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:20260404T110914Z UID:itaca/1 DESCRIPTION:Title: Mackey 2-functors\nby Ivo Dell'Ambrogio (Université Lille) as pa rt of ItaCa Fest 2021\n\n\nAbstract\nMathematicians of different stripes l ike to have groups act on different sorts of objects: vector spaces\, topo logical spaces\, C*-algebras\, spectra\, and so on. At the heart of all fl avours of “equivariant mathematics” are operations such as restriction s and inductions (and conjugations\, inflations\, etc). The latter have be en successfully axiomatized more than half a century ago (at least for fin ite groups) by the algebraic notion of Mackey functors. But Mackey functor s take values in abelian groups\, and the operations are modeled by homomo rphisms between them\; however\, what gives rise to most Mackey functors f ound in Nature is a collection of categories of equivariant objects togeth er with restriction and induction functors between them. These functors en joy properties such as being adjoint\, which are invisible to the classica l axioms. In this talk I will introduce the recent theory of Mackey 2-func tors\, algebraic gadgets similar to additive derivators whose purpose is p recisely to capture this higher-categorical layer of information. In order to motivate our 2-categorical flavour of axiomatic representation theory\ , I will 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 settings. This is joint work with Paul Balmer.\n LOCATION:https://stable.researchseminars.org/talk/itaca/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Paweł Sobociński (Tallinn University of Technology) DTSTART:20210419T133000Z DTEND:20210419T143000Z DTSTAMP:20260404T110914Z UID:itaca/2 DESCRIPTION:Title: Rewriting Modulo Symmetric Monoidal Structure\nby Paweł Sobociń ski (Tallinn University of Technology) as part of ItaCa Fest 2021\n\n\nAbs tract\nString diagrams are an elegant\, convenient and powerful syntax for arrows of symmetric monoidal categories. In recent years\, they have been used as compositional descriptions of computational systems from various fields\, including quantum foundations\, linear algebra\, control theory\, automata theory\, concurrency theory\, and even linguistics. All of these applications rely on diagrammatic reasoning\, which is to string diagrams as equational reasoning is to ordinary terms.\n\nIf we are to take string diagrams out of research papers and into more practical applications\, we need to ask ourselves about how to implement diagrammatic reasoning. This is the focus of my talk.\n\nIt turns out that there is a tight correspond ence between symmetric monoidal categories where every object has a cohere nt special Frobenius algebra structure and categories of cospans of hyperg raphs. The correspondence\, therefore\, takes us from a topological unders tanding of string diagrams to a combinatorial data-structure-like descript ion. Moreover\, diagrammatic reasoning translates via this correspondence exactly to DPO rewriting with interfaces.\n\nGiven the above\, a natural q uestion is how much of this correspondence survives if we drop the assumpt ion about Frobenius structure: i.e. can we use this correspondence to impl ement diagrammatic reasoning on vanilla symmetric monoidal categories. The answer is yes\, but we need to restrict the kinds of cospans we consider: the underlying hypergraph has to be acyclic and satisfy an additional tec hnical condition called monogamy. Moreover\, we must restrict the DPO rewr iting mechanism to a variant that we call convex DPO rewriting. The good n ews is that none of these modifications come with a significant algorithmi c cost.\n\nThe material in this talk is with Filippo Bonchi\, Fabio Gadduc ci\, Aleks Kissinger and Fabio Zanasi\, and has been published in a series of papers:\n\n- “Rewriting modulo symmetric monoidal structure”\, Pro ceedings of LiCS 2016\n\n- “Confluence of Graph Rewriting with Interface s”\, Proceedings of ESOP 2017\n\n- “Rewriting with Frobenius”\, Proc eedings of LiCS 2018\n LOCATION:https://stable.researchseminars.org/talk/itaca/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Walter Tholen (York University) DTSTART:20210520T123000Z DTEND:20210520T133000Z DTSTAMP:20260404T110914Z UID:itaca/3 DESCRIPTION:Title: Revisiting Burroni's T-categories\nby Walter Tholen (York Univers ity) as part of ItaCa Fest 2021\n\n\nAbstract\nFollowing the appearance of Lambek’s multicategories and Barr’s presentation of topological space s as relational T-algebras\, Albert Burroni introduced T-categories and T- functors in 1971. They provide an overarching environment for the general study of algebras and spaces\, which encompasses elements of monad theory\ , internal category theory\, and categorical topology.