BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Ivo Dell'Ambrogio (Université Lille) DTSTART:20210419T123000Z DTEND:20210419T133000Z DTSTAMP:20260422T225723Z UID:itaca/1 DESCRIPTION:Title: Ma ckey 2-functors\nby Ivo Dell'Ambrogio (Université Lille) as part of I taCa Fest 2021\n\n\nAbstract\nMathematicians of different stripes like to have groups act on different sorts of objects: vector spaces\, topological spaces\, C*-algebras\, spectra\, and so on. At the heart of all flavours of “equivariant mathematics” are operations such as restrictions and i nductions (and conjugations\, inflations\, etc). The latter have been succ essfully axiomatized more than half a century ago (at least for finite gro ups) by the algebraic notion of Mackey functors. But Mackey functors take values in abelian groups\, and the operations are modeled by homomorphisms between them\; however\, what gives rise to most Mackey functors found in Nature is a collection of categories of equivariant objects together with restriction and induction functors between them. These functors enjoy pro perties such as being adjoint\, which are invisible to the classical axiom s. In this talk I will introduce the recent theory of Mackey 2-functors\, algebraic gadgets similar to additive derivators whose purpose is precisel y to capture this higher-categorical layer of information. In order to mot ivate our 2-categorical flavour of axiomatic representation theory\, I wil l evoke exemples from throughout mathematics and I will outline our first notable applications. For instance\, we can export results from the usual theory of linear representations to more geometric and topological setting s. This is joint work with Paul Balmer.\n LOCATION:https://researchseminars.org/talk/itaca/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Paweł Sobociński (Tallinn University of Technology) DTSTART:20210419T133000Z DTEND:20210419T143000Z DTSTAMP:20260422T225723Z UID:itaca/2 DESCRIPTION:Title: Re writing Modulo Symmetric Monoidal Structure\nby Paweł Sobociński (Ta llinn University of Technology) as part of ItaCa Fest 2021\n\n\nAbstract\n String diagrams are an elegant\, convenient and powerful syntax for arrows of symmetric monoidal categories. In recent years\, they have been used a s compositional descriptions of computational systems from various fields\ , including quantum foundations\, linear algebra\, control theory\, automa ta theory\, concurrency theory\, and even linguistics. All of these applic ations rely on diagrammatic reasoning\, which is to string diagrams as equ ational reasoning is to ordinary terms.\n\nIf we are to take string diagra ms out of research papers and into more practical applications\, we need t o ask ourselves about how to implement diagrammatic reasoning. This is the focus of my talk.\n\nIt turns out that there is a tight correspondence be tween symmetric monoidal categories where every object has a coherent spec ial Frobenius algebra structure and categories of cospans of hypergraphs. The correspondence\, therefore\, takes us from a topological understanding of string diagrams to a combinatorial data-structure-like description. Mo reover\, diagrammatic reasoning translates via this correspondence exactly to DPO rewriting with interfaces.\n\nGiven the above\, a natural question is how much of this correspondence survives if we drop the assumption abo ut Frobenius structure: i.e. can we use this correspondence to implement d iagrammatic reasoning on vanilla symmetric monoidal categories. The answer is yes\, but we need to restrict the kinds of cospans we consider: the un derlying hypergraph has to be acyclic and satisfy an additional technical condition called monogamy. Moreover\, we must restrict the DPO rewriting m echanism to a variant that we call convex DPO rewriting. The good news is that none of these modifications come with a significant algorithmic cost. \n\nThe material in this talk is with Filippo Bonchi\, Fabio Gadducci\, Al eks Kissinger and Fabio Zanasi\, and has been published in a series of pap ers:\n\n- “Rewriting modulo symmetric monoidal structure”\, Proceeding s of LiCS 2016\n\n- “Confluence of Graph Rewriting with Interfaces”\, Proceedings of ESOP 2017\n\n- “Rewriting with Frobenius”\, Proceedings of LiCS 2018\n LOCATION:https://researchseminars.org/talk/itaca/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Walter