Abstract: Tom Leinster explained long ago how to define infinity-algebras in a symmetric monoidal category with a notion of weak equivalence. I will present an analogous notion of linear infinity-operads, defined by replacing the familiar Segal condition by an additional structure map. These operads are parametrised by a category of trees, which admits a bar-cobar (or "Koszul") duality between its presheaves and the presheaves on the opposite category. Some other categories turn out to have this property as well, although it is not clear (to me, yet) what characterises such categories. (The talk is based on ongoing work with Eric Hoffbeck, Paris.)

Abstract: Equipments, a special kind of double categories, have shown to be a powerful environment to express formal category theory. We build a model structure on the category of double categories and double functors whose fibrant objects are the equipments, and combine this together with Makkai’s early approach to equivalence invariant statements in higher category theory via FOLDS (First Order Logic with Dependent Sorts) and Henry’s recent connection between model structures and formal languages, to show a result on the equivalence invariance of formal category theory.

Back in the late ’60s, Moss Sweedler introduced the concept of a “measuring kcoalgebra” as a space of generalized k-algebra maps. A particular case is that of the finite dual of a k-algebra, namely a coalgebra with the property that coalgebra maps into it naturally correspond to algebra maps into the classical linear dual of a coalgebra. Gavin Wraith was the first to observe that measuring coalgebras induce an enrichment of the category of k-algebras in k-coalgebras. Anel and Joyal first referred to the (tensored and cotensored) enrichment of dg-algebras in dg-coalgebras along with involved structures related to the bar-cobar construction as “Sweedler theory”. In this talk, we will investigate how this fact of an enrichment of monoids in comonoids, established in a broader context of locally presentable and braided monoidal closed categories, can lead to a manyobject generalization in the setting of monoidal double categories. In the process of extending such results in other double categories of interest, it turns out that the structure of an “oplax” monoidal double category is required. Analogous results therein are envisioned to provide insight to further cases of interest, for example that of symmetric coloured operads and cooperads.

Abstract: Compact closed categories are symmetric monoidal categories in which every object has a dual. Natural examples arise in algebra (the category of finite dimensional vector spaces), topology (the category of cobordisms), and logic (examples arising in the categorical quantum mechanics of Abramsky-Coecke). While studying a 2-categorical analogue of Tannaka reconstruction, it became clear that not much beyond the definition of a compact closed bicategory existed in the literature. This talk will present a definition of compact closed 2-category and some basic machinery one can use to do calculations within such a structure. Along the way I will discuss some aspects of the non-compact case as well. This is joint work with Juan Orendain and David Yetter.

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Abstract: Diagrammatic sets are a topologically sound alternative to polygraphs and strict ω-categories: we present the latest advances of the theory and some key points that place them as a natural candidate to bridge the gap between algebraic and non-algebraic models of higher categories.

Abstract: Mazel-Gee's localization theorem asserts that Rezk's classification diagrams compute the localization of relative (∞,1)-categories. This powerful theorem has many applications in detecting nontrivial localizations of (∞,1)-categories. However, Mazel-Gee's proof of the theorem, as he himself acknowledged, is unsatisfyingly complicated. In this talk, we will explain how we can deduce this theorem using the formalism of marked simplicial spaces, due to Rasekh, and an adaptation of classical results of Joyal and Tierney to the marked setting.

Abstract: Factorization methods are of great interest in mathematical physics, especially in higher quantization, presenting key technical features such as: local-to-global principle, built-in (co)filtrations and Poincaré/Koszul duality, that make them particularly appealing for calculations. This talk provides the audience with an overview on the basic tools employed in factorization homology and on the main results. It also keeps pragmatic applications into account with the purpose of elucidating some aspects of the theory.
No prerequisite knowledge of physics is required.

Abstract: We will report on a joint work with Bastiaan Cnossen and Tashi Walde on a type-theoretic foundation for synthetic category theory and propose a definition of an elementary topos in this new context (after a jest of André Joyal).
We will discuss in particular the role of set theory (or, equivalently, of the theory of strict 1-categories, constructively or not), and possibly reconsider what (categorical) logic is about in a set-free and purely univalent formal system with the expressive power of category theory.