\n\n In this talk w e have a fresh look at Burroni’s paper and point to the Street-Walters c omprehensive factorization system for functors and the (antiperfect\, perf ect) factorization system for continuous maps of Tychonoff spaces to demon strate that\, despite its generality\, Burroni’s setting allows for the establishment of non-trivial results and the discovery of unexpected conne ctions between seemingly unrelated theorems. \n\n(Joint work with Leila Ye ganeh)\n LOCATION:https://stable.researchseminars.org/talk/itaca/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Amar Hadzihasanovic (Tallinn University of Technology) DTSTART:20210520T133000Z DTEND:20210520T143000Z DTSTAMP:20260404T110914Z UID:itaca/4 DESCRIPTION:Title: The smash product of monoidal theories\nby Amar Hadzihasanovic (T allinn University of Technology) as part of ItaCa Fest 2021\n\n\nAbstract\ nThe smash product of pointed spaces is a classical construction of topolo gy. The tensor product of props\, which extends both the Boardman-Vogt pro duct of symmetric operads and the tensor product of Lawvere theories\, see ms firmly like a piece of universal algebra. \n\n In this talk\, we will s ee that the two are facets of the same construction: a “smash product of pointed directed spaces”. Here\, “directed spaces” are modelled by combinatorial structures called diagrammatic sets\, developed as a homotop ically sound foundation for diagrammatic rewriting in higher dimensions\, while the cartesian product of spaces is replaced by a form of Gray produc t.\n Most interestingly\, the smash product applies to presentations of hi gher-dimensional theories and systematically produces oriented equations a nd higher-dimensional coherence data (oriented syzygies). This introduces a synthetic\, 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://stable.researchseminars.org/talk/itaca/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Olivia Caramello (Università dell'Insubria) DTSTART:20210615T123000Z DTEND:20210615T133000Z DTSTAMP:20260404T110914Z UID:itaca/5 DESCRIPTION:Title: Relative topos theory via stacks\nby Olivia Caramello (Universit à dell'Insubria) as part of ItaCa Fest 2021\n\n\nAbstract\nIn this talk\, based on joint work with Riccardo Zanfa\, we shall introduce new foundati ons for relative topos theory based on stacks. One of the central results in our theory 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-in dexed categories. This represents a wide generalization of the classical a djunction between presheaves on a topological space and bundles over it\, and allows one to interpret several constructions on sheaves and stacks in a geometrical way\; in particular\, it leads to fibrational descriptions of direct and inverse images of sheaves and stacks\, as well as to a geome tric understanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a ‘petit’ topos associated wi th a ‘gros’ topos\, thereby providing an answer to a problem posed by Grothendieck in the seventies.\n LOCATION:https://stable.researchseminars.org/talk/itaca/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Paolo Saracco (Université Libre de Bruxelles) DTSTART:20210615T133000Z DTEND:20210615T143000Z DTSTAMP:20260404T110914Z UID:itaca/6 DESCRIPTION:Title: Globalization for geometric partial comodules\nby Paolo Saracco ( Université Libre de Bruxelles) as part of ItaCa Fest 2021\n\n\nAbstract\n The study of partial symmetries (e.g. partial dynamical systems\, (co)acti ons\, (co)representations\, comodule algebras) is a relatively recent rese arch area in continuous expansion\, whose origins can be traced back to th e study of C*-algebras generated by partial isometries. One of the central questions in the field is the existence and uniqueness of a so-called glo balization or enveloping (co)action.