Thesis Projects

This page presents a curated selection of potential thesis projects for students interested in working with me as their advisor. I continue to receive an increasing number of requests for supervision — particularly from fellow Italians — often driven by a shared frustration: doing category theory in Italy can be disheartening.

It is unfortunate that much of Italian academia perpetuates the narrow and misguided belief that category theory is merely a tool of algebraic geometry or homological algebra —or worse, ignores the language of category theory altogether. Across the years, this reductive view misled students and researchers alike; it is a deeply troubling perspective —not just intellectually, but ethically— and I have always strongly opposed the mindset that sustains it.

This list is meant to counter the narrative. Instead, I'd like to convey the idea that no matter where you come from, and what Mathematics you care about, category theory is useful for that. Yes, for that as well. And for that too. Here you'll find a selection of diverse topics in category theory that I believe are worth exploring, and that I would be happy to supervise (drop me a line at my email address). This can also be taken as a list of references for self-study, and I am always happy to give advice.

Each project is briefly described and marked with its suitable level — 📚 Bachelor's, 🎓 Master's, or 🧬 PhD (this is the research I am myself conducting right now: this means essentially that whatever you end up doing in that direction has a high chance of becoming a publication) —along with a list of references for what catches your eye.

The Theory of Combinatorial Species | 📚 Bachelor thesis ★★★☆☆

The theory of combinatorial species applies category theory to enumerative combinatorics through the study of the category of functors $\mathbf{Bij}\to\mathbf{Set}$, where $\mathbf{Bij}$ is the category of bijections between finite sets. It's a flourishing area with connections to algebra, topology, logic, computer science and pure category theory.
Further Reading:
Coalgebraic Techniques | 📚 Bachelor thesis ★★☆☆☆

Coalgebras for endofunctors are a category-theoretic tool to describe state-based systems and their behaviour. The category of coalgebras for a functor $F \colon \mathcal{C} \to \mathcal{C}$ is the category of pairs $(X, \alpha)$ where $X$ is an object and $\alpha\colon X \to FX$ is a morphism.
Further Reading:
Factorization Systems in Algebra and Geometry | 📚 Bachelor thesis ★★☆☆☆

A category $\mathcal{C}$ is equipped with a factorization system if there's a way to decompose every morphism of $\mathcal{C}$ as composition of morphisms in classes that are, in a suitable sense, orthogonal. For example, every function $f \colon X\to Y$ between sets can be factored as an epimorphism, followed by a monomorphism. Such an extremely general notion pervades mathematics in its entirety.
Further Reading:
Linear and Set-Theoretic Operads | 📚 Bachelor thesis ★★★☆☆

Operads are a foundational tool to capture the notion of structure; introduced by Peter May in algebraic topology, they rapidly found their way as a general tool to talk about operations and representation, and nowadays they find application in algebra, logic, computer science, mathematical physics, representation theory, combinatorics, and more. The theory of operads is a vast field, and this project can focus on the already vast area of linear operad theory (where all objects involved are vector spaces).
Further Reading:
Yoneda Structures | 📚 Bachelor thesis ★★★★☆

Formal category theory attempts to outline the properties of a 2-category that make it behave like the "concrete" 2-category of categories, functors, and natural transformations; much like category theory tries to unravel the formal content of mathematics, formal category theory tries to unravel the formal content of category theory. A Yoneda structure on a 2-category $\mathcal K$ allows to reason with an internal Yoneda lemma, and derive the theorems of category theory as a consequence. This project aims at providing an introduction to YS unifying a body of scattered references and results.
Further Reading:
The Theory of Relative Monads | 🎓 Master thesis ★★★☆☆

A relative monad is something that behaves exactly like a monad, besides the fact that it is not an endofunctor.
Further Reading:
Coend Calculus in Indexed Category Theory | 🎓🧬 Master/PhD thesis ★★★☆☆

It should be possible to develop coend calculus inside the language of indexed category theory. To my knowledge, no one attempted to do this; partly, I believe because this is an exercise in style, but having at our disposal coend calculi for other flavours of formal category theory, I find it a worthy project. What makes this project a bit different from the others is that the question is rather open-ended; the student will have to figure out in what sense the general claim is true.
Further Reading:
Grothendieck Derivators | 🎓 Master thesis ★★★☆☆