Abstract: Lax limits can be expressed as certain weighted 2-limits. In this talk, I will present an explicit formula for the weight associated with a lax limit using a double categorical Grothendieck construction. I will compare this approach to Bird's, which relies on a 2-categorical version of the Grothendieck construction.
Our motivation for such a formula stems from the goal of implementing lax limits in the (∞,2)-categorical setting. In earlier work, we developed a framework for weighted (∞,2)-limits and introduced an ∞-analogue of the double Grothendieck construction. Building on this setup, I will explain how the strict formula for weights of lax limits can be adapted to define lax (∞,2)-limits in our framework.
This is joint work with Nima Rasekh and Martina Rovelli.

Abstract: There is an established notion of strict n-category and of (∞,n)-category for finite n. We will describe how there are two possibly competing ways of making sense of these for n=∞, depending on whether one takes the inductive or coinductive approach. The inductive and coinductive homotopy theories obtained from the strict homotopy theory of n-categories turn out to be equivalent, and are modeled by ω-categories. Instead, we will discuss how the inductive and coinductive homotopy theories obtained from the homotopy theory of (∞,n)-categories are not expected to be equivalent, mention what is known, and touch on work in progress with Viktoriya Ozornova towards modeling them through appropriate notions of complicial sets.

Abstract: The classifying diagram construction, originally due to Rezk, provides a way to produce a complete Segal space from an ordinary category, in a way that refines the usual nerve construction. It can be generalized to an analogous construction for double categories when the horizontal and vertical categories have shared isomorphisms. In joint work with Shapiro and Zakharevich, we identify when a bisimplicial set is the classifying diagram of a category, and when a trisimplicial set is the classifying diagram of a double category with shared isomorphisms. The motivation for this result is identifying the pointed stable double categories that correspond to CGW categories, thus providing the relationship between two general inputs for algebraic K-theory constructions.

Abstract: When a weak factorization system is built from a generating set of left maps using the small object argument, the resulting class of left maps is the least "saturated" class containing the generators: its closure under coproducts, cobase changes, transfinite compositions, and retracts. I will sketch a generalization to weak factorization systems generated by a small category of left maps. This is work in progress with Christian Sattler.

Abstract: Motivated by the study of weak identity structures in higher category theory, we recover Kock’s fat Delta category as the arities of a monad on the category of relative graphs. This immediately gives rise to a nerve theorem for relative semicategories as well as an active-inert factorisation system on fat Delta. In addition, this factorisation system makes fat Delta into a hypermoment category in the sense of Berger.

Abstract: The completion functor of Rezk is a key tool for understanding complete Segal spaces as (∞,1)-categories and was originally used to characterize weak equivalences in this model. The completion can also be used to refine nerves and various other constructions in the complete Segal space model and relates to Rezk completion in homotopy type theory. We generalize the completion functor and the characterization of the weak equivalences to the Θ_n-space model of  (∞,n)-categories.

Abstract: A sketch consists of a category equipped with specified collections of cones and cocones, and its models are functors to the category of sets that map these cones and cocones to limits and colimits. Sketches formalize the concept of a theory by describing logical operations via limits and colimits. In this talk, we present a homotopy-coherent generalization of sketches in the setting of ∞-categories. We show that numerous ∞-categories, including complete Segal spaces, ∞-operads, E_∞-algebras, spectra, and higher sheaves, can be constructed as ∞-categories of models of limit sketches. Moreover, we establish higher-categorical analogues of the classical correspondences with presentable and accessible ∞-categories.

Abstract: The technical notion of stability conditions on a triangulated category, introduced by Tom Bridgeland, plays a pivotal role in contemporary algebraic geometry and has numerous significant applications. A key result demonstrates that the set of stability conditions, equipped with a generalized metric, forms a complex manifold, known as the stability manifold.
In this talk, we revisit the construction of the stability manifold from a topos-theoretic perspective, offering a more conceptual and natural approach. Our method unfolds in stages, introducing intermediate structures of independent interest. First, we use logical tools to construct the "space" of t-structures as an étendue. Next, we redefine the space of slicings—certain families of t-structures parametrized by real numbers—as a function space derived from the étendue of t-structures. By leveraging a generalized Stone-Gelfand duality, we provide a natural explanation for the existence of a metric structure. Finally, combining these constructions, we obtain the metric space of stability conditions and reinterpret its manifold structure as a geometric morphism between toposes.

