Last updated: Aug 03, 2022, 19:20 CEST
Conference Titles, Abstracts, Slides,
and Videos
All in-person talks will take place in
Aula 3, Edificio de Ciencias de la Salud
Mohamed Barakat, Categorizing computer algebra: Building and compiling towers of categories
Talk cancelled
Abstract: While set theory is by design a foundational theory for mathematics where complex structures are built a posteriori out of the unstructured notion of a set, category theory is a highly expressive meta-theory of structures that keep popping up in a wide range of mathematical disciplines and at arbitrary high abstraction levels with the same syntax but with completely different semantics. In this talk, I will show how we use towers of category constructors to reorganize computer algebra in the spirit of category theory. The categorical organization of computer algebra helped us design algorithms that would have never been possible to directly implement. However, building such towers comes at the cost of runtime. A category-theory-aware compiler becomes indispensable for producing a highly performant code that compiles away the abstraction introduced by the different layers of the tower. I will demonstrate this modern, unorthodox style of computer algebra by examples.
Vladimir Bavula, The PBW Theorem and simplicity criteria for the Poisson enveloping algebra and the algebra of Poisson differential operators.
Abstract: I will explain some of my recent results on Poisson algebras: The PBW Theorem holds for the Poisson enveloping algebra (PEA) of a Poisson algebra (an affirmative answer to a long standing problem). The algebras of Poisson differential operators (APDO) are introduced.
Explicit sets of generators and defining relations are given for PEAs and APDOs as well as simplicity criteria. Explicit formulae for the Gelfand-Kirillov dimension of the above algebras are given. In the case when the Poisson algebra is a regular domain of essentially finite type an explicit simplecticity criterion for PEA is found and a criterion is presented for PEA to be isomorphic to APDO.
Isaac Bird, Definable functors between triangulated categories.
Abstract: I will discuss definable functors between compactly generated triangulated categories. As in the finitely accessible case, these are the functors which preserve pure triangles and pure injective objects. I will give several examples, and explain the relationship between definable functors between triangulated categories and definable functors between their corresponding functor categories. This is based on ongoing joint work with Jordan Williamson
Simion Breaz, Change of scalars functors and silting complexes.
Abstract: I will present results about the transfer of the (co)silting property of an object from the derived category of a module category by using the (co)extension of scalars functors induced by morphisms of commutative rings. In particular, we will see that the derived versions of the (co)extension of scalars functors preserve (co)silting objects of (co)finite type. In the last part of the talk, I will discuss some cases when these functors reflect the (co)silting property. The talk is based on a joint work with Michal Hrbek and George-Ciprian Modoi.
Thomas Brüstle, Homological approximations in persistence theory.
Abstract: Multiparameter persistence modules are defined over a wild algebra and therefore they do not admit a complete discrete invariant. One thus tries in persistence theory to “approximate” such a module by a more manageable class of modules. Using that approach we define a class of invariants for persistence modules based on ideas from homological algebra. This is a report on joint work with Benjamin Blanchette and Eric Hanson.
Manuel Cortés Izurdiaga, Rooted functor categories.
Abstract: First, we will see how the usual description of the category of modules over a triangular matrix ring can be extended to the category of all additive functors from a small preadditive category to the category of abelian groups. Then, we will define rooted small preadditive categories, so that we will be able to apply methods from the representations (by modules) of rooted quivers to the aforementioned functor category. Finally, we will see how these methods can be extended to study other functor categories.
Samuel Dean, Infinitary model theory of modules.
Abstract: Let λ be a regular infinite cardinal. Using abelian functor categories, I will show that finitary pp formulas for right R-modules, i.e. the classical objects of study in the finitary model theory of modules, are sufficient to describe the λ-ary model theory of a locally presentable additive category C, where R is the category of λ-presented objects in C. The cost is that we must interpret pp formulas only among the right R-modules which preserve λ-small products.
Alberto Facchini, Multiplicative lattices, commutators, braces.
Abstract: The multiplicative lattices we will consider are those defined in the paper [3], published in February 2022. Multiplicative lattices yield the natural setting in which several basic mathematical questions concerning algebraic structures find their answer. We will consider the particular cases of braces ([2] and [3]).