Derivators arose in the late work of A. Grothendieck to address the many shortcomings of triangulated categories; instead of considering just one derived category $\mathbf{D}(\mathcal A)$ attached to an abelian category, one can consider the pseudofunctor $J\mapsto \mathbf{D}(\mathcal A^J)$, and more in general any pseudofunctor of type $\mathbf{Cat}^\text{op} \to \mathbf{CAT}$. This theory expresses all facts about derived categories, in a neat, functorial way, avoiding the drawbacks of focusing on the "base" category $\mathbf{D}(1)$ alone.
Further Reading:
Bisimulation and Homotopy Theory | 🎓🧬 Master/PhD thesis ★★★★★

I have thought for quite a bit of time already that "bisimulation" in logic and CS is a relation on transition systems / automata that resembles a lot a notion of homotopy equivalence in abstract homotopy theory. This project aims at exploring this connection, and possibly developing a theory of bisimulation in the style of homotopy theory. I consider this a difficult project to tackle, especially for a Master's thesis, as it's very open-ended and requires a lot of background reading.
Further Reading:
The Theory of Model 2-Categories | 🎓🧬 Master/PhD thesis ★★★★★

Some 2-categories have the property that each hom-category $\mathbf{Hom}(X,Y)$ is a model category, and composition $\_\circ\_ \colon \mathbf{Hom}(Y,Z) \times \mathbf{Hom}(X,Y) \to \mathbf{Hom}(X,Z)$ is a left Quillen bifunctor. Examples are abundant and draw from diverse branches of mathematics. What is the general theory of such categories?
Further Reading:
Monoidal Topology | 🎓 Master thesis ★★☆☆☆

Monoidal topology applies techniques of categorical algebra to general topology and order theory. It starts from a (now old) result of Barr characterizing (all!) topological spaces as algebras for a certain monad; monoidal topology regards this result as a particular instance of the fact that "all" metric, topological and order-theoretic structures can be seen as enriched categories.
Further Reading:
Contravariant Monads and Their Applications | 🎓🧬 Master/PhD thesis ★★★★☆

A contravariant monad is exactly like a monad, besides the fact that it's a contravariant functor.
Further Reading:
Shape theory via profunctors | 🎓 Master thesis ★★★☆☆

Shape theory is the homotopy theory of pathological spaces, that defines homotopy-invariants like cohomology and homotopy groups for spaces which are not CW complexes. Surprisingly, shape theory is encoded in a branch of 2-category theory. This project would survey on both the algebraic topology and the categorical side of shape theory, and provide a survey of the current state of the art. Such a manuscript would be immensely valuable for the community (and easily publishable), as the current literature is scarce, scattered and sometimes inaccessible.
Further Reading:
The Fibration of Wedges and Cowedges | 🎓🧬 Master/PhD thesis ★★★★☆

The project started as a spinoff of what my PhD student Andrea Laretto has been doing for the last few years; coend calculus turns out to be a promising way to give a semantics for quantifiers of directed type theory. What is the fibrational side of this story? One can cook up a certain bifibration (the "fibration of wedges" of the title), and study its categorical properties. This is somewhat of a hard problem. A prospective goal would be to taxonomize the properties of this fibration, and to provide semantic backup for the syntax developed by Andrea.
Further Reading:
Fibrations Associated to Parametric Functors | 🧬 PhD project ★★★★☆

Every functor $F\colon \mathcal{A}\times \mathcal{X}\to\mathcal{X}$ defines a fibered category "of algebras" $\mathcal A\ltimes_F \mathcal X$ over $\mathcal{A}$, where all $F_A$-algebra structures are considered coherently and compatibly. Such situations should be thought of as generalized instances of semidirect products, hence the notation. Representation theory provides a rich source of examples, as well as categorical logic, topology, geometry. I have plenty of side projects that can be developed in this direction.
Further Reading:
2-Categories of Automata | 🧬 PhD project ★★★★★

Since the end of the '70s people in the category theory community explored connections between formal language theory and formal category theory. The kind of structure in which processes naturally organize is a natural instance of a monoidal bicategory in the sense of Gordon-Power-Street's famous Memoir Coherence of Tricategories.
Further Reading:
Derivations on a Category | 🧬 PhD project ★★★★★

What is a category equipped with an endofunctor that is "linear and Leibniz"? The answer will shock you.
Further Reading: