Pacific Category Theory Seminar
The Pacific Category Theory (PCT) seminar is an online seminar for the category theory community in the Asia-Pacific time zone and beyond. We aim to cover topics in all areas of pure and applied category theory in relationship with other disciplines.
The seminar will run once a month on Friday at 10am JST/11am AED (1am UTC). Here is the zoom link to partipate. Previous talks are available on the seminar's youtube channel.
Previous talks
September 11, 2026
- Speaker: Yuki Maehara (Tokyo Metropolitan University)
Title: Towards a homotopy theory of algebraic weak ω-categories
Abstract: Mainstream models of weak infinite-dimensional categories (such as quasi-categories or complicial sets) are based on certain non-globular shapes (such as simplices). This allows one to encode the desired category-like *structure* as an existence *property*. For example, composition of 1-cells can be encoded as "for any composable pair f and g, there exists a triangle two of whose edges are f and g", and such a triangle exhibits the third edge as a composite of f and g. We don't require this triangle to be unique for given f and g, so composition is not well defined on the nose. This makes it impossible even to state strict axioms, and hence the resulting model is necessarily weak. But it is not too weak — composition is still "essentially" unique, satisfies the unit and associativity laws "up to equivalence", and so on — thanks to similar existence properties of higher-dimensional shapes.
What happens if we instead use the familiar globes so that, for example, the boundary of a 2-cell is given by a parallel pair of 1-cells rather than a triangle? Well, the 2-globe (or in fact the n-globe for any n) doesn't contain a composable pair of 1-cells, so now we must encode composition as an actual algebraic operation rather than as an existence property; this is what "algebraic" means in the title of this talk. I will give an overview of a joint project with Soichiro Fujii and Keisuke Hoshino on such algebraic weak ω-categories, namely those of Batanin and Leinster, which are the Eilenberg–Moore algebras for a suitable monad on the category of globular sets.
Compared with the mainstream models (formulated in the language of model categories or ∞-categories), this definition has the downside that it does not naturally come equipped with a homotopy theory, which makes it difficult to capture up-to-equivalence phenomena and constructions. On the other hand, it has the upside that these algebraic weak ω-categories feel much closer to strict ω-categories, particularly because not only the shapes of the cells but also the algebraic structure is designed to mirror the familiar behaviour of their strict counterparts. The overarching theme of this project is therefore to develop a suitable homotopy theory of algebraic weak ω-categories, drawing on the strict case for as much intuition — and even proof strategies — as possible.
Recording and slides
August 28, 2026
- Speaker: Alyssa Renata (Imperial College London)
Title: The Game Spectrum of a Monad?
Abstract: In recent work joint with Richard Garner and my supervisor Nicolas Wu, inspired by the computational effects interpretation of monads and algebraic theories due to Moggi, Plotkin and Power, we constructed an adjunction between the category of monads on set and the category of small categories and retrofunctors/cofunctors. Under the computational effects interpretation, the monad encodes computations which may interact with or affect some environment. The left adjoint to this construction assigns to each monad what is called its **behaviour category**, which may be seen as a model of the environment itself: the objects are the possible states, and the morphisms transitions between states. This adjunction ought to be seen as a kind of duality, with the behaviour category acting as the spectrum of a monad. The first half of the talk will be dedicated to reviewing this construction and its surrounding adjunction.
The second half of the talk is about my recent attempt with Richard at rectifying a defect of the theory: many monads have trivial behaviour category! This is undesirable because of an analogy we want to develop with schemes: the monads replace the (commutative) rings, while the behaviour category should replace the affine schemes. But of course, an affine scheme/spectrum should faithfully encode its corresponding ring/monad, and this is evidently not so for the behaviour category. Our proposal then is to replace the behaviour category by a particular 2-player game (between the program and the environment), from which the data of the behaviour category can be extracted by inspecting its (winning) strategies. The point is that the behaviour category is trivial when there are no winning strategies, but this need not imply triviality of the game itself. To be more precise, we will investigate the relationship with the template games of Paul-André Melliès, which he used to build higher-dimensional models of linear logic.