[1] D. Bourn, A. Facchini and M. Pompili, Aspects of the Category SKB of Skew Braces, submitted for publication, available in arXiv, 2022
[2] A. Facchini, Algebraic structures from the point of view of complete multiplicative lattices, accepted for publication in "Rings, Quadratic Forms, and their Applications in Coding Theory", Contemporary Math., 2022, available at: http://arxiv.org/abs/2201.03295
[3] A. Facchini, C. A. Finocchiaro and G. Janelidze, Abstractly constructed prime spectra, Algebra universalis 83(1) (2022).
Grigory Garkusha, Recollements for derived categories of enriched functors and triangulated categories of motives.
Abstract: This is joint work with Darren Jones. We investigate certain categorical aspects of Voevodsky's triangulated categories of motives. For this, various recollements for Grothendieck categories of enriched functors and their derived categories are established. In order to extend these recollements further with respect to Serre's localization, the concept of the (strict) Voevodsky property for Serre localizing subcategories is introduced. This concept is inspired by the celebrated Voevodsky theorem on homotopy invariant presheaves with transfers. As an application, it is shown that Voevodsky's triangulated categories of motives fit into recollements of derived categories of associated Grothendieck categories of Nisnevich sheaves with specific transfers.
Lorna Gregory, Model theory of modules over Prüfer domains and dimensions on lattice ordered abelian groups.
Abstract: Much of the model theory of modules over a Prüfer domain R is captured by its value group, that is, the group of fractional ideals of R ordered by reverse inclusion. This group is a lattice ordered abelian group and all lattice ordered abelian groups occur as the value group of a Prüfer, or even Bézout, domain.
In this talk I will explain how the principal invariants of model theory of modules, m-dimension and breadth of the lattice of pp-formulae, can be calculated (or bounded) for Prüfer domains using their value groups. I will explain the intended meaning of these invariants and, if there is time, explain how to show that for Prüfer domains, they measure what they are supposed to measure.
Alina Iacob, Ding injective envelopes.
Abstract: Among the generalizations of the Gorenstein injective modules, the Ding injectives were probably studied the most. The existence of the Ding injective envelopes is known over coherent rings. We prove that, in fact, the Ding injective modules form an enveloping class over any ring. We also show that the result carries to the category of complexes: for any ring R, the class of Ding injective complexes is enveloping in Ch(R). This is joint work with J. Gillespie.
Miodrag Iovanov, On some homological categories and their underlying quantum group classification.
Abstract: Several categories, such as that of bounded or arbitrary chain compelexes, double chain complexes or other diagrams are at the core of homological algebra. Many of these have natural monoidal rigid tensor category structure on them, and they can in fact be regarded as the category of comodules ("rational representations") of a suitable quantum group, AKA pointed Hopf algebras. It turns out such Hopf algebras posses the interesting feature that "locally" they are of finite representation type. We pursue the question of what are the pointed Hopf algebras which have this property, equivalently, pointed tensor categories so that there are only finitely many objects having a fixed dimension vector. Important classical quantum groups such as the Taft algebras and their generalization are of these types. Our main result is a classification of these, which morally says that, up to deformations of all these examples, these are all the examples.
Mamuka Jibladze, Lawvere distributions, pseudolinear algebra and factorization structures.
Abstract: There are currently at least two approaches to the algebraic description of structures arising in field theory. Both of them use a certain "distortion" of monoidal structures.
One approach uses several variations of multicategory structures (like the Borcherds category Fin≢ , colored operads, Beilinson-Drinfeld pseudo-tensor categories, factorization monoids of Kapranov-Vasserot, and more recent refined version of factorization algebras by Costello and Gwilliam).
Another approach appeared in the work of Bakalov, D'Andrea and Kac, where the formalism of pseudo-linear algebra was developed to describe structures like dynamical Poisson brackets using a version of Yetter-Drinfeld modules.
On the other hand, much earlier Lawvere worked on what would be nowadays called categorification of the calculus of distributions and the notion of Radon-Nicodym derivative in the non-linear (set theoretic) context of the combination of co- and contravariant set-valued functors, using his concept of extensive and intensive quantities.
