| Homepage. |
| Programme. |
| Participants. |
| Local information. |
All times are EEST (Eastern European Summer Time), UTC+3.
We define torsion theories in a bicategory equipped with a bizero object, bikernels and bicokernels, following the classical approach of torsion theories in pointed categories. We discuss examples for symmetric monoidal categories, categorical groups, crossed modules and abelian butterflies. When relevant, we will compare with recent examples of homotopy torsion theories studied with Sandra Mantovani and Mariano Messora. We will also show that, passing to the bicategorical setting, some relevant examples of 1-dimensional torsion theories collapse. This is a work in progress with Mariano Messora.
A metric compact Hausdorff space is a Lawvere metric space equipped with a compatible compact Hausdorff topology (which does not need to be the induced topology). These spaces maintain many important features of compact metric spaces, but the resulting category is much better behaved.
In the category of separated metric compact Hausdorff spaces, we characterise the regular monomorphisms as the embeddings and the epimorphisms as the surjective morphisms. Moreover, we show that epimorphisms out of an object X can be encoded internally to X by their kernel metrics, which are characterised as the continuous metrics below the metric on X. Finally, as the main result, we prove that its dual category has an algebraic flavour: it is Barr-exact. While we show that it cannot be a variety of finitary algebras, it remains open whether it is an infinitary variety.
Based on M. Abbadini, D. Hofmann. Barr-coexactness for metric compact Hausdorff spaces. Theory and Applications of Categories, 44(6):196-226 (2025).
The full, reflective subcategory of locales spanned by so called measurable locales provides a wonderful place for measure and probability theory. It is anti-equivalent to the category of commutative Von-Neumann algebras via a measurable version of Gelfand duality. A central object for measure theory is the measurable locale of Lebesgue reals, which is obtained as the complete Boolean algebra of measurable subsets of the reals modulo null sets. Its topos of sheaves provides a natural home for Lebesgue integration, in which almost everywhere equivalence is automatic. We will give a simplified description of the Lebesgue reals, by showing that it is obtained from a Grothendieck topology on the poset of compact subsets, and show that it is determined as a locale by a universal property. We give a general construction on how to do measure theory, starting from small data, that circumvents the traditional method of using sigma-algebras.
The manifold of stability conditions on a triangulated category plays a pivotal role in contemporary algebraic geometry and has numerous significant applications. 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 space-like structures of independent interest. We conclude and reinterpret the manifold of stability conditions as a topos-theoretic spectrum.
Grothendieck Abelian categories have many features analogous to toposes. Categories of quasicoherent sheaves of O-modules on schemes supply an important class of examples. Any quasicompact quasiseparated scheme can be reconstructed up to an isomorphism from its category of quasicoherent sheaves (Gabriel-Rosenberg theorem). For an affine scheme, by the affine Serre's theorem, the quasicoherent sheaves correspond to the modules over the coordinate ring (global sections of the structure ring). Noncommutative algebraic geometry may be formulated as studying 'spaces' represented by Abelian categories which are glued from local models for which one usually takes the categories of modules over (noncommutative) ring(oid)s. I will sketch two developments from this point of view. Firstly, how to introduce noncommutative analogues of torsors in wide generality, using properties of certain adjunctions. Secondly, how to describe a 2-category of noncommutative spaces by defining such spaces by gluing representable 2-presheaves (of categories) on an appropriate (analogue of a) site of noncommutative affine schemes.
In my talk I describe a comma 2-comonad on the 2-category whose objects are functors, 1-cell are colax squares and 2-cells are their transformations. I give a complete description of the Eilenberg-Moore 2-category of colax coalgebras, colax morphisms between them and their transformations and I show how many fundamental constructions in formal category theory like adjoint triples, distributive laws, comprehension structures, Frobenius functors etc. naturally fit in this context. Then I proceed to describe various pseudo distributive laws between a comma 2-comonad and its cousins - the associated split fibration 2-monad and the associated split cofibration 2-monad. The former is an instance of a pseudodistributive law which Garner used in his description of Szabo’s polycategories, and the pseudoalgebras for the latter are Beck-Checalley fibrations. Then I proceed to show how this contexts is related to Bunge and Funk admissible 2-monads whose Eilenberg-Moore 2-category of algebras are characterised in terms of (co)completeness. Finally I describe the Kleisi 2-category of the associated split fibration 2-monad by means of its bifibrations which are defined by a certain bicomma object condition and the corresponding comprehensive factorization for those 1-cells which have an admissible domain.
