BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Robert Paré (Dalhousie University) DTSTART:20221128T173000Z DTEND:20221128T183000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/1 DESCRIPTION:Title: Some things about double categories\nby Robert Paré (Dalhousie University) as part of Virtual Double Categories Worksho p\n\n\nAbstract\nThe title says it all. I will look at some examples as an excuse to introduce and motivate the basic concepts of double category th eory. This way I hope to bring them to life so that\, in the end\, they mi ght be considered as friends and not mere acquaintances (or worse). No pri or knowledge of double categories will be assumed.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /1/ END:VEVENT BEGIN:VEVENT SUMMARY:Matthew Di Meglio (University of Edinburgh) DTSTART:20221128T183000Z DTEND:20221128T193000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/2 DESCRIPTION:Title: Recognising retromorphisms retrospectively\nby Matthew Di Meglio (University of Edinburgh) as part of Virtual Double Cat egories Workshop\n\n\nAbstract\nWhat do categories\, preordered sets\, met ric spaces\, and topological spaces have in common? They all have an under lying set\, and\, when viewed in the right way\, they all have some additi onal data that relates the elements of the underlying set. For each of the se kinds of mathematical objects\, there are two natural kinds of morphism s. The usual kind of morphisms\, which we will refer to merely as morphism s\, consist of a function between the underlying sets\, and a mapping of t he additional data in the same direction as the function. For example\, fu nctors and continuous maps are morphisms of categories and topological spa ces\, respectively. The other kind of morphisms\, which we will refer to a s retromorphisms\, consist of a function between the underlying sets\, and a lifting of the additional data in the opposite direction to that of the function. For example\, cofunctors and open maps are retromorphisms of ca tegories and topological spaces\, respectively. In all cases\, the morphis ms and retromorphisms assemble into a double category whose cells capture a notion of compatibility between the two kinds of morphisms.\n\nActually\ , for any double category with chosen companions\, there is an associated double category of monad morphisms and monad retromorphisms\, and all of t he above examples arise in this way. My talk will give a gentle introducti on to these double-categorical concepts\, focusing on how the above exampl es and several more fit into the general theory. No prior knowledge about double categories will be assumed. This extends recent work (arXiv:2209.01 144) on enriched cofunctors beyond the setting of enriched category theory .\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /2/ END:VEVENT BEGIN:VEVENT SUMMARY:Paula Verdugo (Macquarie University) DTSTART:20221128T193000Z DTEND:20221128T203000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/3 DESCRIPTION:Title: An homotopical way to compare 2-categories and dou ble categories\nby Paula Verdugo (Macquarie University) as part of Vir tual Double Categories Workshop\n\n\nAbstract\nIt has proven useful to stu dy some aspects of 2-categories within the framework of double categories – for example\, there are some 2-categorical universal properties that e lude a purely 2-categorical description but show themselves naturally expr essed in the double categorical world. In this talk we present ways to com pare these two 2-dimensional worlds homotopically by means of model catego ries. If time permits\, we will explore how we may generalize this idea to higher dimensions. This is joint work with Lyne Moser and Maru Sarazola. Basic knowledge about (2- and double) categories will be useful\, together with the idea behind model categories.