It seems that the Lawvere calculus relates to both of the above approaches, which might provide further insight into vertex algebras and the operator product expansion formalism through his categorified calculus of distributions.
We will try to briefly survey all three notions and show parallels between them. If time permits, we will also address what could be the Lawvere calculus viewpoint on the Chas-Sullivan string topology and planar algebras of V. F. R. Jones.
Ilias Kaperonis, K-Absolutely pure complexes.
Abstract: Projective and injective resolutions of modules play an important role in the study of Homological Algebra. However we can view the category of modules over a ring as a subcategory of the complexes of the ring. Spaltenstein in 1972 introduces the classes of K-Projective and K-Injective complexes which can be used to generalize common resolutions. In this talk we will introduce the class of K-Absolutely pure complexes, we will study their basic properties and we will compare them with the class of K-Flat complexes of Spaltenstein.
George Ciprian Modoi, Not necessarily compact approximability via silting theory.
Abstract: D be a triangulated category with coproducts. The (quite technical) definition of approximability of D involves a compact generator G and a t-structure (see Definition 5.1 of A. Neeman, Approximable triangulated categories, preprint, arXiv:1806.06995[math.CT]). One of the main examples of approximable triangulated categories is obtained when G is assumed, in addition, to be silting (case in which the t-structure can be chosen to be the associated silting t-structure). In this talk I will present an ongoing project about a generalization of approximability in triangulated categories via silting theory.
George Nadareishvili, Homological algebra in noncommutative topology.
Abstract: We will give a brief introduction to homological algebra in nonabelian setting of triangulated categories of C*-algebras. Next, using these techniques, we will explain how Universal Coefficient Theorems are a homological phenomenon.
Juan Orendain Almada, Reconstructing double categories from End-indexings.
Abstract: Framed bicategories are double categories satisfying a generalized operation of base change. Symmetric monoidal structures on framed bicategories descend to symmetric monoidal structures on horizontal bicategories. The axioms defining symmetric monoidal double categories are significantly more tractable than those defining symmetric monoidal bicategories. It is thus convenient to study ways of lifting a given bicategory into a framed bicategory along an appropriate category of vertical morphisms. Solutions to the problem of lifting bicategories to double categories have classically been useful in expressing Kelly and Street's mates correspondence and in proving the 2-dimensional Seifert-van Kampen theorem of Brown et. al., amongst many other applications.
Globularly generated double categories are minimal solutions to lifting problems of bicategories into double categories along given categories of vertical arrows. Globularly generated double categories form a coreflective sub-2-category of general double categories. This, together with an analysis of the internal structure of globularly generated double categories yields a numerical invariant on general double categories. We call this invariant the length. The length of a double category C measures the complexity of lifting decorated bicategories into C.
It is conjectured that framed bicategories are of length 1. Motivated by this I present a general method for constructing globularly generated double categories of length 1 through extra data in the form of what I will call End-monoidal indexings of decoration categories. The methods presented are related to Moeller and Vasilakopoulou's monoidal Grothendieck construction, to Shulman's construction of framed bicategories from monoidal fibrations on cocartesian categories, and in the case of strict single object and single horizontal morphism 2-groupoids decorated by groups, specialize to semidirect products.
Sebastian Posur, How to classify objects in Krull-Schmidt categories.
Abstract: Krull-Schmidt categories are ubiquitous in representation theory. In such categories every object can be written as a finite direct sum of indecomposables in an essentially unique way. In this talk, I discuss helpful tools for the determination of the isomorphism classes in Krull-Schmidt categories. These tools are direct generalizations from ring theory to category theory, like filtrations, gradings, or Galois descent. With the help of these tools, I show how to classify the objects in Deligne's interpolation category for the symmetric groups in arbitrary characterstic, and how to classify the objects in Khovanov–Sazdanovic's recent generalization of Deligne's interpolation category. This is joint work with Johannes Flake and Robert Laugwitz.
Mike Prest, Definability in monoidal additive categories.