\n\nIn the framework of partial action s of groups\, any global action of a group on a set induces a partial acti on of the group on any subset by restriction. The idea behind the concept of globalization of a given partial action is to find a (universal) global action such that the initial partial action can be realized as the restri ction of this global one. The importance of this procedure is testified by the numerous globalization results already existing in the literature whi ch\, however\, are based on some ad hoc constructions\, depending on the n ature of the objects carrying the partial action.\n\nWe propose here a uni fied approach to globalization in a categorical setting\, explaining sever al of the existing results from the literature and\, at the same time\, pr oviding a procedure to construct globalizations in concrete contexts of in terest. Our approach relies on the notion of geometric partial comodules ( recently introduced by Hu and Vercruysse in [HV]) which –unlike classica l partial actions\, that exist only for (topological) groups and Hopf alge bras– can be defined over any coalgebra in an arbitrary monoidal categor y with pushouts.\n\n[HV] J. Hu\, J. Vercruysse\, Geometrically partial act ions. Trans. Amer. Math. Soc. 373 (2020)\, no. 6\, 4085–4143.\n\n[PJ] P. Saracco\, J. Vercruysse\, Globalization for geometric partial comodules. Preprint (2020).\n LOCATION:https://stable.researchseminars.org/talk/itaca/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Mauro Porta (IRMA\, Université de Strasbourg) DTSTART:20210928T123000Z DTEND:20210928T133000Z DTSTAMP:20260404T110914Z UID:itaca/7 DESCRIPTION:Title: Pro and Ind-categories in Algebra and Geometry\nby Mauro Porta (I RMA\, Université de Strasbourg) as part of ItaCa Fest 2021\n\n\nAbstract\ nIn this talk we are going to discuss some natural instances of pro and in d categories in algebraic and geometric contexts\, highlighting the import ance 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 questions\, with the intent of determining what is the “correct” object to consider in these contexts\, so as to optimize the generalizati on/applicability trade-off.\n LOCATION:https://stable.researchseminars.org/talk/itaca/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Tim Van der Linden (Université catholique de Louvain) DTSTART:20210928T133000Z DTEND:20210928T143000Z DTSTAMP:20260404T110914Z UID:itaca/8 DESCRIPTION:Title: Algebras with representable representations\nby Tim Van der Linde n (Université catholique de Louvain) as part of ItaCa Fest 2021\n\n\nAbst ract\n(Joint work with Xabier García-Martínez\, Matsvei Tsishyn and Core ntin Vienne)\n\nJust like group actions are represented by group automorph isms\, Lie algebra actions are represented by derivations: up to isomorphi sm\, a split extension of a Lie algebra B by a Lie algebra X corresponds t o a Lie algebra morphism B$\\to\\mathbf{Der}$(X) from B to the Lie algebra $\\mathbf{Der}$(X) of derivations on X. The aim of this talk is to elabor ate on the question\, whether the concept of a derivation can be extended to other types of non-associative algebras over a field $\\mathbf{K}$\, in such a way that these generalised derivations characterise the $\\mathbf{ K}$-algebra actions. We prove that the answer is no\, as soon as the field $\\mathbf{K}$ is infinite. In fact\, we prove a stronger result: already the representability of all abelian actions – which are usually called r epresentations or Beck modules – suffices for this to be true. Thus we c haracterise the variety of Lie algebras over an infinite field of characte ristic different from 2 as the only variety of non-associative algebras wh ich is a non-abelian category with representable representations. This emp hasises the unique role played by the Lie algebra of linear endomorphisms $\\mathbf{gl}$(V) as a representing object for the representations on a ve ctor space V.