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /3/ END:VEVENT BEGIN:VEVENT SUMMARY:Seerp Roald Koudenburg (Middle East Technical University\, Norther n Cyprus Campus) DTSTART:20221129T150000Z DTEND:20221129T160000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/4 DESCRIPTION:Title: Formalising size in formal category theory\nby Seerp Roald Koudenburg (Middle East Technical University\, Northern Cypru s Campus) as part of Virtual Double Categories Workshop\n\n\nAbstract\nFor malising the classical notion of Yoneda embedding necessarily includes for malising the notion of "locally small functor"\, that is functors $f\\colo n A\\to C$ such that all homsets $C(fa\, c)$ are small. A Yoneda structure on a 2-category for instance includes the notion of "admissible morphism" which formalises that of locally small functor.\n\nIn a (generalised) dou ble categorical approach to formal Yoneda embeddings\, instead of postulat ing a notion of "locally small morphism"\, it is natural to regard *all* h orizontal (say) morphisms to be "locally small". This gives an intrisic no tion of local smallness which fruitfully interacts with other formal notio ns such as that of adjunct vertical morphism\, fully faithful vertical mor phism and restriction of horizontal morphisms.\n\nThe aim of this talk is to introduce a double categorical approach to formal Yoneda embeddings wit hout presuming familiarity with formal category theory. Motivated by obtai ning a common formalisation of the classical notions of generic subobject (of topos theory) and Yoneda embedding\, we are led to consider "augmented virtual double categories" as the right setting for doing so. These gener alise double categories by allowing cells with paths of horizontal morphis ms as horizontal source and\, as horizontal target\, either a single horiz ontal morphism or an empty path. We conclude by looking at several formal results that involve formal notions of smallness such as\, for instance\, a description of the sense in which a formal presheaf object $P$\, defined by a formal Yoneda embedding $y\\colon A\\to P$\, is cocomplete.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /4/ END:VEVENT BEGIN:VEVENT SUMMARY:Christina Vasilakopoulou (National Technical University of Athens\ , School of Applied Mathematics and Physical Sciences) DTSTART:20221129T160000Z DTEND:20221129T170000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/5 DESCRIPTION:Title: Monoidal Kleisli double categories\nby Christi na Vasilakopoulou (National Technical University of Athens\, School of App lied Mathematics and Physical Sciences) as part of Virtual Double Categori es Workshop\n\n\nAbstract\nIn this talk\, we will describe how to formally extend the so-called arithmetic product of symmetric sequences to colored symmetric sequences\, namely their many-object analogue. This requires ge neral results on extending monoidal structures to Kleisli constructions\, specifically in the world of double categories. Some basic knowledge of fi brant double categories is assumed\, but involved concepts and tools shall be in general described throughout the talk.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /5/ END:VEVENT BEGIN:VEVENT SUMMARY:Simona Paoli (University of Aberdeen) DTSTART:20221201T160000Z DTEND:20221201T170000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/6 DESCRIPTION:Title: Double categories and weak units\nby Simona Pa oli (University of Aberdeen) as part of Virtual Double Categories Workshop \n\n\nAbstract\nSegal-type models of weak 2-categories are simplicial mode ls of bicategories\, and they comprise the Tamsamani model and the more re cent weakly globular double categories\, introduced by Paoli and Pronk. Fa ir 2-categories\, introduced by J. Kock\, model weak 2-categories with str ictly associative compositions and weak unit laws. This model has some fea tures in common with the Segal-type models\, but with the simplicial delta replaced by the fat delta\, which encodes weak units.\n\nI will illustrat e a direct comparison of fair 2-categories with weakly globular double cat egories: this result sheds new light on the notion of weak globularity as encoding weak units and has potential for higher dimensional generalizatio ns.