Abstract: Model theory and definability in additive, including triangulated, categories is well-developed. I discuss some of what we get when there is tensor structure on the categories. In particular I will present some 2-category equivalences due to Rose Wagstaffe.
Job Rock, Continuous Nakayama algebras.
Abstract: We construct a continuous analogue of Nakayama algebras using representations of the real line and of the circle. This is done with a combination of (pre-)Kupisch functions (a continuous analogue of Kupisch series) and functor categories. We classify the orthogonal components of categories of Nakayama representations with regard to connectedness using some analytic properties of the (pre-)Kupisch functions. We conclude by relating continuous Nakayama representations to their discrete counter part and to dynamical systems. Joint with Shijie Zhu.
arXiv link
Philipp Rothmaler, Universal models and Bass modules.
Abstract: For the purpose of this talk, I call a module Bass module if it is the direct limit of a chain of order type of the natural numbers of finitely presented modules. This term is suggested by work of Trlifaj et al on Bass type phenomena. Puninski suggested a model-theoretic proof of the Bass-Björk result that all flat (left) modules are projective (i.e., the ring is left perfect) if and only if it satisfies the dcc on (pp formulas defining) finitely generated right ideals. In this talk on joint work with Anand Pillay I show how this can be done with any kind of descending chain of pp formulas and their free realizations (i.e., associated f.p. modules) in order to show that the corresponding Bass module is pure-projective precisely when the category of left modules enjoys the dcc on the pp chains in question. This gives a self-contained proof of Bass’ theorem (and Björk’s) in case all the f.p. modules in question are flat. The other end of the spectrum, when all f.p. modules are considered at once, one obtains a well-known result of Daniel Simson that a ring is left pure-semisimple iff all left modules are Sigma-pure-injective iff all left modules are Mittag-Leffler. I will explain how this investigation may be viewed as a study of universal models in certain classes of structures, resp., of certain theories. This talk is dedicated to the memory of Professor Simson.
Kamal Saleh, Constructive category theory and tilting equivalences via strong exceptional sequences.
Abstract: In this presentation, we'll demonstrate how to implement categories on a computer using our software project CAP - Categories, Algorithms, and Programming. We will discuss how to construct the bounded homotopy category of an additive or Abelian category, as well as the obstacles encountered and their practical categorical solutions.
We use these homotopy categories to demonstrate some applications to homological algebra: Given a bounded homotopy category that is finitely generated as a triangulated category, we can verify whether a given set of objects forms a complete strong exceptional sequence and, if so, to implement the associated tilting equivalences.
Manuel Saorín, Hearts which are Grothendieck categories: a study through functor categories.
Abstract: Given a triangulated category with arbitrary coproducts and a t-structure in it, it is a well-known fact that the heart of the latter is an abelian category. A problem that has deserved a lot of attention in recent times is that of identifying the t-structures whose heart is a Grothendieck category. In this talk we will give a short survey on the most important related results, and will present some recent ones based on joint work with Jan Stovicek, where the use of functor categories is crucial.
Alex Sorokin, All concepts are defects.
Abstract: In the mid 1960s Auslander introduced a notion of the defect of a finitely presented functor on a module category. In 2021 Martsinkovsky generalized it to arbitrary additive functors. In this talk I will show how to define a defect of any enriched functor with a codomain a cosmos. Under mild assumptions, the covariant (contravariant) defect functor turns out to be a left covariant (right contravariant) adjoint to the covariant (contravariant) Yoneda embedding. Both defects can be defined for any profunctor enriched in a cosmos V. They happen to be adjoints to the embeddings of V-Cat in V-Prof. Moreover, the Isbell duals of a profunctor are completely determined by the profunctor's covariant and contravariant defects. Furthermore, Kan extensions can be described as defects of certain profunctors, showing the fundamental role of the defects. These results are based on (co)end calculus.
Ashish Srivastava, On a quotient of face algebra.
Abstract: Hayashi introduced face algebras as a new class of quantum groups inspired by the quantum inverse scattering method and solvable lattice models of face type. In this talk, we will discuss a quotient algebra of Hayashi's face algebra over a quiver Q and its connection with the Leavitt path algebra over Kronecker square of Q. We study structural connection between Leavitt path algebra over a quiver and its Kronecker square and category equivalence of modules over such algebras.
