This tutorial will be an introduction to the notion of c onnection introduced in [1] in the setting of tangent categories and its e quivalent characterizations in [2]. Building on two formulations of conne ctions on vector bundles that are due to Ehresmann and to Patterson [3]\, respectively\, Cockett and Cruttwell [1] defined a notion of connection in the abstract setting of tangent categories. Equivalent definitions of co nnections in tangent categories were developed in [2] using biproducts in the additive category of differential bundles over an object of a tangent category\, leading also to an economical definition of connections in tang ent categories as vertical connections with the property that a certain co ne is a limit cone. In this tutorial\, we shall survey these equivalent d efinitions of connection and some aspects of their equivalence.\n\n
[1] J. R. B. Cockett and G. S. H. Cruttwell\, Connections in tangent categorie s\, Theory Appl. Categ. 32 (2017)\, 835-888.\n\n
[2] R. B. B. Lucyshyn-W right\, On the geometric notion of connection and its expression in tangen t categories\, Theory Appl. Categ. 33 (2018)\, 832-866.\n\n
[3] L.-N. Pa tterson\, Connexions and prolongations\, Canad. J. Math. 27 (1975)\, 766-7 91.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Richard Garner (Macquarie University) DTSTART:20210615T220000Z DTEND:20210615T224500Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/12 DESCRIPTION:Title: The free tangent category on an affine connection\nby Richard Ga rner (Macquarie University) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\n
The purpose of this talk is to s ketch a construction of the free tangent category containing an object M w ith a connection on its tangent bundle. It turns out that the maps of this tangent category are completely determined by the calculus of multilinear maps on the tangent\nbundle\; and that this calculus can be encoded by a certain kind of operad\, which comes endowed with an operation of covarian t derivative O(n)->O(n+1) and constants T in O(2) (torsion) and R in O(3) (curvature)\, with as axioms the chain rule\, the two Bianchi identities\ nand the Ricci identity. Any such operad generates a tangent category\; t he free such operad generates the free tangent category on an affine conne ction.\n\n
This is work-in-progress with Geoff Cruttwell.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Ben MacAdam (University of Calgary) DTSTART:20210615T230000Z DTEND:20210615T234500Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/13 DESCRIPTION:Title: New tangent structures for Lie algebroids and Lie groupoids\nby Ben MacAdam (University of Calgary) as part of BIRS workshop : Tangent Cat egories and their Applications\n\n\nAbstract\nThe tangent bundle on a smoo th manifold is\, in a sense\, sufficient structure for Lagrangian mechanic s. In a famous note from 1901\, Poincare reformulated Lagrangian mechanics by replacing the tangent bundle with a Lie algebra acting on a smooth man ifold [1\, 2]. Poincare's formalism leads to the Euler-Poincare equations\ , which capture the usual Euler-Lagrange equations as a specific example. In 1996\, Weinstein sketched out a general program building on Poincare's ideas to formulate mechanics on Lie groupoids using Lie algebroids [3]\, w hich motivates the work of Martinez et al. [4\,5]\, Libermann [6]\, and th e recent thesis by Fusca [7].\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Tom Goodwillie (Brown University) DTSTART:20210616T150000Z DTEND:20210616T154500Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/14 DESCRIPTION:Title: Functor calculus\nby Tom Goodwillie (Brown University) as part o f BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\ nFunctor calculus is a way of organizing the interplay between homotopy th eory and stable homotopy theory. Its name reflects an analogy with differe ntial calculus. There are derivatives\, $n$th derivatives\, and Taylor pol ynomials in functor calculus. \n\nFunctors between homotopical categories (categories which\, like the category of topological spaces\, have a suita ble structure for “doing homotopy theory”) can be thought of as resemb ling smooth maps between manifolds. Homotopical categories that are stable correspond to manifolds that are vector spaces. I will sketch the high po ints of functor calculus with this geometric analogy in mind. \n\nUntil re cently the relation with smooth geometry has existed mostly as a suggestiv e analogy. It is being pursued in detail now by Bauer\, Burke\, Ching\, an d others using the framework of tangent structures on categories.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Brenda Johnson (Union College) DTSTART:20210616T160000Z DTEND:20210616T164500Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/15 DESCRIPTION:Title: An example of a cartesian differential category from functor calculu s\nby Brenda Johnson (Union College) as part of BIRS workshop : Tangen t Categories and their Applications\n\n\nAbstract\nIn this talk\, I will p rovide an introduction to abelian functor calculus\, a version of functor calculus inspired by classical constructions of Dold and Puppe\, and of Ei lenberg and Mac Lane. I will then explain how the analog of a directional derivative in abelian functor calculus gives rise to the structure of a c artesian differential category for a particular category of functors of ab elian categories.