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /6/ END:VEVENT BEGIN:VEVENT SUMMARY:Brandon Shapiro (Topos Institute) DTSTART:20221129T183000Z DTEND:20221129T193000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/7 DESCRIPTION:Title: Double Presheaf Categories via Polynomial Functors \nby Brandon Shapiro (Topos Institute) as part of Virtual Double Categ ories Workshop\n\n\nAbstract\nThe category Poly of polynomial functors on sets and natural transformations between them forms a monoidal category un der composition. By results of Ahman–Uustalu and Garner\, the bicategory of comonoids and bicomodules in Poly has as objects small categories and as morphisms from C to D familial (aka parametric right adjoint) functors from C-Set to D-Set\, the categories of Set-valued functors from C and D. Familial functors between (co)presheaf categories are defined as disjoint unions of representables on each component\, and include the free category monad on graphs and data migration functors between categories of databas es.\n\nPolynomial functors on categories also have a monoidal structure by composition\, where the comonoids are precisely small double categories s atisfying a factorization condition that includes most double categories i n common use. Bicomodules between such double categories \\(\\bf C\\) and \\(\\bf D\\) induce "familial" double functors from \\({\\bf C}^t\\)-Set t o \\({\\bf D}^t\\)-Set\, the copresheaf double categories of lax double fu nctors from the transpose of \\(\\bf C\\) and \\(\\bf D\\) to the double c ategory of sets\, functions\, spans\, and maps of spans.\n\nI will introdu ce polynomial functors\, comonoids and bicomodules\, familial functors\, a nd double copresheaf categories to build up to this result (joint with Dav id Spivak) and its early implications for double category theory and categ orical databases\, assuming background only in double categories and ordin ary (co)presheaf categories.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /7/ END:VEVENT BEGIN:VEVENT SUMMARY:Chad Nester (Tallinn University of Technology) DTSTART:20221130T150000Z DTEND:20221130T160000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/8 DESCRIPTION:Title: The Free Cornering of a Monoidal Category\nby Chad Nester (Tallinn University of Technology) as part of Virtual Double C ategories Workshop\n\n\nAbstract\nThe free cornering of a monoidal categor y is a single-object double category obtained by adding companion and conj oint structure to it. If the morphisms of the monoidal category admit inte rpretation as processes\, then this interpretation extends to the cells of the free cornering\, which then admit interpretation as interacting proce sses. In this talk I will introduce the free cornering construction\, expl ain its interactive interpretation\, and survey a number of results surrou nding the free cornering from both published and ongoing work. \n\nSingle- object double categories are much simpler than their more general counterp arts. I expect that most of the talk will be accessible to anyone with eve n a surface understanding of monoidal categories\, and in particular their string diagrams.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /8/ END:VEVENT BEGIN:VEVENT SUMMARY:John Bourke (Masaryk University) DTSTART:20221130T160000Z DTEND:20221130T170000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/9 DESCRIPTION:Title: A double-categorical look at different flavours of factorisation system\nby John Bourke (Masaryk University) as part of Virtual Double Categories Workshop\n\n\nAbstract\nOrthogonal factorisation systems are a simple and natural categorical structure that is not hard t o grasp. Algebraic weak factorisation systems\, which generalise the orth ogonal ones\, are not so straightforward\, involving monads and comonads a nd quite a lot of structure to keep track of. However these structures ca n be approached via double categories and this double-categorical perspect ive is very useful. I will tell a bit of the story of the double-categori cal approach to these structures.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /9/ END:VEVENT BEGIN:VEVENT SUMMARY:Nicolas Behr (CNRS\, IRIF\, Université Paris Cité) DTSTART:20221130T173000Z DTEND:20221130T183000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/10 DESCRIPTION:Title: Double-categorical Compositional Rewriting Theory \nby Nicolas Behr (CNRS\, IRIF\, Université Paris Cité) as part of V irtual Double Categories Workshop\n\n\nAbstract\nReporting on recent resul ts of joint work with R. Harmer\, P.