Recording and slides
August 7, 2026
- Speaker: Sveta Makarova (ANU College of Science)
Title: Contractions and flops via moduli of non-pure sheaves
Abstract: The goal of the talk is to convince the audience that 2-categorical enhancements of some classical geometric objects yield interesting geometric constructions. Consider the stack of ideal sheaves of l points on a smooth projective variety X. Its good moduli space is the well-known Hilbert scheme of l points Hilb^l(X). In this talk, I will define a certain enlargement U of this stack and prove that it admits a good moduli space. Furthermore, I will describe a different open substack of U that admits a good moduli space and explain how it yields a surgery diagram via non-GIT wall-crossing. As an application in the toy case l=1, I will explain how this construction helps contract rational curves on higher-dimensional varieties and, time permitting, use the interpretation of the surgery as a fine moduli of sheaves to prove instances Kawamata’s DK-hypothesis in this setting.
This is joint work with Andres Fernandez Herrero.
Recording and slides
July 10, 2026
- Speaker: Lili Shen (Sichuan University)
Title: Towards the categorical foundation of quantale-valued sets
Abstract: A classical set may be regarded as a set equipped with an equality relation valued in the two-element Boolean algebra. Fourman and Scott generalized this idea by introducing Ω-sets, where equality is valued in a frame Ω; this construction is closely related to sheaves and topos theory. Höhle and his collaborators later developed the theory of quantale-valued sets, replacing the frame Ω by a quantale Q.
This talk provides an overview of the theory of quantale-valued sets and the ongoing efforts to establish its categorical foundation. I will explain why Q-sets form a natural and interesting generalization of Fourman–Scott Ω-sets, and then present two recent results that clarify the extent to which the category of Q-sets behaves like the category of sets or the category of Ω-sets.
First, for a commutative and divisible quantale Q, the category of Q-sets is a topos if and only if Q is a frame. Second, for the unit interval [0,1] equipped with a continuous t-norm *, the category of ([0,1], *)-sets is cartesian closed if and only if * is the minimum t-norm.
Recording and slides
June 5, 2026
- Speaker: Thomas Seiller (CNRS, JFLI)
Title: From Statistical Data to Geometry and Logic via Isbell Nuclei
Abstract: The impressive results obtained by generative language models suggest that statistical information contained in a sufficiently large corpus can be used to recover significant linguistic structure. In particular, Levy and Goldberg (2014) showed that word embeddings can be recovered from low-rank factorisations of word co-occurrence matrices, as obtained in practice through singular value decomposition. While this provides an important first step towards understanding mathematically the structure exploited by language models, it remains limited, notably in its treatment of compositionality.
In this talk I will present a categorical refinement of this perspective, drawing on enriched category theory and realisability methods for linear logic. Starting from a real-valued measure (M : C x D -> R), one studies the Isbell nucleus of the induced enriched adjunction. This construction simultaneously generalises classical formal concepts defined from binary relations and singular value decomposition of linear operators.
While every nucleus carries a canonical lattice structure, in the statistical setting this is only part of a considerably richer picture. On the one hand, the nucleus admits a tropical geometric structure, with row and column spaces appearing as dual presentations of the same object, equipped with a canonical projective metric and polyhedral decomposition. On the other hand, when the underlying data is additionally equipped with concatenation, it carries a type-theoretic structure arising from linear realisability and closely related to substructural logics.
Recording and slides
May 8, 2026
- Speaker: Ross Street (Macquarie University)
Title: Homodular pseudofunctors as objective invariants
Abstract: As Riemann proved, a lot can come out of an 8 page paper! There are two techniques used in the paper [André Joyal, Calcul intégral combinatoire et homologie des groupes symétriques, C.R. Acad. Sci. Canada VII(6) (Dec. 1985) 337--342] which fascinate me.
The author's goal is to prove something about how the homology of the symmetric group on n symbols sits in that on n+1 symbols. Rather than specify a particular homological functor, his first technique is to construct a universal one and prove the result for that. The property in question is preserved by additive functors and so holds for any homology.
The second technique is to use his theory of (virtual) species of structure where passing from n to n+1 gives differentiation. My goal is to do something similar for the general linear groups over a fixed finite field. I have begun the adaptation of the two techniques and hope the results so far will be of independent interest. The two strands have yet to conflow into the desired application.