\n LOCATION:https://stable.researchseminars.org/talk/itaca/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Todd Trimble (Western Connecticut State University) DTSTART:20211021T133000Z DTEND:20211021T143000Z DTSTAMP:20260404T110914Z UID:itaca/9 DESCRIPTION:Title: Grothendieck groups of 2-rigs as lambda rings\nby Todd Trimble (W estern Connecticut State University) as part of ItaCa Fest 2021\n\n\nAbstr act\nThis talk will report on recent joint work with John Baez and Joe Moe ller. We introduce a notion of 2-rig as a way of categorifying the usual n otion of rig (ring without negatives)\; examples include categories of gro up representations\, categories of vector bundles over spaces\, categories of coherent sheaves over projective varieties\, and many others. We descr ibe Schur functors on general 2-rigs\, and indicate how the free 2-rig on one generator encodes Schur functors on 2-rigs. Finally\, we indicate how the Grothendieck group of a 2-rig yields a lambda ring\, where the theory of lambda-rings is a “plethory” obtained by decategorifying a conceptu ally simple 2-plethory that is associated with the free 2-rig.\n LOCATION:https://stable.researchseminars.org/talk/itaca/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Riccardo Zanfa (Università degli studi dell'Insubria) DTSTART:20211021T123000Z DTEND:20211021T133000Z DTSTAMP:20260404T110914Z UID:itaca/10 DESCRIPTION:Title: Generalized presheaf-bundle adjunctions\nby Riccardo Zanfa (Univ ersità degli studi dell'Insubria) as part of ItaCa Fest 2021\n\n\nAbstrac t\nWe present two new results generalizing the well-known presheaf-bundle adjunction for topological spaces\, which relate indexed categories (and p resheaves) over a site (C\,J) with toposes over the topos Sh(C\,J). The co ntent of this seminar can be found in a joint work with Olivia Caramello t itled Relative topos theory via stacks\, and currently available on arXiv. \n LOCATION:https://stable.researchseminars.org/talk/itaca/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Domenico Fiorenza (Università degli Studi di Roma "La Sapienza") DTSTART:20211118T133000Z DTEND:20211118T143000Z DTSTAMP:20260404T110914Z UID:itaca/11 DESCRIPTION:Title: Categorical shadows lurking behind integral formulas for genera\ nby Domenico Fiorenza (Università degli Studi di Roma "La Sapienza") as p art 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 the complex cobordism ring to the field Q of rational numbers\, you h ave an integral formula expressing it.\n\nThe Categorist: If a formula is true\, it must be expressed by a commutative diagram.\n\nProfessor Friedri ch: But my formula is true!\n\nThe Categorist: Then it must be expressed b y a commutative diagram.\n\nProfessor Friedrich: Show me.\n\nThe Categoris t: Let us consider the category of spectra…\n\nProfessor Friedrich: And so?\n\nThe Categorist: Well… I don’t see a commutative diagram here.\n \nProfessor Friedrich: And so?\n\nThe Categorist: And so your formula must be false.\n\nProfessor Friedrich: But my formula is true!\n\nThe Categori st: Impossible.\n\nSir Michael: There is something I think I know on the S panier-Whitehead dual of a smooth manifold that may happen to be of some r elevance here.\n\n(Emil Ionesco\, Triceratops)\n LOCATION:https://stable.researchseminars.org/talk/itaca/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Federico Olimpieri (University of Leeds) DTSTART:20211118T143000Z DTEND:20211118T153000Z DTSTAMP:20260404T110914Z UID:itaca/12 DESCRIPTION:Title: Categorifying Intersection Types\nby Federico Olimpieri (Univers ity of Leeds) as part of ItaCa Fest 2021\n\n\nAbstract\nWe study a family of distributors-induced bicategorical models of lambda-calculus\, proving that they can be syntactically presented via intersection type systems. We first introduce a class of 2-monads whose algebras are monoidal categorie s modelling resource management. We lift these monads to distributors and define a parametric Kleisli bicategory\, giving a sufficient condition for its cartesian closure. In this framework we define a proof-relevant seman tics: the interpretation of a term associates to it the set of its typing derivations in appropriate systems. We prove that our model characterize s olvability\, adapting reducibility techniques to our setting. We conclude by describing wo examples of our construction.\n LOCATION:https://stable.researchseminars.org/talk/itaca/12/ END:VEVENT END:VCALENDAR