-A. Melliès and N. Zeilberger\, I wil l present a novel formalization of compositional rewriting theory via doub le categories. For a given rewriting theory\, individual rewriting steps a re formalized as 2-cells of a double category. One of the crucial aspects of compositionally consists then in providing a set of axioms that the dou ble category of the rewriting system must satisfy in order to ensure the e xistence of concurrency and associativity theorems.\, which are quintessen tial for developing important applications of rewriting systems such as In combinatorics and Markov chain theory. Another concept central to this en d\, I.e.\, “counting modulo universal properties”\, may be implemented via a certain presheaf and coend calculus. Time permitting\, I will sketc h how the counting calculus then leads to a categorification of the concep t of rule algebras (which capture the combinatorics of interactions of rew riting steps).\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /10/ END:VEVENT BEGIN:VEVENT SUMMARY:Susan Niefield (Union College\, Schenectady\, NY) DTSTART:20221130T183000Z DTEND:20221130T193000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/11 DESCRIPTION:Title: Cartesian Closed Double Categories\nby Susan Niefield (Union College\, Schenectady\, NY) as part of Virtual Double Cate gories Workshop\n\n\nAbstract\nIn this talk\, we consider approaches to ex ponentiability in double categories. For a 1-category $\\mathcal D$\, one can define cartesian closure via a pointwise or a 2-variable adjunction\, and arrive at equivalent definitions. The pointwise approach requires the existence of an exponential object $Z^Y$\, for every object $Y$ of $\\math cal D$\, whereas for the latter\, the exponentials are given by a bifuncto r ${\\mathcal D}^{\\rm op}\\times {\\mathcal D}\\to \\mathcal D$. We will see that these two approaches differ for a double categories $\\mathbb D$. In particular\, a bifunctor ${\\mathbb D}^{\\rm op}\\times {\\mathbb D}\\ to {\\mathbb D}$ would involve not only objects of $\\mathbb D_0$\, but al so those of $\\mathbb D_1$\, i.e.\, vertical morphisms of $\\mathbb D$.\n\ nWe will assume some familiarity with double categories\, but recall the d efinitions and properties of adjoints on double categories\, as well as th e relevant examples.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /11/ END:VEVENT BEGIN:VEVENT SUMMARY:Lyne Moser (University of Regensburg) DTSTART:20221201T150000Z DTEND:20221201T160000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/12 DESCRIPTION:Title: Representation theorem for enriched categories\nby Lyne Moser (University of Regensburg) as part of Virtual Double Cate gories Workshop\n\n\nAbstract\nUniversal properties play an important role in mathematics\, as they allow us to make many constructions such as (co) limits\, Kan extensions\, adjunctions\, etc. In particular\, a universal p roperty is often formulated by requiring that a certain presheaf is repres entable. The representation theorem gives a useful characterization of the se representable presheaves in terms of initial objects in their category of elements. Going one dimension up and considering 2-categories\, with ts lil clingman we showed that the same results is true for Cat-valued preshe aves if one considers their "double category of elements". In this talk\, I will explain how to generalize the representation theorem to the more ge neral framework of V-enriched categories\, where V is a cartesian closed c ategory. This is joint work with Maru Sarazola\, and Paula Verdugo. Only b asic knowledge of ordinary category theory will be assumed.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /12/ END:VEVENT BEGIN:VEVENT SUMMARY:Bojana Femić (Mathematical Institute of Serbian Academy of Scienc es and Arts) DTSTART:20221129T173000Z DTEND:20221129T183000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/13 DESCRIPTION:Title: Gray-type monoidal product and Bifunctor Theorem for double categories\nby Bojana Femić (Mathematical Institute of Ser bian Academy of Sciences and Arts) as part of Virtual Double Categories Wo rkshop\n\n\nAbstract\nIn the introductory part we will recall what double categories are and we will\nshow some examples that illustrate why double categories present a