In this talk, I will speak about joint work with Chris Kapulkin on a new proof of McCord's theorem, formulated in the language of abstract homotopy theory.

Traditionally, one studies a Grothendieck topos with respect to an elementary topos $\mathcal{S}$ called the base. A Grothendieck topos with respect to this base consists of another elementary topos with a bounded geometric morphism to $\mathcal{S}$. This approach has its drawbacks, though, since geometric morphisms are not the most natural notion of morphism between elementary toposes.

On the other hand, morphisms of arithmetic universes (AUs) are much closer to geometric morphisms. Vickers suggested that it might be possible to develop the theory of Grothendieck toposes using AUs. This idea has seen some success: for instance, Vickers recently developed an analogue for classifying toposes using AUs.

In this talk, I'll examine the special case of extending an AU $\mathcal{A}$ by a single indeterminate object $X$. I'll show that, just like for Grothendieck toposes, this yields the category $\operatorname{CoPsh}(\textbf{Fin}_\mathcal{A})$ of internal copresheaves on finite sets. This is the first step in giving a more concrete description of classifying toposes over an arithmetic universe, analogous to the case of a base elementary topos.

This work is the result of PhD research under the supervision of Simon Henry and Philip J. Scott, and is based on a conjecture of Simon Henry.

Towards showing this result Makkai introduced the notion of ultracategories and ultrafunctors, these are categories equipped with an ultraproduct functor (plus data and coherence). More recently Lurie reintroduced ultracategories and showed a similar in spirit version of conceptual completeness, stating that there is an equivalence between a certain class of functors called left ultrafunctors from the category of points of a coherent topos to $\mathsf{Set}$ and the Topos itself.

We want to extend Makkai and Lurie's results to any topos with enough points, the first obstruction is that the category of points of such toposes do not have a canonical notion of ultraproduct. Toward this, we introduce a new notion of generalised ultracategories, where the ultraproduct may not exist, but we may find instead the ``representable'' at the ultraproduct. We use this new structure to show a conceptual completeness theorem for geometric logic, stating that for any two toposes with enough points $E$ and $E^{'}$ there is an equivalence between $\mathrm{Geom}(E,E^{'})$ and $\mathrm{Left}$-$\mathrm{ultrafunctors}(\mathrm{Points}_E,\mathrm{Points}_{E^{'}})$. This result reduces to a one similar to Lurie's one if we replace the topos $E^{'}$ by the classifying topos of the theory of objects.

I will present a general notion of "polygraphs for a generalized algebraic theory" that cover all the known examples and capture this general idea of free models. At this level of generality the category of polygraphs we obtain have very poor properties, so the key point is to understand how various assumptions on the theory allows to recover various classically expected or desired properties of these category of polygraphs - I will in particular start discussing the question of when these are presheaves categories.

The long term goal is to develop a more general understanding of how properties of the category of polygraphs relates to homotopy theoretic properties of the corresponding higher structures through coherence and strictification problems, but there is still a lot of work left to get there.

(Work in progress, joint with Daniel De Almeida Souza)

When studying models of dependent type theory, we have a new parameter: thanks to the work of Voevodsky, we know that each such model is canonically equipped with a class of equivalences. It thus makes sense to consider a small category $D$ also equipped with a class of equivalences and a subcategory of the presheaf category $[D, \mathsf{E}]$ consisting of those presheaves that take equivalences to equivalences. Such constructions are again of fundamental importance, as they were used for example in the proof of Voevodsky's conjecture on homotopy canonicity of homotopy type theory.

In this talk, I will report on joint work with Fiore and Li (arXiv:2410.11728) on identifying sufficient conditions on a category $D$ with a class of equivalences so that $[D, \mathsf{E}]$ and its subcategory of equivalence-preserving presheaves again form a model of dependent type theory.

A discrete dynamical system is a set $X$ with a function $f\colon X \to X$. Thus, the category of discrete dynamical systems is $\mathsf{Set}^{B\mathbb{N}}$. We can (functorially) associate to a discrete dynamical system its state space, which is a specific type of directed graph. To study state spaces, we introduce the notion of a cycle set. We can assign a cycle set to a directed graph. The composition taking a discrete dynamical system to its cycle set is a right adjoint, so in particular preserves limits. As a proof of concept of our methods, we give a conceptual proof of a generalization of a decomposition theorem of Kadelka et al. (2023).