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Eric Finster (University of Cambridge) DTSTART:20210616T170000Z DTEND:20210616T174500Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/16 DESCRIPTION:Title: The Nilpotence Tower\nby Eric Finster (University of Cambridge) as part of BIRS workshop : Tangent Categories and their Applications\n\n\n Abstract\nMuch like the theory of affine schemes and commutative rings\, t he\ntheory of (higher) topoi leads a dual life: one algebraic and one\ngeo metric. In the geometric picture\, a topos is a kind of\ngeneralized spac e whose points carry the structure of a category.\nDually\, in the algebra ic point of view\, a topos may be thought of\nas the "ring of continuous f unctions on a generalized space with\nvalues in homotopy types".\n\nIn thi s talk\, I will explain the connection between Goodwillie's\ncalculus of f unctors and this algebro-geometric picture of the\ntheory of higher topoi. Specifically\, I will describe how one can\nview the topos of n-excisive functors as an analog of the\ncommutative k-algebra k[x]/xⁿ⁺¹\, free ly generated by a nilpotent\nelement of order n+1.\n\nMore generally\, I w ill show how every left exact localization E → F\nof topoi may be extend ed to a tower of such localizations\n\nE ⋯ → Fₙ → Fₙ₋₁ → ⋯ F₀ = F\n\nwhich we refer to as the Nilpotence Tower\, and whose valu es at an\nobject of E may be seen as a generalized version of the Goodwill ie\ntower of a functor with values in spaces. Under the analogy with\nsch eme theory described above\, this construction corresponds to the\ncomplet ion of a commutative ring along an ideal\, or\, geometrically\,\nto the fi ltration of the formal neighborhood of a subscheme by it's\nn-th order sub -neighborhood. I will also explain how\, in addition\nto the homotopy cal culus\, the orthogonal calculus of Michael Weiss\ncan be seen as an instan ce of this same construction.\n\nThis is joint work with M. Anel\, G. Bied ermann and A. Joyal.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Kristine Bauer (University of Calgary) DTSTART:20210616T210000Z DTEND:20210616T214500Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/17 DESCRIPTION:Title: Tangent Infinity Categories\nby Kristine Bauer (University of Ca lgary) as part of BIRS workshop : Tangent Categories and their Application s\n\n\nAbstract\nThis is joint work with M. Burke and M. Ching. In this t alk\, I will present the definition of a tangent infinity category as a ge neralization of Leung's presentation of tangent categories as Weil-modules . A key example of a tangent structure on the infinity category of infini ty categories is an extension of Lurie’s tangent bundle functor. We cal l this the Goodwillie tangent structure\, since it encodes the theory of G oodwillie calculus. The differential objects in this tangent infinity category are precisely the stable infinity categories. Following Cockett- Cruttwell these form a cartesian differential category. I will explain th at the derivative in this CDC is the same as the BJORT derivative for abel ian functor calculus\, showing that the Goodwillie tangent structure is an extension of BJORT.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Ching (Amherst College) DTSTART:20210616T220000Z DTEND:20210616T224500Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/18 DESCRIPTION:Title: Dual tangent structures for infinity-toposes\nby Michael Ching ( Amherst College) as part of BIRS workshop : Tangent Categories and their A pplications\n\n\nAbstract\nI will describe two tangent infinity-categories whose objects are the infinity-toposes: one algebraic and one geometric. The algebraic version is a restriction to infinity-toposes of the Goodwill ie tangent structure defined by Bauer\, Burke and myself\, in which the ta ngent bundle consists of the stabilizations of slice infinity-toposes. The geometric structure is dual to the algebraic with tangent bundle functor given by an adjoint to that of the Goodwillie structure. There is a useful analogy to tangent structures on the category of commutative rings and it s opposite (the category of affine schemes).\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/18/ END:VEVENT BEGIN:VEVENT SUMMARY:André Joyal (Université du Québec à Montréal) DTSTART:20210616T230000Z DTEND:20210616T233000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/19 DESCRIPTION:Title: The (higher) topos classifying ∞-connected objects\nby André Joyal (Université du Québec à Montréal) as part of BIRS workshop : Tan gent Categories and their Applications\n\n\nAbstract\nJoint work with Math ieu Anel\, Georg Biedermann and Eric Finster.