more\nsuitable framework to work in than bicategories . We proceed by describing the\nconstruction of a Gray type monoidal produ ct for double categories for two\nversions of morphisms: strict and lax do uble functors. Along the way we\ncharacterize (lax double) quasi-functors( in analogy to "quasi-functors of two\nvariables" for 2-categories of Gray ). We introduce their 2-category\n$q\\text{-}\\operatorname{Lax}_{hop}(\\m athbb{A}\\times\\mathbb{B}\,\\mathbb{C})$\nand construct a 2-functor\n$\\m athcal{F}: q\\text{-}\\operatorname{Lax}_{hop}(\\mathbb{A}\\times\\mathbb{ B}\,\\mathbb{C}) \\to \\operatorname{Lax}_{hop}(\\mathbb{A}\\times\\mathbb {B}\,\\mathbb{C})$\nto the 2-category of lax\ndouble bifunctors. This is a double category version of the Bifunctor Theorem\,\nrecently proved for 2 -categories. We will show when this 2-functor $\\mathcal{F}$\nrestricts to a 2-equivalence. For a consequence we derive 2-functors known as\ncurryin g and uncurrying functors in Computer Science\, in the context of double\n categories. We finish by showing the application of the 2-functor $\\mathc al{F}$ on\n2-monads. Concretely: we obtain that $\\mathcal{F}$ is a genera lization to non-trivial\ndouble categories of the classical Beck's result\ , that a distributive law\nbetween two monads makes it possible for them t o compose. It turns out that a\ndistributive law is a specification of a ( lax double) quasi-functor to trivial\ndouble categories. No prior knowledg e of double categories is necessary\, but\nknowing 2-categories\, their fu nctors\, transformations and modifications is\nhelpful.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /13/ END:VEVENT BEGIN:VEVENT SUMMARY:Juan Orendain (Case Western Reserve University) DTSTART:20221201T173000Z DTEND:20221201T183000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/14 DESCRIPTION:Title: How long does it take to frame a bicategory?\ nby Juan Orendain (Case Western Reserve University) as part of Virtual Dou ble Categories Workshop\n\n\nAbstract\nFramed bicategories are double cate gories having all companions and conjoints. Many structures naturally orga nize into framed bicategories\, e.g. open Petri nets\, polynomials functor s\, polynomial comonoids\, structured cospans\, etc. Symmetric monoidal st ructures on framed bicategories descend to symmetric monoidal structures o n bicategories. The axioms defining symmetric monoidal double categories a re more tractible\, and arguably more natural\, than those defining symmet ric 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 been useful in expressing Kelly and Street's mates correspondence and in proving the higher dimensional Seifert-van Kampen t heorem of Brown et. al.\, amongst many other applications. We consider lif ting problems in their full generality.\n\nGlobularly generated double cat egories are minimal solutions to lifting problems of bicategories into dou ble categories along given categories of vertical arrows. Globularly gener ated double categories form a coreflective sub-2-category of general doubl e categories. This\, together with an analysis of the internal structure o f globularly generated double categories yields a numerical invariant on g eneral double categories. We call this number the vertical length. The ver tical length of a double category C measures the complexity of composition s of globular and horizontal identity squares of C and thus provides a mea sure of complexity for lifting problems on the horizontal bicategory of C. I will explain recent results on the theory of globularly generated doubl e categories and the vertical length invariant. Minimal knowledge of doubl e categories and bicategories will be assumed.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /14/ END:VEVENT BEGIN:VEVENT SUMMARY:Dorette Pronk (Dalhousie University\, Department of Mathematics an d Statistics) DTSTART:20221201T183000Z DTEND:20221201T193000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/15 DESCRIPTION:Title: Fibrations and doubly lax colimits of double cate gories\nby Dorette Pronk (Dalhousie University\, Department of Mathema tics and Statistics) as part of Virtual Double Categories Workshop\n\n\nAb stract\nClassically\, the Grothendieck construction\, or the category of e lements\, of a pseudo functor \\(F\\colon {\\mathbf D}\\to{\\bf Cat}\\) (a lso called the indexing functor) has two properties that define it (up to equivalence):\n\n(A) it is the object part of the adjoint equivalence bet ween the category of such pseudo functors\, also called indexing functors\ , and the category of fibrations over \\({\\mathbf D}\\)\;\n\n(B) it is th e lax colimit for the diagram defined by \\(F\\).