content, either in terms of minimal description length (such as Kolmogorov complexity a.k.a. algorithmic information, which featured prominently in my Licentiate thesis), or in terms of coarse-graining: “information is that which is lost through coarse-graining”. This perspective contributes to my interest in model theory: the equivalence classes given by logical indistinguishability of models constitute a form of logical or language-based coarse-graining.