\n\nI will present an applica tion of Goodwillie’s calculus to higher topos theory. The (higher) topos which classifies $\\infty$-connected objects is formally the "dual" of th e (higher) logos $S[U_\\infty]$ freely generated by an $\\infty$-connected object $U_\\infty$. The logos $S[U_\\infty]$ is a left exact topological localisation of the logos $S[U] = Fun[Fin\, S]$ freely generated by an obj ect $U$. We show that a functor\n$ Fin \\to S$ belongs to $S[U_\\infty]$ if and only if it is $\\infty$-excisive if and only if it is the right Kan extension of its restriction to the subcategory of finite n-connected spa ces $C_n \\subset Fin$ for every $n \\geq 0$. There is a morphism of logoi from $S[U_\\infty]$ to the category of Goodwillie towers of functors $Fin \\to S$\, but we do not know if it is an equivalence of categories. We a lso consider the logos $S[U_\\infty′ ]$ freely generated by a pointed $\ \infty$-connected object $U'_\\infty$ .\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Jonathan Gallagher (Dalhousie University) DTSTART:20210617T150000Z DTEND:20210617T154500Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/20 DESCRIPTION:Title: Differential programming\nby Jonathan Gallagher (Dalhousie Unive rsity) as part of BIRS workshop : Tangent Categories and their Application s\n\n\nAbstract\nDifferential and tangent categories have been applied to providing the\nsemantics of differential programming languages. As intere st in\ndifferential programming langauges continues to grow due to\napplic ations in machine learning\, many differential programming\nlanguages are being extended with features for probabilistic\nprogramming and in some ca ses quantum programming. In this talk\, we\nwill investigate structures o n top of differential and tangent\ncategories that allow modelling probab ilistically extended programming\nlanguages. To do this\, we will develop some of the basics of\nfunctional analysis and distribution theory in the context of\ndifferential categories. We will also develop different appr oaches to\nencoding probabilistic computations in a differential language. \n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Bruno Gavranovic (University of Strathclyde) DTSTART:20210617T160000Z DTEND:20210617T164500Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/21 DESCRIPTION:Title: Categorical Foundations of Gradient-Based Learning\nby Bruno Gav ranovic (University of Strathclyde) as part of BIRS workshop : Tangent Cat egories and their Applications\n\n\nAbstract\nWe propose a categorical fou ndation of gradient-based machine learning algorithms in\nterms of lenses\ , parametrised maps\, and reverse derivative categories.\n\nThis foundatio n provides a powerful explanatory and unifying framework: it encompasses a variety of gradient\ndescent algorithms such as ADAM\, AdaGrad\, and Nest erov momentum\,\nas well as a variety of loss functions such as as MSE and Softmax cross-entropy\, shedding new light on their similarities and diff erences.\nOur approach also generalises beyond neural networks (modelled i n categories of smooth maps)\,\naccounting for other structures relevant t o gradient-based learning such as boolean circuits.\n\nFinally\, we also d evelop a novel implementation of gradient-based learning in\nPython\, info rmed by the principles introduced by our framework.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Mario Alvarez-Picallo (Huawei Research) DTSTART:20210617T170000Z DTEND:20210617T174500Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/22 DESCRIPTION:Title: Soundness for automatic differentiation via string diagrams\nby Mario Alvarez-Picallo (Huawei Research) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\nReverse-mode automatic d ifferentiation\, especially in the presence of complex language\nfeatures\ , is notoriously hard to implement correctly\, and most implementations fo cus on\ndifferentiating straight-line imperative first-order code. General isations exist\, however\,\nthat can tackle more advanced features\; for e xample\, the algorithm described by Pearlmutter\nand Siskind in their 2008 paper can differentiate (pure) code containing closures.\n\nWe show that AD algorithms can benefit enormously from being translated into the langua ge\nof string diagrams in two steps: first\, we rephrase Pearlmutter and S iskind's algorithm as\na set of rules for transforming hierarchical graphs \; rules which can -and indeed have been-\nbe implemented correctly and ef ficiently in a non-trivial language. Then\, we sketch a proof\nof soundnes s for it by reducing its transformations to the axioms of Cartesian revers e\ndifferential categories\, expressed as string diagrams.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Dorette Pronk (Dalhousie University) DTSTART:20210617T210000Z DTEND:20210617T212000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/23 DESCRIPTION:Title: Exponentials and Enrichment for Orbispaces\nby Dorette Pronk (Da lhousie University) as part of BIRS workshop : Tangent Categories and thei r Applications\n\n\nAbstract\nOrbifolds are defined like manifolds\, by lo cal charts. Where manifold charts are open subsets of Euclidean space\, or bifold charts consist of an open subset of Euclidean space with an action by a finite group (thus allowing for local singularities). However\, a mor e useful way to represent them is in terms of proper étale groupoids (whi ch we will call orbispaces) and the maps between them are obtained as a bi category of fractions of the 2-category of proper étale groupoids with re spect to the class of essential equivalences. In recent work with Bustillo and Szyld we have shown that in any bicategory of fractions the hom-categ ories form a pseudo colimit of the hom categories of the original bicatego ry.