The concept of monadicity is fundamental in category theory, permitting us to view the objects of one category as objects in a base category that are equipped with algebraic structure. There are several methods for verifying monadicity, but one of the most important is the monadicity theorem, which provides necessary and sufficient criteria that are typically straightforward to check. In many situations, however, we are not interested in arbitrary algebraic structure, but only that which can be axiomatised in terms of operations with particular arity: for example, in universal algebra, we are typically interested in algebraic structure with finitary operations. This prompts the study of relative monadicity.
In this talk, I will introduce the concept of relative monadicity, and present two relative monadicity theorems. The first is an analogue of the classical monadicity theorems due to Beck and Paré. The second is an analogue of the pasting law for pullbacks. As a application, I will derive a monadicity theorem for the monads with arities of Berger–Melliès–Weber and the nervous monads of Bourke–Garner.
This talk is based on the paper "Relative monadicity" with Dylan McDermott.
Skew monoidal categories generalise monoidal categories by relaxing the requirement that the associativity and unit constraints be invertible, while also fixing a suitable orientation of these morphisms. An early example of such a structure was described by Altenkirch, Chapman, and Uustalu (2010) in the context of functor categories, where they showed that monoids in this laxified monoidal structure correspond to relative monads. The term skew monoidal category was later introduced by Szlachányi (2012), who used it to describe structures arising from bialgebroids over a ring. A simple example is obtained by taking a bialgebra B in Vect and defining the tensor as XY=X⊗B⊗Y.
It turns out that both of these motivating examples can be understood as special cases of a more general construction. In his later work, Szlachányi already mentions that monoidal actions, together with the existence of a certain family of adjoints, give rise to skew monoidal structures. In this talk, I will present a result closely related to this observation: under the assumption of the existence of a suitable adjoint, one can equip an actegory with a skew monoidal structure.
This construction not only encompasses many known examples of skew monoidal categories but also allows one to generate new ones from some existing examples of monoidal actions. Moreover, it generalises several insights of the aforementioned authors. The adjunction involved can be shown to be monoidal with respect to the original and the newly defined tensor products. In particular, the correspondence between relative monads and certain ordinary monads described by Altenkirch, Chapman, and Uustalu emerges as a special case of this more general observation.
Operads are typically defined to be equipped with actions of the symmetric groups on their sets of n-ary operations, equivariant with respect to the operadic composition. We can also define operads without such actions or with actions governed by other families of groups, such as braid groups, ribbon braid groups, or more unfamiliar families such as cactus groups.
Every operad also has an associated monad and in the 1-dimensional case we can consider those monads which are commutative, for which the category of algebras inherits a closed monoidal structure as shown by Anders Kock. One dimension up we can consider 2-monads which are pseudo-commutative, for which the 2-category of strict algebras and pseudomorphisms inherits a pseudo-closed structure as described by Hyland and Power.
In this talk I will first introduce action operads and discuss some well-known examples of these, as well as hint at some surprising non-examples which includes many skew-simplicial groups. I will also describe conditions on Cat-enriched operads governed by such action operads in order for the associated 2-monad to possess a pseudo-commutativity. This is immediate in the case of the symmetric operad and we are able to show that is also true in the case of the braid operad using some geometric arguments. However, it does not appear to be possible to do this in a straightforward way in the case of the cactus operad, as shown by some simple algebraic manipulation of the pseudo-commutativity conditions and the cactus operad structure.
This is joint work with Nick Gurski.