\n\nThese two results hav e been shown to hold for suitably defined Grothendieck constructions for b icategories and \\(\\infty\\)-categories and has been conjectured for weak \\(n\\)-categories. So when one is interested in colimits of double categ ories or fibrations between double categories we want to look for a suitab le Grothendieck construction.\n\nWhen we replace the category \\({\\mathbf D}\\) above by a double category \\({\\mathbb D}\\)\, the first question is what the codomain of the indexing functor should be. The issue is that \\({\\bf DblCat}\\)\, the category of double categories\, is not a double category. However\, it is enriched over double categories: it has two clas ses of transformations (which I will call inner and outer) a s local arrows and modifications as local double cells. In this talk I wi ll present two replacements for \\({\\bf DblCat}\\) in the definition of i ndexing functor. I will replace it with the quintet double category of the 2-category of double categories with double functors and outer transforma tions and I will replace it by the double 2-category \\({\\mathbb S}\\math rm{pan(Cat)}\\) (and we will work with all lax functors). The first case h as the advantage that the objects are still double categories. The second case is inspired by the fact that a lax double functor from the terminal d ouble category into \\({\\mathbb S}\\mathrm{pan(Cat)}\\) corresponds to a double category.\n\nThe last type of indexing functor has an associated Gr othendieck construction that can be viewed as a Grothendieck construction that is done in two layers and leads to a category object in a suitable ca tegory of fibrations: a double fibration.\nThe first type of indexing func tor leads us to a new notion of what we call a doubly lax transformatio n between double functors of the form \\({\\mathbb A}\\to{\\mathbb Q}{ \\mathcal D}_{\\mathrm{outer}}\\) where \\({\\mathcal D}\\) is a \\({\\bf DblCat}\\)-enriched category. These give then rise to a notion of doubl y lax colimit for diagrams of double categories. These doubly lax coli mits can be constructed using the double Grothendieck construction I will introduce\, and have a universal property that is expressed in term s of both classes of 2-cells of \\({\\bf DblCat}\\). Furthermore\, it also has a canonical projection double functor to \\({\\mathbb D}\\)\, the ind exing double category\, and this has the properties of a double fibration in one direction and a weaker fibration property in the other direction.\n \nWe will discuss how both constructions generalize various colimit and fi bration constructions for 2-categories and bicategories and finally\, we w ill consider how the two are related. A partial answer to this last questi on is given by the construction of a lax double functor \\({\\mathbb Q}{\\ bf DblCat}_{\\mathrm{outer}}\\to {\\mathbb S}\\mathrm{pan(Cat)}\\).\n\nThi s talk is based on work done with Geoff Cruttwell\, Michael Lambert and Ma rtin Szyld in [1] and work done with Marzieh Bayeh and Martin Szyld in [2] .\n\nBackground needed for this talk: mainly\, the basics of double catego ries - double functors\, and horizontal and vertical transformations (here referred to as outer and inner transformations)\; familiarity with the Gr othendieck construction and its properties is also useful.\n\n[1] G.S.H. C ruttwell\, M.J. Lambert\, D.A. Pronk\, M. Szyld\, Double Fibrations\, arXi v:2205.15240\, and to appear in Theory and Applications of Categories (2022)\n\n[2] Marzieh Bayeh\, Dorette Pronk\, Martin Szyld\, The Grothe ndieck Construction for Double Categories\, in progress.