Recording and slides
April 24, 2026
- Speaker: Dusko Pavlovic (University of Hawaii)
Title: From concept mining to categorical nuclei and tight completions
Abstract: While students learn from teachers and textbooks, machines learn from datasets crawled on the web. Either way, the concepts arise as invariants of the matrices of term-situation contexts. The process of learning is therefore construed as an instance of spectral decomposition through nuclear spaces of latent concepts. When the data are not just counted and averaged, but stored as data sets, the matrix entries are not numbers but sets. The induced matrices of sets form profunctors (distributors) under the actions of the categories of previously mined concepts. This gives rise to the task of spectral decomposition of profunctors, and the quest for the induced nuclear adjunctions.
Some special cases are well-known and widely used. The Formal Concept Analysis (FCA) mines concepts from relational and posetal contexts. Latent Semantic Analysis (LSA) mines the latent concepts from given numeric contexts, capturing the cumulative correlations as bimodules. The family of linear concept analysis algorithms is among the most run on the web, since it underlies all personalized recommendation and profiling systems. The feedback loops inherent in such systems cause the information cascades and the dreaded "echo chambers”. The engineering mitigations led to the context matrices of sets and suggested the construction of the categorical nucleus, which reopened and answered a long abandoned fundamental question.
This presentation includes work driven by ongoing collaborations with Dominic Hughes.
Recording and slides
March 27, 2026
- Speaker: Rose Kudzman-Blais (RIMS, Kyoto University)
Title: (Bi)categorical Semantics for Non-Commutative Linear Logic
Abstract: Girard introduced a sub-structural logic, without contraction and weakening, in 1987 known as linear logic. Linear logic was initially introduced as a commutative logic, however its sophisticated structural rules allowed the further introduction of non-commutative variants. Of note are Lambek’s classical bilinear logic and Yetter’s cyclic linear logic. Both are non commutative variants of multiplicative linear logic, wherein tensor and par are non-commutative connectives, but the former considers right and left versions of linear negation, while the latter has only one coherent version. In this talk, we shall consider both these variants and do a deep dive into their categorical and bicategorical semantics as developed by authors Barr, Cockett, Kowslowski and Seely over the years.
Recording and slides
February 27, 2026
- Speaker: Yuki Imamura (RIMS, Kyoto University)
Title: A formal category theoretic approach to the homotopy theory of dg categories
Abstract: A dg category is a category enriched over the category of complexes of modules. Arising from the homotopy theory of complexes up to quasi-isomorphism, dg categories admit a natural homotopy theory in
their own right, in which the weak equivalences are the quasi-equivalences.
In this talk, I present an approach to the homotopy theory of dg categories from the viewpoint of formal category theory. Concretely, I construct a proarrow equipment in the sense of Wood that captures the homotopy theory of dg categories, and study the behavior of homotopy limits in dg categories within this framework.
Recording and slides
January 16, 2026
- Speaker: Taichi Uemura (Nagoya University)
Title: A direct-categorical approach to opetopic sets and opetopes
Abstract: Opetopes and opetopic sets were introduced by Baez and Dolan as a combinatorial approach to weak ω-categories. Since its birth, several equivalent definitions have been proposed.
Recently, Leclerc gave a posetal definition of opetopes, where an opetope is encoded as a poset of cells ordered by the subcell relation. This seems to be the most elementary and simple definition of opetopes, but there is some complication related to loops.
In this talk, I propose another elementary definition of opetopes, encoding an opetope as a direct category rather than a poset. Loop issues are resolved by allowing distinct parallel morphisms, and the theory of opetopic sets gets simplified.
Recording and slides
December 12, 2025
- Speaker: Richard Garner (Macquarie University)
Title: Universal enrichments
Abstract: For a given category C, there are all sorts of things we might enrich it in. For example, the category of complex vector spaces can be enriched in commutative monoids, or abelian groups, or real vector spaces, or complex vector spaces. In this talk, we explain how, for any locally presentable category C, there is a universal locally presentable monoidal category V in which it can be enriched. The fun part is trying to calculate V for particular choices of C; in general, it is rather intractable but sometimes we get lucky!
Recording and slides