Abstract: As an elementary topos, Hyland’s Effective topos Eff models extensional Martin-L”of type theory with an impredicative universe of propositions. It also contains another, non-posetal, impredicative universe of “modest” sets. In joint work with Frey and Speight, we proposed a higher-dimensional version of this model with an impredicative, univalent universe, in order to give system F style, impredicative encodings of some (higher) inductive types. This model was based on cubical assemblies and exploited the constructive character of the cubical model structures introduced by Coquand. As subsequently shown by Swan and Uemura, however, the 0-types in this model are *not* equivalent to Eff, but to a “larger” topos, requiring a localization in order to recover Eff.
Toward the goal of an elementary infinity-topos containing Eff as the sub-topos of 0-types, and having an impredicative, univalent universe, we here consider the 2-truncated case of groupoidal assemblies, and some variations thereon, and establish some preliminary results.
This is joint work in progress with several researchers: Mathieu Anel, Reid Barton, Jacopo Emmenegger, Jonas Frey, and Pino Rosolini.

Abstract: It is a well-known result of Ahlbrandt and Ziegler that certain theories can be ‘reconstructed’ from the topological automorphism groups of certain models. Recently, this result has been extended by Ben Yaacov using topological groupoids.
We will show that such results can be understood as special cases of equivalences of topoi. We will explore descriptions of the (bi-)category of topoi using localic (i.e. point-free) groupoids and topological groupoids.

Abstract: Inspired by a recent characterisation of coherent topoi as a class of Kan injectives, we provide a tentative definition of fragment of geometric logic. Logics can be treated as mathematical objects, and one can study them from several points of view. Our main motivation is to reproduce Lindstrom-type characterisations of first order logic. We take the opportunity to discuss several aspects of Kan injectivity in the bicategory of topoi.

Abstract: Simplicial completion is a classical construction that turns a Quillen model category into a Quillen equivalent simplicial model category. It turns out that this construction can be applied more generally to a "premodel category" which is just a category with two weak factorization systems, without any relevant homotopy theory, and yet still outputs a model category.
I will present a new result that gives a model-independent description of the resulting infinity-category associated to this new model category.
As a special case, we obtain a general strictification theorem for dependently typed algebraic theories, which I will explain in detail. A generalization of our earlier result can also be seen as a general formulation of the connection between "coherence theorems" - expressing that certain freely generated weak models of an algebraic theory are set-truncated - and "strictification theorems" - expressing that weak models are equivalent to strict ones.
(joint work with Chaitanya Leena Subramaniam)

Abstract: While homotopy type theory (HoTT) has proved to be a powerful framework to reason synthetically about spaces and homotopy theory it is inherently problematic to use it to capture higher categories due to its lack of directed homomorphism types. Riehl--Shulman introduced simplicial HoTT as an extension where one can define hom-types using a directed interval. While this enables an extensive development of certain aspects of (∞,1)-category theory (such adjoint functors, (co)limits, and cartesian fibrations) it is missing categorical universes such as the (∞,1)-category of spaces. I will present recent work to further extend simplicial type theory to a framework called triangulated type theory that makes it possible to construct the (∞,1)-category of spaces internally and prove that it satisfies a directed univalence axiom: its homomorphisms correspond to functors between ∞-groupoids. I'll give an overview of this construction and point out some applications. This is joint work with Daniel Gratzer and Ulrik Buchholtz: https://arxiv.org/abs/2407.09146

Abstract: The semantic study of logical theories in its most general form should identify theories as fibrations of models over some bases. For instance, a classical framework developed by Ghilardi and Zawadowski allows a very detailed study of Heyting algebras by identifying them as sheaf of models on Kripke frames. More concretely, take the site (K,J) of (finite) Kripke frames with canonical topology. Any Heyting algebra will induce a sheaf on (K,J) by taking models on each Kripke frame, and this gives us a dual embedding of the category of (finitely presented) Heyting algebras HAf into the sheaf category Sh(K,J). This embedding is then used to show a lot of interesting result about Heyting algebras, e.g. the opposite of HAf is in fact itself a Heyting category, which implies Pitt’s celebrated result of second order quantifier elimination in intuitionistic propositional logic. In this talk, I will talk about the possibility of applying this philosophy to study first-order intuitionistic theories. In particular, I will discuss whether there could be a stack representation of Heyting pretoposes.

