The purpose of this talk is to sketch a construction of the free tangent category containing an object M with a connection on its tan gent bundle. It turns out that the maps of this tangent category are compl etely determined by the calculus of multilinear maps on the tangent\nbundl e\; and that this calculus can be encoded by a certain kind of operad\, wh ich comes endowed with an operation of covariant derivative O(n)->O(n+1) and constants T in O(2) (torsion) and R in O(3) (curvature)\, with as axio ms the chain rule\, the two Bianchi identities\nand the Ricci identity. A ny such operad generates a tangent category\; the free such operad generat es the free tangent category on an affine connection.\n\n

This is work-i n-progress with Geoff Cruttwell.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/12/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Ben MacAdam (University of Calgary) DTSTART;VALUE=DATE-TIME:20210615T230000Z DTEND;VALUE=DATE-TIME:20210615T234500Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca /live\n\nAbstract\nThe tangent bundle on a smooth manifold is\, in a sense \, sufficient structure for Lagrangian mechanics. In a famous note from 19 01\, Poincare reformulated Lagrangian mechanics by replacing the tangent b undle with a Lie algebra acting on a smooth manifold [1\, 2]. Poincare's f ormalism leads to the Euler-Poincare equations\, which capture the usual E uler-Lagrange equations as a specific example. In 1996\, Weinstein sketche d out a general program building on Poincare's ideas to formulate mechanic s on Lie groupoids using Lie algebroids [3]\, which motivates the work of Martinez et al. [4\,5]\, Libermann [6]\, and the recent thesis by Fusca [7 ].\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/13/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Tom Goodwillie (Brown University) DTSTART;VALUE=DATE-TIME:20210616T150000Z DTEND;VALUE=DATE-TIME:20210616T154500Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z UID:BIRS_21w5251/14 DESCRIPTION:by Tom Goodwillie (Brown University) as part of BIRS workshop : Tangent Categories and their Applications\n\nView-only livestream: http: //www.birs.ca/live\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/14/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Brenda Johnson (Union College) DTSTART;VALUE=DATE-TIME:20210616T160000Z DTEND;VALUE=DATE-TIME:20210616T164500Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z UID:BIRS_21w5251/15 DESCRIPTION:by Brenda Johnson (Union College) as part of BIRS workshop : T angent Categories and their Applications\n\nView-only livestream: http://w ww.birs.ca/live\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/15/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Eric Finster (University of Cambridge) DTSTART;VALUE=DATE-TIME:20210616T170000Z DTEND;VALUE=DATE-TIME:20210616T174500Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z UID:BIRS_21w5251/16 DESCRIPTION:by Eric Finster (University of Cambridge) as part of BIRS work shop : Tangent Categories and their Applications\n\nView-only livestream: http://www.birs.ca/live\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/16/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Kristine Bauer (University of Calgary) DTSTART;VALUE=DATE-TIME:20210616T210000Z DTEND;VALUE=DATE-TIME:20210616T214500Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca/live\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/17/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Michael Ching (Amherst College) DTSTART;VALUE=DATE-TIME:20210616T220000Z DTEND;VALUE=DATE-TIME:20210616T224500Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca/live\n\nAbstract\n I will describe two tangent infinity-categories whose objects are the infi nity-toposes: one algebraic and one geometric. The algebraic version is a restriction to infinity-toposes of the Goodwillie tangent structure define d by Bauer\, Burke and myself\, in which the tangent bundle consists of th e stabilizations of slice infinity-toposes. The geometric structure is dua l to the algebraic with tangent bundle functor given by an adjoint to that of the Goodwillie structure. There is a useful analogy to tangent structu res on the category of commutative rings and its opposite (the category of affine schemes).\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/18/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:André Joyal (Université du Québec à Montréal) DTSTART;VALUE=DATE-TIME:20210616T230000Z DTEND;VALUE=DATE-TIME:20210616T233000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z UID:BIRS_21w5251/19 DESCRIPTION:by André Joyal (Université du Québec à Montréal) as part of BIRS workshop : Tangent Categories and their Applications\n\nView-only livestream: http://www.birs.ca/live\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/19/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Jonathan Gallagher (Dalhousie University) DTSTART;VALUE=DATE-TIME:20210617T150000Z DTEND;VALUE=DATE-TIME:20210617T154500Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca/live\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/20/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Bruno Gavranovic (University of Strathclyde) DTSTART;VALUE=DATE-TIME:20210617T160000Z DTEND;VALUE=DATE-TIME:20210617T164500Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca /live\n\nAbstract\nWe propose a categorical foundation of gradient-based m achine learning algorithms in\nterms of lenses\, parametrised maps\, and r everse derivative categories.\n\nThis foundation provides a powerful expla natory and unifying framework: it encompasses a variety of gradient\ndesce nt algorithms such as ADAM\, AdaGrad\, and Nesterov momentum\,\nas well as a variety of loss functions such as as MSE and Softmax cross-entropy\, sh edding new light on their similarities and differences.\nOur approach also generalises beyond neural networks (modelled in categories of smooth maps )\,\naccounting for other structures relevant to gradient-based learning s uch as boolean circuits.\n\nFinally\, we also develop a novel implementati on of gradient-based learning in\nPython\, informed by the principles intr oduced by our framework.