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /15/ END:VEVENT BEGIN:VEVENT SUMMARY:Matteo Capucci (University of Strathclyde) DTSTART:20221202T160000Z DTEND:20221202T170000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/16 DESCRIPTION:Title: From categorical systems theory to categorical cy bernetics\nby Matteo Capucci (University of Strathclyde) as part of Vi rtual Double Categories Workshop\n\n\nAbstract\nMyers' categorical system theory is a double categorical yoga for describing the compositional struc ture of open dynamical systems. It unifies and builds on previous work on operadic notions of system theory\, and provides a strong conceptual scaff olding for behavioral system theory. However\, some of the most interestin g systems out have a richer compositional structure than that of dynamical systems. These are cybernetic systems\, or in other words\, interactive c ontrol systems. Notable and motivating examples are strategic games and ma chine learning models. In this talk I’m going to introduce the tools and language of categorical system theory and outline how categorical cyberne tics theory might look like. At the end\, we will briefly venture into the triple dimension.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /16/ END:VEVENT BEGIN:VEVENT SUMMARY:Claudio Pisani (Independent researcher) DTSTART:20221202T150000Z DTEND:20221202T160000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/17 DESCRIPTION:Title: Operads as double functors\nby Claudio Pisani (Independent researcher) as part of Virtual Double Categories Workshop\n\ n\nAbstract\nThe language of double categories provides a direct abstract approach to colored\noperads (that is\, symmetric multicategories)\, enlig htening and simplifying\nseveral classical facts and notions. Namely\, an operad \\(\\mathcal{O}\\)\, in its\nnon-skeletal form\, is a product-prese rving lax double functor \n\\[\n(\\mathbb{P}\\mathrm{b}{\\mathbf{Set_f}})^ {\\mathrm{op}} \\to \\mathbb{S}\\mathrm{et}\n\\tag{1}\n\\]\nfrom the dual of the double category of pullback squares in\nfinite sets to the double c ategory of mappings and spans. This\ncharacterization can be rephrased in two ways. First\, by a universal property\nof the monoid construction one can consider instead normal functors \\((\\mathbb{P}\\mathrm{b}{\\mathbf{S et_f}})^{\\mathrm{op}} \\to \\mathbb{C}\\mathrm{at}\\). Second\, by a doub le category of elements\nconstruction [1\,2]\, operads are certain discret e double\nfibrations \\(d:\\mathbb{D} \\to \\mathbb{P}\\mathrm{b}{\\mathbf {Set_f}}\\). The idea is that the proarrow part\nof \\(\\mathbb{D}\\) is t he category of families of objects and of arrows in \\(\\mathcal{O}\\)\n(i ndexed by \\({\\mathbf{Set_f}}\\)) and that families of arrows in \\(\\mat hcal{O}\\) can be\nreindexed along pullback squares in \\({\\mathbf{Set_f} }\\).\n\nSequential operads [3]\, symmetric monoidal categories and commut ative\nmonoids are those operads \\(\\mathcal{O}\\) for which the vertical part of \\(d\\) is a\nfibration\, an opfibration or a discrete opfibratio n\, respectively. In\nparticular\, we get the accompanying characterizatio n of commutative monoids as\nproduct-preserving functors \\((\\mathbb{P}\\ mathrm{b} {\\mathbf{Set_f}})^{\\mathrm{op}} \\to \\mathbb{S}\\mathrm{q}{\\ mathbf{Set}}\\).\n\nThis approach suggests a notion of generalized operad obtained by replacing in\n(1) the category of finite sets with any catego ry \\(\\mathcal{C}\\). For instance\,\na category with small products (or sums) gives a generalized monoidal category\nwith \\(\\mathcal{C} = {\\mat hbf{Set}}\\)\, along with the corresponding generalized monoid of\nisomorp hism classes. Furthermore\, one can define cartesian operads in this more\ ngeneral setting and prove therein the equivalence between tensor products \,\nuniversal products and algebraic products (as in the well-known case o f\ncategories enriched in commutative monoids).\n\nWe assume a basic knowl edge of the language of double categories and of\nfibrations\; some acquai ntance with operads or multicategories would be helpful\,\neven if the bas ic ideas will be recalled.\n\n[1] M. Lambert\, Discrete double fibrations\ , TAC 37 (2021).\n\n[2] R. Paré\, Yoneda theory for double categories\, T AC 17 (2011).\n\n[3] C. Pisani\, Sequential multicategories\, TAC 29 (2014 ).\n\n[4] C. Pisani\, Fibered multicategory theory\, arXiv (2022).