Tholen (York University) DTSTART:20210520T123000Z DTEND:20210520T133000Z DTSTAMP:20260422T225723Z UID:itaca/3 DESCRIPTION:Title: Re visiting Burroni's T-categories\nby Walter Tholen (York University) as part of ItaCa Fest 2021\n\n\nAbstract\nFollowing the appearance of Lambek ’s multicategories and Barr’s presentation of topological spaces as re lational T-algebras\, Albert Burroni introduced T-categories and T-functor s in 1971. They provide an overarching environment for the general study o f algebras and spaces\, which encompasses elements of monad theory\, inter nal category theory\, and categorical topology.\n\n In this talk we have a fresh look at Burroni’s paper and point to the Street-Walters comprehe nsive factorization system for functors and the (antiperfect\, perfect) fa ctorization system for continuous maps of Tychonoff spaces to demonstrate that\, despite its generality\, Burroni’s setting allows for the establi shment of non-trivial results and the discovery of unexpected connections between seemingly unrelated theorems. \n\n(Joint work with Leila Yeganeh)\ n LOCATION:https://researchseminars.org/talk/itaca/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Amar Hadzihasanovic (Tallinn University of Technology) DTSTART:20210520T133000Z DTEND:20210520T143000Z DTSTAMP:20260422T225723Z UID:itaca/4 DESCRIPTION:Title: Th e smash product of monoidal theories\nby Amar Hadzihasanovic (Tallinn University of Technology) as part of ItaCa Fest 2021\n\n\nAbstract\nThe sm ash product of pointed spaces is a classical construction of topology. The tensor product of props\, which extends both the Boardman-Vogt product of symmetric operads and the tensor product of Lawvere theories\, seems firm ly like a piece of universal algebra. \n\n In this talk\, we will see that the two are facets of the same construction: a “smash product of pointe d directed spaces”. Here\, “directed spaces” are modelled by combina torial structures called diagrammatic sets\, developed as a homotopically sound foundation for diagrammatic rewriting in higher dimensions\, while t he cartesian product of spaces is replaced by a form of Gray product.\n Mo st interestingly\, the smash product applies to presentations of higher-di mensional theories and systematically produces oriented equations and high er-dimensional coherence data (oriented syzygies). This introduces a synth etic\, compositional method in rewriting on higher structures.\n \n\nThis talk is based on my preprint arXiv:2101.10361 with the same title.\n LOCATION:https://researchseminars.org/talk/itaca/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Olivia Caramello (Università dell'Insubria) DTSTART:20210615T123000Z DTEND:20210615T133000Z DTSTAMP:20260422T225723Z UID:itaca/5 DESCRIPTION:Title: Re lative topos theory via stacks\nby Olivia Caramello (Università dell' Insubria) as part of ItaCa Fest 2021\n\n\nAbstract\nIn this talk\, based o n joint work with Riccardo Zanfa\, we shall introduce new foundations for relative topos theory based on stacks. One of the central results in our t heory is an adjunction between the category of (relatively small) toposes over the topos of sheaves on a given site (C\, J) and that of C-indexed ca tegories. This represents a wide generalization of the classical adjunctio n between presheaves on a topological space and bundles over it\, and allo ws one to interpret several constructions on sheaves and stacks in a geome trical way\; in particular\, it leads to fibrational descriptions of direc t and inverse images of sheaves and stacks\, as well as to a geometric und erstanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a ‘petit’ topos associated with a ‘ gros’ topos\, thereby providing an answer to a problem posed by Grothend ieck in the seventies.\n LOCATION:https://researchseminars.org/talk/itaca/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Paolo Saracco (Université Libre de Bruxelles) DTSTART:20210615T133000Z DTEND:20210615T143000Z DTSTAMP:20260422T225723Z UID:itaca/6 DESCRIPTION:Title: Gl obalization for geometric partial comodules\nby Paolo Saracco (Univers ité Libre de Bruxelles) as part of ItaCa Fest 2021\n\n\nAbstract\nThe stu dy of partial symmetries (e.g. partial dynamical systems\, (co)actions\, ( co)representations\, comodule algebras) is a relatively recent research ar ea in continuous expansion\, whose origins can be traced back to the study of C*-algebras generated by partial isometries. One of the central questi ons in the field is the existence and uniqueness of a so-called globalizat ion or enveloping (co)action.