\n\n \n\nWe will show that this result can be extended to our topologic al context: for topological groupoids the hom-groupoids can again be topol ogized and under suitable conditions on the spaces these groupoids form bo th exponentials and enrichment. We will show that taking the appropriate p seudo colimit of these hom-groupoids within the 2-category of topological groupoids gives us a notion of hom-groupoids for the bicategory of orbispa ces. When the domain orbispace is orbit compact\, we see show that this gr oupoid is proper and satisfies the conditions to be an exponential. When w e further cut back our morphisms between orbispaces to so-called admissibl e maps\, we obtain a proper étale groupoid that is essentially equivalent to the pseudo colimit and hence is also the exponential. Furthermore\, we show that the bicategory of orbit-compact orbispaces is enriched over orb ispaces: the composition is given by a map of orbispaces rather than a con tinuous functor.\n\nThis work rephrases the result from [Chen] in terms of groupoid representations for orbifolds and strengthens his result on enri chment: he expressed this in terms of a map between the quotient spaces of the mapping orbispaces\, where we are able to give this in terms of a map between the orbispaces.\n\nI will end the talk with several examples of m apping spaces. This is joint work with Laura Scull and started out as a pr oject of the first Women in Topology workshop.\n\n[Chen] Weimin Chen\, On a notion of maps between orbifolds I: function spaces\, Communications in Contemporary Mathematics 8 (2006)\, pp. 569-620.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Susan Niefield (Union College) DTSTART:20210617T213000Z DTEND:20210617T215000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/24 DESCRIPTION:Title: Linear Bicategories: Quantales and Quantaloids\nby Susan Niefiel d (Union College) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\nLinear bicategories were introduced by Cockett \, Koslowski and Seely as the\nbicategorical version of linearly distribut ive categories. Such a bicategory B\nhas two forms of composition related by a linear distribution. In this talk\, we\nconsider locally ordered line ar bicategories of the form Q-Rel\, i.e.\, relations\nvalued in a quantale Q\; as well as those B which are Girard bicategories.\nThe latter provide examples which are not locally ordered\; and they have\nthe same relation to linear bicategories as ∗-autonomous categories have to\nlinearly dis tributive categories. Examples include the bicategories Quant and\nQtld\, whose 1-cell are bimodules and objects are quantales and quantaloids\,\nre spectively.\n\nThis is joint work with Rick Blute.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Priyaa Srinivasan (University of Calgary) DTSTART:20210617T220000Z DTEND:20210617T222000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/25 DESCRIPTION:by Priyaa Srinivasan (University of Calgary) as part of BIRS w orkshop : Tangent Categories and their Applications\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Bryce Clarke (Macquarie University) DTSTART:20210617T223000Z DTEND:20210617T225000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/26 DESCRIPTION:Title: Lenses as algebras for a monad\nby Bryce Clarke (Macquarie Unive rsity) as part of BIRS workshop : Tangent Categories and their Application s\n\n\nAbstract\nLenses are a family of mathematical structures used in co mputer science to specify bidirectional transformations between systems. I n many instances\, lenses can be understood as morphisms equipped with add itional algebraic structure\, and admit a characterisation as algebras for a monad on a slice category. For example\, very well-behaved lenses betwe en sets were shown to be algebras for a monad on Set / B\, while c-lenses between categories (better known as split opfibrations) are algebras for a monad on Cat / B. Delta lenses are another kind of lens between categorie s which generalise both of these previous examples\, however they have onl y been previously characterised as certain algebras for a semi-monad. In t his talk\, I will improve this result to show that delta lenses also arise as algebras for a monad\, and discuss several interesting consequences of this characterisation.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Anders Kock (Aarhus\, Denmark) DTSTART:20210618T150000Z DTEND:20210618T152000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/27 DESCRIPTION:Title: Barycentric calculus\, and the log-exp bijection\nby Anders Kock (Aarhus\, Denmark) as part of BIRS workshop : Tangent Categories and thei r Applications\n\n\nAbstract\nIn terms of synthetic differential geometry\ , it makes sense to compare the infinitesimal structure of a space and of its tangent bundle. This hinges of the possibility to form certain affine combinations (barycentic calculus) of the algebra maps from A to B\, where A and B are arbitrary commutative rings.