\n\n[5] C. Pisani\, Operads as double functors\, arXiv (2022).\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /17/ END:VEVENT BEGIN:VEVENT SUMMARY:David Jaz Myers (Center for Topological and Quantum Systems\, NYU Abu Dhabi) DTSTART:20221202T173000Z DTEND:20221202T183000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/18 DESCRIPTION:Title: The Para Construction as a Distributive Law\n by David Jaz Myers (Center for Topological and Quantum Systems\, NYU Abu D habi) as part of Virtual Double Categories Workshop\n\n\nAbstract\n(Joint with Matteo Capucci)\n\nThe Para construction takes a monoidal category $M $ and gives a category $\\mathrm{Para}(M)$ where a morphism $a \\to b$ is a pair $(c\, f : c \\otimes a \\to b)$ of a "parameter space" $c$ and a pa rameterized map f in M. This construction formalizes the idea of separatin g inputs into special "control variables" or "parameters" which will be se t separately from the other inputs to a process. The Para construction has played an important role in categorical accounts of deep learning — whe re it was first described by Fong\, Spivak\, and Tuyeras — open games\, and cybernetics.\n\nThe Para construction has been generalized in a number of ways. First\, it can take an action of a monoidal category $M$ on a ca tegory $C$ (an "actegory"). And second\, the resulting category can be see n as the shadow of a bicategory where 2-cells are reparameterizations. In this talk\, we will see a further generalization of the scope of the Para construction — we will take an actegory $\\otimes : M \\times C \\to C$ and produce a double category $\\mathbb{P}\\mathrm{ara}(\\otimes)$ whose v ertical morphisms are parameterized by objects of $M$ and whose horizontal morphisms are those of $C$.\n\nWe will show that in this guise\, the Para construction arises as a (pseudo)distributive law between the action doub le category of $\\otimes : M \\times C \\to C$ and the double category of arrows of $C$\, each seen as (pseudo)monads in a tricategory of spans in $ \\mathsf{Cat}$. Our construction is abstract and applies in any suitably c omplete 2-category $\\mathbb{K}$\, in particular in the 2-category of doub le categories with vertical transformations. This lets us construct a trip le category $\\mathrm{Para}(\\mathrm{Arena})$ whose morphisms are lenses\, charts\, and parameterized lenses respectively. The cubes in this triple category give representable behaviors of Capucci-Gavranovic-Hedges-Rischel cybernetic systems\, and one of the resulting face double categories is a variant of Shapiro and Spivak's double category $\\mathbb{O}\\mathrm{rg}$ . The method of proof also suggests generalizing the domain of the Para co nstruction to take any "dependent actegory"\, and at this level of general ity the double category of spans in a cartesian category is revealed to be a sort of Para construction.\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /18/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Lambert (University of Massachusetts Boston) DTSTART:20221202T183000Z DTEND:20221202T193000Z DTSTAMP:20260409T093541Z UID:VirtualDoubleCategoriesWorkshop/19 DESCRIPTION:Title: Double Categories of Relations\nby Michael La mbert (University of Massachusetts Boston) as part of Virtual Double Categ ories Workshop\n\n\nAbstract\nHow can we tell whether a double category is a double category of relations on a regular category? Any such double cat egory is at least an equipment and is cartesian\, but what else is needed? This talk aims to present a characterization theorem that describes these further conditions. This result should be seen as a double-categorical ve rsion of the development due to Carboni and Walters that showed which bica tegories occur as bicategories of relations on a regular category. This ta lk should be accessible to anyone who knows the basics of bicategories and double categories. Along the way\, we will review the work of Carboni and Walters\, and define what it means for a double category to be cartesian and to be an equipment. We will also talk about tabulators and in what way a double category can be seen as "functionally complete".\n LOCATION:https://researchseminars.org/talk/VirtualDoubleCategoriesWorkshop /19/ END:VEVENT END:VCALENDAR