\n\nIn the framework of partial actions of gr oups\, any global action of a group on a set induces a partial action of t he group on any subset by restriction. The idea behind the concept of glob alization of a given partial action is to find a (universal) global action such that the initial partial action can be realized as the restriction o f this global one. The importance of this procedure is testified by the nu merous globalization results already existing in the literature which\, ho wever\, are based on some ad hoc constructions\, depending on the nature o f the objects carrying the partial action.\n\nWe propose here a unified ap proach to globalization in a categorical setting\, explaining several of t he existing results from the literature and\, at the same time\, providing a procedure to construct globalizations in concrete contexts of interest. Our approach relies on the notion of geometric partial comodules (recentl y introduced by Hu and Vercruysse in [HV]) which –unlike classical parti al actions\, that exist only for (topological) groups and Hopf algebras– can be defined over any coalgebra in an arbitrary monoidal category with pushouts.\n\n[HV] J. Hu\, J. Vercruysse\, Geometrically partial actions. T rans. Amer. Math. Soc. 373 (2020)\, no. 6\, 4085–4143.\n\n[PJ] P. Saracc o\, J. Vercruysse\, Globalization for geometric partial comodules. Preprin t (2020).\n LOCATION:https://researchseminars.org/talk/itaca/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Mauro Porta (IRMA\, Université de Strasbourg) DTSTART:20210928T123000Z DTEND:20210928T133000Z DTSTAMP:20260422T225723Z UID:itaca/7 DESCRIPTION:Title: Pr o and Ind-categories in Algebra and Geometry\nby Mauro Porta (IRMA\, U niversité de Strasbourg) as part of ItaCa Fest 2021\n\n\nAbstract\nIn thi s talk we are going to discuss some natural instances of pro and ind categ ories in algebraic and geometric contexts\, highlighting the importance of working with objects in Ind(Cat$_\\infty$) and Pro(Cat$_\\infty$) instead of their Cat$_\\infty$ realizations. Towards the end we will raise some q uestions\, with the intent of determining what is the “correct” object to consider in these contexts\, so as to optimize the generalization/appl icability trade-off.\n LOCATION:https://researchseminars.org/talk/itaca/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Tim Van der Linden (Université catholique de Louvain) DTSTART:20210928T133000Z DTEND:20210928T143000Z DTSTAMP:20260422T225723Z UID:itaca/8 DESCRIPTION:Title: Al gebras with representable representations\nby Tim Van der Linden (Univ ersité catholique de Louvain) as part of ItaCa Fest 2021\n\n\nAbstract\n( Joint work with Xabier García-Martínez\, Matsvei Tsishyn and Corentin Vi enne)\n\nJust like group actions are represented by group automorphisms\, Lie algebra actions are represented by derivations: up to isomorphism\, a split extension of a Lie algebra B by a Lie algebra X corresponds to a Lie algebra morphism B$\\to\\mathbf{Der}$(X) from B to the Lie algebra $\\mat hbf{Der}$(X) of derivations on X. The aim of this talk is to elaborate on the question\, whether the concept of a derivation can be extended to othe r types of non-associative algebras over a field $\\mathbf{K}$\, in such a way that these generalised derivations characterise the $\\mathbf{K}$-alg ebra actions. We prove that the answer is no\, as soon as the field $\\mat hbf{K}$ is infinite. In fact\, we prove a stronger result: already the rep resentability of all abelian actions – which are usually called represen tations or Beck modules – suffices for this to be true. Thus we characte rise the variety of Lie algebras over an infinite field of characteristic different from 2 as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasises the unique role played by the Lie algebra of linear endomorphisms $\\math bf{gl}$(V) as a representing object for the representations on a vector sp ace V.