The construction of 2-limits in internal category theory is simple; they are just pointwise limits in the ambient category. On the other hand, 2-colimits are more subtle; whilst the construction of coproducts and copowers by the walking arrow are known, the construction of coequalisers of internal categories is a folklore result.
In this talk, I will give a recipe for calculating coequalisers of parallel pairs of internal functors. This process uses the construction of free internal categories on internal graphs, extensivity and some exactness properties of the ambient category. I will show that these assumptions are necessary and sufficient.
If you don't like internal category theory, then you can treat this talk as if it were about small category theory and get a different perspective on this case. Otherwise, you may wish to work internal to a locally cartesian closed pretopos with natural numbers object; in particular, this gives a proof of this for categories when working in type-theoretic foundations of mathematics.
There is a wild analogue of the notion of sequential convergence in a topological space; for a family of points and an ultrafilter on the indexing set one can define what it means for a point to be a limit of this sequence. It has been noticed by Barr [1] that this notion of convergence is enough to recover all the topological structure of a space. From this point of view the topological structure is now a generalized poset on the set of point extending the usual specialization order.
We categorify this story by replacing topological spaces with Grothendieck topoi. We can define a convergence notion indexed by ultrafilters that extends the categorical structure on the points of the topos. In the coherent case this convergence notion relates with the usual ultraproduct of models of a first-order theory and connects with Makkai's ultracategories [2].
We will prove that a Grothendieck topos with enough points can be recovered from this generalized convergence structure, thus extending the Makkai–Lurie duality between coherent topoi and ultracategories to all topoi with enough points. The proof will go through the Butz–Moerdijk representation of Grothendieck topoi [3].
[1] M. Barr, (1970), Relational algebras, 10.1007/BFb0060439
[2] M. Makkai, (1987), Stone duality for first order logic, 10.1016/0001-8708(87)90020-X.
[3] C. Butz, I. Moerdijk, (1998), Representing topoi by topological groupoids, 10.1016/S0022-4049(97)00107-2.
TBA
We introduce contextads and the Ctx construction, unifying various structures and constructions in category theory dealing with context and contextful arrows -- comonads and their Kleisli construction, actegories and their Para construction, adequate triples and their Span construction. Contextads are defined in terms of Lack--Street wreaths, suitably categorified for pseudomonads in a tricategory of spans. This abstract approach can be daunting, so in this talk we will work with a lower-dimensional version of contextads which is relevant to capture dependently graded comonads arising in functional monadic programming. In fact we show that many side-effects monads can be dually captured by discrete contextads, seen as dependently graded comonads, and gesture towards a general result on the 'transposability' of parametric right adjoint monads to dependently graded comonads.
This talk concerns distributive monoidal categories that are traced on the coproduct and posses a distinguished object N that is isomorphic to I+N. Such categories, which I propose to call Elgot categories, contain all that is necessary to represent arbitrary partial recursive functions, in the sense that e.g., natural numbers objects imply representability of the primitive recursive functions. There is an initial Elgot category, whose morphisms correspond to a lightly modified version of Lambek's abacus programs. This initial Elgot category strongly represents (in a sense, generates) all and only the partial recursive functions. A preprint is available.
Compositional reasoning for processes with shared memory requires understanding how individual process properties combine via shared resources. Traditional separation logic [Rey02] is typically applied to processes with localized state—while its standard formulation handles disjoint memory regions, variants (e.g., probabilistic separation logic) extend the framework to systems with additional structure.
In this talk, I present a sheaf-theoretic framework for capturing program logics, with separation logic as a guiding example. Unlike traditional approaches where the separating conjunction glues only disjoint regions, we use the gluing axiom of sheaves to merge compatible, possibly overlapping, memory regions. This forms the basis for reasoning about interacting processes, as processes confined to disjoint memory regions would nullify the interactions that shared memory is meant to support.