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Kadri Ilker Berktav (Middle East Technical University\, Turkey) DTSTART:20210618T153000Z DTEND:20210618T155000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/28 DESCRIPTION:Title: Higher structures in physics\nby Kadri Ilker Berktav (Middle Eas t Technical University\, Turkey) as part of BIRS workshop : Tangent Catego ries and their Applications\n\n\nAbstract\nThis is a talk on higher struct ures in geometry and physics. We\, indeed\, intend to overview the basics of derived algebraic geometry and its essential role in encoding the forma l geometric aspects of certain moduli problems in physics. Throughout the talk\, we always study objects with higher structures in a functorial pers pective\, and we shall focus on algebraic local models for those structure s. With this spirit\, we will investigate higher spaces and structures in a variety of scenarios. In that respect\, we shall also mention some of ou r works in this research direction.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Rowan Poklewski-Koziell (University of Cape Town) DTSTART:20210618T160000Z DTEND:20210618T162000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/29 DESCRIPTION:Title: Frobenius-Eilenberg-Moore objects in dagger 2-categories\nby Row an Poklewski-Koziell (University of Cape Town) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\nA Frobenius monad on a category is a monad-comonad pair whose multiplication and comultipli cation are related via the Frobenius law. Street has given several equival ent definitions of Frobenius monads. In particular\, they are those monads induced from ambidextrous adjunctions. On a dagger category\, much of thi s comes for free: every monad on a dagger category is equivalently a comon ad\, and all adjunctions are ambidextrous. Heunen and Karvonen call a mona d on a dagger category which satisfies the Frobenius law a dagger Frobeniu s monad. They also define the appropriate notion of an algebra for such a monad\, and show that it captures quantum measurements and aspects of reve rsible computing. In this talk\, we will show that these definitions are e xactly what is needed for a formal theory of dagger Frobenius monads\, wit h the usual elements of Eilenberg-Moore object and completion of a 2-categ ory under such objects having dagger counterparts. This may pave the way f or characterisations of categories of Frobenius objects in dagger monoidal categories and generalisations of distributive laws of monads on dagger c ategories.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Tarmo Uustalu (Reykjavik University) DTSTART:20210618T163000Z DTEND:20210618T165000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/30 DESCRIPTION:Title: Monad-comonad interaction laws (co)algebraically\nby Tarmo Uusta lu (Reykjavik University) as part of BIRS workshop : Tangent Categories an d their Applications\n\n\nAbstract\nI will introduce monad-comonad interac tion laws as mathematical objects to describe how an effectful computation (in the sense of functional programming) can run in an environment servin g its requests. Such an interaction law is a natural transformation typed\ n\n$T X \\times DY \\to R (X \\times Y)$\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Nicolas Blanco (University of Birmingham) DTSTART:20210618T170000Z DTEND:20210618T172000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/31 DESCRIPTION:Title: Bifibrations of polycategories and MLL\nby Nicolas Blanco (Unive rsity of Birmingham) as part of BIRS workshop : Tangent Categories and the ir Applications\n\n\nAbstract\nPolycategories are structures generalising categories and multicategories by letting both the domain and codomain of the morphisms to be lists of objects. This provides an interesting framewo rk to study models of classical multiplicative linear logic. In particular the interpretation of the connectives ise given by objects defined by uni versal properties in contrast to their interpretation in a *-autonomous ca tegory.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Simona Paoli (Leicester University) DTSTART:20210618T173000Z DTEND:20210618T175000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/32 DESCRIPTION:Title: Weakly globular double categories and weak units\nby Simona Paol i (Leicester University) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\nWeakly globular double categories are a model of weak 2-categories based on the notion of weak globularity\, and they are known to be suitably equivalent to Tamsamani 2-categories. Fair 2 -categories\, introduced by J. Kock\, model weak 2-categories with strictl y associative compositions and weak unit laws. In this talk I will illustr ate how to establish a direct comparison between weakly globular double ca tegories and fair 2-categories and prove they are equivalent after localis ation with respect to the 2-equivalences. This comparison sheds new light on weakly globular double categories as encoding