\n LOCATION:https://researchseminars.org/talk/itaca/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Todd Trimble (Western Connecticut State University) DTSTART:20211021T133000Z DTEND:20211021T143000Z DTSTAMP:20260422T225723Z UID:itaca/9 DESCRIPTION:Title: Gr othendieck groups of 2-rigs as lambda rings\nby Todd Trimble (Western Connecticut State University) as part of ItaCa Fest 2021\n\n\nAbstract\nTh is talk will report on recent joint work with John Baez and Joe Moeller. W e introduce a notion of 2-rig as a way of categorifying the usual notion o f rig (ring without negatives)\; examples include categories of group repr esentations\, categories of vector bundles over spaces\, categories of coh erent sheaves over projective varieties\, and many others. We describe Sch ur functors on general 2-rigs\, and indicate how the free 2-rig on one gen erator encodes Schur functors on 2-rigs. Finally\, we indicate how the Gro thendieck group of a 2-rig yields a lambda ring\, where the theory of lamb da-rings is a “plethory” obtained by decategorifying a conceptually si mple 2-plethory that is associated with the free 2-rig.\n LOCATION:https://researchseminars.org/talk/itaca/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Riccardo Zanfa (Università degli studi dell'Insubria) DTSTART:20211021T123000Z DTEND:20211021T133000Z DTSTAMP:20260422T225723Z UID:itaca/10 DESCRIPTION:Title: G eneralized presheaf-bundle adjunctions\nby Riccardo Zanfa (Università degli studi dell'Insubria) as part of ItaCa Fest 2021\n\n\nAbstract\nWe p resent two new results generalizing the well-known presheaf-bundle adjunct ion for topological spaces\, which relate indexed categories (and presheav es) over a site (C\,J) with toposes over the topos Sh(C\,J). The content o f this seminar can be found in a joint work with Olivia Caramello titled R elative topos theory via stacks\, and currently available on arXiv.\n LOCATION:https://researchseminars.org/talk/itaca/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Domenico Fiorenza (Università degli Studi di Roma "La Sapienza") DTSTART:20211118T133000Z DTEND:20211118T143000Z DTSTAMP:20260422T225723Z UID:itaca/11 DESCRIPTION:Title: C ategorical shadows lurking behind integral formulas for genera\nby Dom enico Fiorenza (Università degli Studi di Roma "La Sapienza") as part of ItaCa Fest 2021\n\n\nAbstract\nProfessor Friedrich: And so\, if you have a complex genus taking rational values\, i.e.\, a ring homomorphism from th e complex cobordism ring to the field Q of rational numbers\, you have an integral formula expressing it.\n\nThe Categorist: If a formula is true\, it must be expressed by a commutative diagram.\n\nProfessor Friedrich: But my formula is true!\n\nThe Categorist: Then it must be expressed by a com mutative diagram.\n\nProfessor Friedrich: Show me.\n\nThe Categorist: Let us consider the category of spectra…\n\nProfessor Friedrich: And so?\n\n The Categorist: Well… I don’t see a commutative diagram here.\n\nProfe ssor Friedrich: And so?\n\nThe Categorist: And so your formula must be fal se.\n\nProfessor Friedrich: But my formula is true!\n\nThe Categorist: Imp ossible.\n\nSir Michael: There is something I think I know on the Spanier- Whitehead dual of a smooth manifold that may happen to be of some relevanc e here.\n\n(Emil Ionesco\, Triceratops)\n LOCATION:https://researchseminars.org/talk/itaca/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Federico Olimpieri (University of Leeds) DTSTART:20211118T143000Z DTEND:20211118T153000Z DTSTAMP:20260422T225723Z UID:itaca/12 DESCRIPTION:Title: C ategorifying Intersection Types\nby Federico Olimpieri (University of Leeds) as part of ItaCa Fest 2021\n\n\nAbstract\nWe study a family of dist ributors-induced bicategorical models of lambda-calculus\, proving that th ey can be syntactically presented via intersection type systems. We first introduce a class of 2-monads whose algebras are monoidal categories model ling resource management. We lift these monads to distributors and define a parametric Kleisli bicategory\, giving a sufficient condition for its ca rtesian closure. In this framework we define a proof-relevant semantics: t he interpretation of a term associates to it the set of its typing derivat ions in appropriate systems. We prove that our model characterize solvabil ity\, adapting reducibility techniques to our setting. We conclude by desc ribing wo examples of our construction.\n LOCATION:https://researchseminars.org/talk/itaca/12/ END:VEVENT END:VCALENDAR