This talk is based on joint work with D. Carranza, C. Kapulkin, R. Laubenbacher, and M. Wheeler.

This talk is partly based on joint work with Keisuke Hoshino (Kyoto Univ) as well as my master's thesis.

Classical models deal with relative categories, categories equipped with a class of weak equivalences. These generalize the notion of homotopy equivalences in the category of topological spaces. This implies that constructions and invariants should be studied up to these weak equivalences instead of isomorphisms. Examples include model categories and cofibration categories.

Modern models rely on the idea that the hom-sets are replaced by spaces of morphisms. This has various implications, such as that compositions of morphisms are defined only up to contractible spaces, or that morphisms should not be compared by equality, but rather by homotopies, themselves subject to comparisons by higher homotopies. Examples include quasicategories and complete Segal spaces.

The former approach is best suited for constructing universal objects whereas the latter approach is used for working with universal properties. Therefore, one is interested in comparing these two approaches. Concretely, we would like to know how classical models translate into modern ones and vice versa.

This talk compares two models of the theory of finitely cocomplete \(\infty\)-categories: cofibration categories and finitely cocomplete quasicategories. The equivalence of their theories has originally been proved by Karol Szumiło. We will give an alternative proof that does not rely on a specific choice of a functorial localization and avoids the construction of a quasi-inverse. Instead, we exploit the finite completeness of both homotopy theories.

Recently, Par\'{e} (Outstanding Contributions to Logic 20, Springer, 2021) considered adjoints and Cauchy completeness in double categories, and showed that an $(R,Q)$-bimodule $M$ has a right adjoint in the double category of commutative rings if and only if it is finitely generated and projective as a left $R$-module. Subsequently (TAC 43, 2025), we incorporated this adjoint result for double categories into a version of the 2017 theorem with Wood, which we then applied to rings and quantales. However, the proofs of the latter were again separate due to the finiteness condition on rings.

In this talk, adding additional conditions on $\cal V$, we introduce a notion of {\it projective} module over a monoid in $\cal V$ which includes finiteness for rings and, when added to our theorem characterizing adjoints in a double category, gives a single proof of the application to rings and quantales.

In the talk, I will introduce and motivate the notion of elementary $\infty$-topos, and I will sketch the progress that has been made so far towards proving the conjecture. I will explain how HoTT presents such an elementary $\infty$-topos via its syntactic category built from the syntax and rules of the type theory. First, I will use the fact that the syntactic category of HoTT has the structure of a tribe in the sense of Joyal. I will extend Joyal's theory of tribes by introducing the notion of a univalent fibration in a tribe. These fibrations exist in particular in the syntactic category of HoTT. In the second step, I will explain how each such tribe presents via its localisation an $\infty$-category and if the tribe has enough univalent fibrations then this $\infty$-category is an elementary $\infty$-topos.

I will discuss the details of this construction and present several examples. I will also present a 2-monad for which the double categories with a DOFS form the algebras, and describe the induced maps between these double categories. I will also discuss the interaction between these factorization systems and double fibrations.

In this talk, I shall introduce dagger Drazin inverses, discuss its properties and examples, and sketch out the connections between dagger Drazin inverse and its close relatives -- Drazin inverse and Moore-Penrose inverse.

This is joint work with Robin Cockett and JS Pacaud Lemay.

Two basic examples are the category of smooth manifolds and the category of commutative rings.

Poon Leung has proven that to make a category $\mathcal{C}$ into a tangent category is equivalent to equip it with a nice monoidal functor from a subcategory of Weil algebras generated by $\mathbb{N}[x]/(x^2)$ to the category $\mathrm{End}(\mathcal{C})$ of endofunctors of $\mathcal{C}$.

I’ll explain how it could be interesting to define the notion of a Weil category as a category $\mathcal{C}$ with a nice monoidal functor from the category of all the Weil algebras to $\mathrm{End}(\mathcal{C})$.

We’ll then see how both the category of smooth manifolds and the category of commutative rings should not only be tangent categories but Weil categories.

This is work in progress. The talk will explain the plan and hopefully suggest some precise definition for the notion of a Weil category.