a strictly associative\, though not strictly unital\, composition\, as well as the category of weak units via the weak globularity condition. \n\nReference: S. Paoli\, Weakl y globular double categories and weak units\, arXiv:2008.11180v1\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Chad Nester (Union College) DTSTART:20210618T190000Z DTEND:20210618T192000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/33 DESCRIPTION:Title: Concurrent Material Histories\nby Chad Nester (Union College) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAb stract\nThe resource-theoretic interpretation of symmetric monoidal catego ries allows us to express pieces of material history as morphisms. In this talk we will see how to extend this to capture concurrent interaction. \n Specifically\, we will see that the resource-theoretic interpretation exte nds to single object double categories with companion and conjoint structu re\, and that in this setting material history may be decomposed into inte racting concurrent components. \nAs an example\, we will show how transiti on systems with boundary (spans of reflexive graphs) can be equipped to ge nerate material history in a compositional way as transitions unfold. Some directions for future work will also be proposed.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Cole Comfort (University of Oxford) DTSTART:20210618T193000Z DTEND:20210618T195000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/34 DESCRIPTION:Title: A graphical calculus for Lagrangian relations\nby Cole Comfort ( University of Oxford) as part of BIRS workshop : Tangent Categories and th eir Applications\n\n\nAbstract\n
Symplectic vector spaces are the phase space of linear mechanical systems. The symplectic form describes\, for ex ample\, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic ve ctor spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on t he evolution -- of various physical systems.\n\n
We give a new presentat ion of the category of Lagrangian relations over an arbitrary field as a ` doubled' category of linear relations. More precisely\, we show that it ar ises as a variation of Selinger's CPM construction applied to linear relat ions\, where the covariant orthogonal complement functor plays of the role of conjugation. Furthermore\, for linear relations over prime fields\, th is corresponds exactly to the CPM construction for a suitable choice of da gger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation\, we prove the equivalence of the prop of affine Lagrangi an relations with the prop of qudit stabilizer theory in odd prime dimensi ons. We hence obtain a unified graphical language for several disparate pr ocess theories\, including electrical circuits\, Spekkens' toy theory\, an d odd-prime-dimensional stabilizer quantum circuits.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Nuiok Dicaire (University of Edinburgh) DTSTART:20210618T200000Z DTEND:20210618T202000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/35 DESCRIPTION:Title: Localization of monads via subunits\nby Nuiok Dicaire (Universit y of Edinburgh) as part of BIRS workshop : Tangent Categories and their Ap plications\n\n\nAbstract\nGiven a “global” monad\, one wishes to obtai n “local” monads such that these locally behave like the global monad. In this talk\, I will provide an overview of how subunits can be used to provide a notion of localisation on monads. I will start by introducing su bunits\, a special kind of subobject of the unit in a monoidal category. A fterwards\, I will provide two equivalent ways of understanding the locali sation of monads. The first involves a strength on subunits\, while the se cond relies on the formal theory of graded monads. I will also explain how to construct one from the other.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Sacha Ikonicoff (University of Calgary) DTSTART:20210618T203000Z DTEND:20210618T205000Z DTSTAMP:20260422T185046Z UID:BIRS_21w5251/36 DESCRIPTION:Title: Divided power algebras with derivation\nby Sacha Ikonicoff (Univ ersity of Calgary) as part of BIRS workshop : Tangent Categories and their Applications\n\n\nAbstract\nClassical divided power algebras are commutat ive associative algebras endowed with `divided power' monomial operations. They were introduced by Cartan in the 1950's in the study of the homology of Eilenberg-MacLane spaces\, and appear in several branches of mathemati cs\, such as crystalline cohomology and deformation theory.\n \nIn this ta lk\, we will investigate divided power algebras with derivation\, and iden tify the most natural compatibility relation between a derivation and the divided power operations. The work of Keigher and Pritchard on formal divi ded power series (also called Hurwitz series) suggests a certain `power ru le'. We will prove\, using the framework of operads\, that this power rule gives a reasonable definition for a divided power algebra with derivation . We will extend this result to a more general notion of divided power alg ebras\, such as restricted Lie algebras\, with derivation.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/36/ END:VEVENT END:VCALENDAR