Jan Šťovíček, The coderived category of a locally coherent dg category.
Abstract: The homotopy category of complexes of injective modules or injective quasi-coherent sheaves (aka the coderived category) received a lot of attention in the last two decades in connection with the Grothendieck duality in algebraic geometry, the Koszul duality or representation theory of finite groups. In the present joint work with Leonid Positselski, we define the coderived category of a locally coherent dg category, which generalizes coderived categories both for complexes over usual locally coherent abelian categories and for curved dg modules over curved dg rings. Moreover, we prove that the coderived category is always compactly generated in this setting and identify the compact objects.
Alberto Tonolo, Injective envelopes of simple modules over a family of Leavitt path algebras.
Abstract: I will quickly recall the definition of Leavitt path algebras (LPA) associated to a graph E, giving some examples and properties. Assuming the graph E is finite and that every vertex in E is the base of at most one cycle, I will explicitly construct the injective envelope of each simple left module. The talk is based on some very recent results obtained in collaboration with Francesca Mantese and Gene Abrams.
Jan Trlifaj, Locality of quasi-coherent sheaves associated with relative Mittag-Leffler modules.
Abstract: The ascent and descent of the Mittag-Leffler property was instrumental in proving locality of the notion of an (infinite dimensional) vector bundle in the classic work of Raynaud and Gruson. More recently, relative Mittag-Leffler modules were employed in the theory of (infinitely generated) tilting modules and the associated quasi-coherent sheaves. Here, we study the ascent and descent of the relative Mittag-Leffler property. As an application, we prove locality of the notions of an f-projective quasi-coherent sheaf for all coherent schemes, and of an i-filtered quasi-coherent sheaf for all integral schemes (based on joint work with Asmae Ben Yassine).
arXiv link
Kristo Väljako, Tensor product rings and Morita equivalence.
Abstract: In my talk I will introduce tensor product rings Q ⊗R P, where QR and RP are R-modules for an arbitrary associative ring R. I will define pseudo-surjective mappings and show that if S is an idempotent ring, then a pseudo-surjectively defined tensor product ring Q ⊗S P is Morita equivalent to S. Then I will define locally injective homomorphisms of rings and strict local isomorphisms of rings. It turns out that a locally injective homomorphism of rings arises naturally from any Morita context between rings.
I will talk about rings of adjoint endomorphisms of modules and their connections to Morita equivalence. I will introduce dual mappings and show how they can be used to describe Morita equivalence. Dual mappings also arise naturally form unitary and surjective Morita context.
Finally, I will give a new necessary and sufficient condition for firm rings to be Morita equivalent. Also I will give a necessary and sufficient condition for s-unital rings to be Morita equivalent, which is a generalization result that two R and S with identity are Morita equivalent if and only if there exists a progenerator QR with S ≅ End(QR).
My talk is based on the article Väljako, K., (2022). Tensor product rings and Rees matrix rings. Comm. in Algebra, published online.
Jordan Williamson, Duality and definability in triangulated categories.
Abstract: In the category of modules over a ring, purity may be viewed as a weakening of splitting - a short exact sequence is pure if and only if it is split exact after applying the character dual. The notion of purity in triangulated categories was introduced by Krause, and it has since been seen to be intimately related to many questions of interest in representation theory and homotopy theory. In this talk, I will explain a framework of duality pairs in triangulated categories which provides an elementary way to check pure closure properties, and describe an application to the study of definable subcategories of triangulated categories. Throughout I will explain some connections of this with tensor-triangular geometry. This is joint work with Isaac Bird.
arXiv link
Robert Wisbauer, An alternative approach to tilting theory.
Comments and Table of Contents:
Abstract: We show that essentially an R-module P is tilting provided a suitable restriction of the comparison functor for Hom(P,-) is fully faithful. The latter constructions work for objects P in any category, thus bringing elements of tilting theory to any categories with suitable completeness conditions, in fact to any adjoint pair of functors.
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