Abstract: I will outline an approach to realizability models of higher type theories wherein realizers themselves carry higher-dimensional structure. I will propose a definition of groupoidal combinatory algebra and discuss the category of groupoidal assemblies built over one of these.

Abstract: Tripos theory provides a very general and flexible framework for topos theory which allows us to see both localic and realizability toposes as instances of a common construction. The notion of tripos is however very high-level if compared to frames and partial combinatory algebras: in this talk, I will present arrow algebras, concrete structures which induce triposes generalising Miquel’s implicative algebras, in such a way to encompass both the localic and realizability examples. I will introduce morphisms of arrow algebras which correspond to appropriate morphisms of the associated triposes and toposes, thus showing how arrow algebras perfectly factor through the two constructions even on a categorical level. Finally, I will present a notion of nuclei on an arrow algebra corresponding to subtoposes of the induced topos, generalising the analogue locale-theoretical result.

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Abstract: We consider a few relatively recent results concerning the ways in which model-theoretic stable independence---both of the classical 2-dimensional variety and in higher dimensions---manifests itself in the context of locally presentable categories. We address, in particular, the fact that stable independence notions correspond precisely to combinatorial weak factorisation systems, and note a few ways in which this simplifies existing category-theoretic arguments, e.g. the proof of closure of combinatorial categories under pseudopullbacks. We note that in this context, any category with stable independence is excellent; that is, it admits stable amalgamation of objects in arbitrary finite dimensions. As excellence is typically connected with extreme niceness, e.g., eventual categoricity, this suggests that the current context is too specialised to allow us to address certain fundamental (and interesting) issues on the model-theoretic side. In particular, we note that even in a stable first-order theory, type-amalgamation may fail even at very low dimensions. We point to an old, moderately category-theoretic analysis of this situation by Goodrick, Kim and Kolesnikov, and suggest that more work is needed along these lines.

Abstract: We start with a motivating description of stability theory, an important subject in model theory. In particular, we discuss the role of independence relations in model theory. Guided by some concrete and simple examples, we will define what an independence relation is and how to approach it category-theoretically. In the final part of the talk I will discuss results on lifting independence relations along a functor. Given a functor F: C → D and an independence relation on D, we can lift the independence relation to one on C. The basic question then is: what properties does this lifted independence relation inherit from the original one?
This is joint ongoing work with Jiří Rosický.

Abstract: This talk will revisit the duality theory for clans presented in arxiv.org/abs/2308.11967 and then discuss the coclan of *finite direct categories*.

Abstract: The power and flexibility of the theory of ∞-categories gives a natural way of describing many classes of higher algebraic structures. Lurie's ∞-operads, for example, allow us to describe theories of operadic type, a class which suffices for many applications in homtopy theory. Generalizing classical Lawvere style functorial semantics to the higher setting in terms of finite limit ∞-categories provides a natural framework for thinking about more general higher algebraic theories. And higher geometric theories can be understood in the language of ∞-topoi.
Classically, all these types of theories have natural syntactic presentations, and these presentations are incredibly useful not only for proving theorems, but for computer manipulation in proof assistants. Cartmell's Generalized Algebraic Theories, for example, codify the basic syntax and structural rules we use in dependent type theories, while having the same expressive power as finite limit theories.
What should a higher Generalized Algebraic Theory be, and what kind of syntax might it have? In this talk, I will suggest some possibilities by generalizing the Baez and Dolan's +-construction for operads, which gives rise to the opetopes, to generalized algebraic theories. The resulting objects, the dependetopes, blend the geometric intuition of the opetopes with the additional syntactic structure to encode the higher coherences of type dependency.