In this direction, we work in the internal category theory of sheaves on a poset. The structure of the predicate logic is provided by discrete fibrations of internal categories. In our setting the representable definition of a discrete fibration in a 2-category [LR20] can be reduced to verifying its lifting condition at only the terminal internal category. This affords a straightforward definition of internal reindexing functors. We obtain separation logics, extensions of our predicate calculus, as monoids for Day convolution. We define an amalgamation operation that uniquely glues compatible local sections. By combining this gluing with the closed monoidal structure we obtain global operators for composing local data.
This talk is based on joint work with Henning Basold and Tanjona Ralaivaosaona.
[LR20] Fosco Loregian and Emily Riehl. Categorical notions of fibration. Expositiones Mathematicae, 38(4):496–514, December 2020. arXiv:1806.06129 [math].
[Rey02] John C. Reynolds. Separation Logic: A Logic for Shared Mutable Data Structures. In 17th IEEE Symposium on Logic in Computer Science (LICS 2002),Proceedings, pages 55–74, 2002.
I will discuss some ongoing work on using the algebraic structure of continuations and double negation monads to understand control effects.
This represents joint work in progress with Umberto Tarantino.
From the perspective of category theory, the pervasive interest in compact Hausdorff spaces is ‘inevitable’. The right Kan extension of the inclusion of finite sets into sets along itself is a monad, the ‘ultrafilter monad’, whose algebras are precisely the compact Hausdorff spaces. Thus, one only needs to think that finite sets, sets, and Kan extensions are important to arrive at the (unsurprising) conclusion that compact Hausdorff spaces are important too.
The goal in this talk is to make ‘ultracategories’ emerge just as naturally.
Ultracategories were introduced by Makkai as a way to encode the logical notion of an ‘ultraproduct’ and were later developed further by Lurie. Most recently, ultracategories were shown to be algebras for a pseudomonad on the category of categories. We will recover the monadicity of ultracategories via a general result concerning ‘relative 2-monads’. Generalising work in the 1-dimensional setting by Altenkirch, Chapman and Uustalu, we show that, under suitable hypotheses, the left Kan extension of a relative 2-monad on categories yields a pseudomonad which, moreover, has the same co-lax algebras.
By observing that ultracategories can be naturally understood as co-lax algebras for a ‘relative 2-monad’, we will deduce our main result: ultracategories are co-lax algebras for the pseudomonad obtained by taking the left Kan extension of the ultrafilter monad along the inclusion of sets into categories. Thus, if finite sets, sets, categories and Kan extensions are important, then so too are ultracategories.
We explore the classification and representation of algebras of the Giry monad. Intuitively, these algebras define spaces where integration is well-defined. The structure map of the algebra assigns to each probability measure its centre of mass. However, characterizing these spaces for a given probability monad is nontrivial. For certain probability monads, representations of the algebras are well understood. Convex compact subsets of locally convex spaces correspond to algebras of the Radon monad, while convex closed subsets of Banach spaces characterize those of the Kantorovich monad. The distribution monad’s algebras can be represented as abstract convex sets, but classifying the algebras of the Giry monad is more challenging.
A key difficulty in classifying Giry algebras is that the algebras of the Giry monad do not always satisfy the cancellation property (C1). We introduce a generalized cancellation property (C2). We endow Giry algebras with a canonical topology and prove that those satisfying (C2) can be represented as relatively closed convex subsets of locally convex spaces. Furthermore, in this setting, the structure map can be described as a Pettis integral, allowing the application of functional analysis techniques to general Giry algebras that satisfy (C2). This leads to an adjunction between Radon algebras and the Giry algebras that satisfy (C2).
For Giry algebras that do not satisfy (C2), we consider those whose associated vector space is finite dimensional. Under this assumption, we show that the conditions (C1) and (C2) are equivalent. We identify specific elements within the algebra as ‘infinite elements’ and introduce a partial order on those elements. Every infinite element corresponds, in an order-preserving way, to a well-behaved subalgebra, where the structure map can be described using Pettis integrals. This provides an algorithmic approach to describing the structure map and the Giry algebra using methods from functional analysis and order theory.