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/21/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Mario Alvarez-Picallo (Huawei Research) DTSTART;VALUE=DATE-TIME:20210617T170000Z DTEND;VALUE=DATE-TIME:20210617T174500Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.bir s.ca/live\n\nAbstract\nReverse-mode automatic differentiation\, especially in the presence of complex language\nfeatures\, is notoriously hard to im plement correctly\, and most implementations focus on\ndifferentiating str aight-line imperative first-order code. Generalisations exist\, however\,\ nthat can tackle more advanced features\; for example\, the algorithm desc ribed by Pearlmutter\nand Siskind in their 2008 paper can differentiate (p ure) code containing closures.\n\nWe show that AD algorithms can benefit e normously from being translated into the language\nof string diagrams in t wo steps: first\, we rephrase Pearlmutter and Siskind's algorithm as\na se t of rules for transforming hierarchical graphs\; rules which can -and ind eed have been-\nbe implemented correctly and efficiently in a non-trivial language. Then\, we sketch a proof\nof soundness for it by reducing its tr ansformations to the axioms of Cartesian reverse\ndifferential categories\ , expressed as string diagrams.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/22/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Dorette Pronk (Dalhousie University) DTSTART;VALUE=DATE-TIME:20210617T210000Z DTEND;VALUE=DATE-TIME:20210617T212000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca/live\n\nAbstrac t\nOrbifolds are defined like manifolds\, by local charts. Where manifold charts are open subsets of Euclidean space\, orbifold charts consist of an open subset of Euclidean space with an action by a finite group (thus all owing for local singularities). However\, a more useful way to represent t hem is in terms of proper étale groupoids (which we will call orbispaces) and the maps between them are obtained as a bicategory of fractions of th e 2-category of proper étale groupoids with respect to the class of essen tial equivalences. In recent work with Bustillo and Szyld we have shown th at in any bicategory of fractions the hom-categories form a pseudo colimit of the hom categories of the original bicategory.\n\n \n\nWe will show th at this result can be extended to our topological context: for topological groupoids the hom-groupoids can again be topologized and under suitable c onditions on the spaces these groupoids form both exponentials and enrichm ent. We will show that taking the appropriate pseudo colimit of these hom- groupoids within the 2-category of topological groupoids gives us a notion of hom-groupoids for the bicategory of orbispaces. When the domain orbisp ace is orbit compact\, we see show that this groupoid is proper and satisf ies the conditions to be an exponential. When we further cut back our morp hisms between orbispaces to so-called admissible 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 o f orbit-compact orbispaces is enriched over orbispaces: the composition is given by a map of orbispaces rather than a continuous functor.\n\nThis wo rk rephrases the result from [Chen] in terms of groupoid representations f or orbifolds and strengthens his result on enrichment: he expressed this i n terms of a map between the quotient spaces of the mapping orbispaces\, w here we are able to give this in terms of a map between the orbispaces.\n\ nI will end the talk with several examples of mapping spaces. This is join t work with Laura Scull and started out as a project of the first Women in Topology workshop.\n\n[Chen] Weimin Chen\, On a notion of maps between or bifolds I: function spaces\, Communications in Contemporary Mathematics 8 (2006)\, pp. 569-620.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/23/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Susan Niefield (Union College) DTSTART;VALUE=DATE-TIME:20210617T213000Z DTEND;VALUE=DATE-TIME:20210617T215000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca/live\n\nAbstract\ nLinear bicategories were introduced by Cockett\, Koslowski and Seely as t he\nbicategorical version of linearly distributive categories. Such a bica tegory B\nhas two forms of composition related by a linear distribution. I n this talk\, we\nconsider locally ordered linear bicategories of the form Q-Rel\, i.e.\, relations\nvalued in a quantale Q\; as well as those B whi ch are Girard bicategories.\nThe latter provide examples which are not loc ally ordered\; and they have\nthe same relation to linear bicategories as ∗-autonomous categories have to\nlinearly distributive categories. Examp les include the bicategories Quant and\nQtld\, whose 1-cell are bimodules and objects are quantales and quantaloids\,\nrespectively.\n\nThis is join t work with Rick Blute.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/24/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Priyaa Srinivasan (University of Calgary) DTSTART;VALUE=DATE-TIME:20210617T220000Z DTEND;VALUE=DATE-TIME:20210617T222000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z UID:BIRS_21w5251/25 DESCRIPTION:by Priyaa Srinivasan (University of Calgary) as part of BIRS w orkshop : Tangent Categories and their Applications\n\nView-only livestrea m: http://www.birs.ca/live\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/25/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Bryce Clarke (Macquarie University) DTSTART;VALUE=DATE-TIME:20210617T223000Z DTEND;VALUE=DATE-TIME:20210617T225000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca/live\n\nAbstract\nLenses are a family of mathematical structures used in computer science to specify b idirectional transformations between systems. In many instances\, lenses c an be understood as morphisms equipped with additional algebraic structure \, and admit a characterisation as algebras for a monad on a slice categor y. For example\, very well-behaved lenses between sets were shown to be al gebras 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 le nses are another kind of lens between categories which generalise both of these previous examples\, however they have only been previously character ised as certain algebras for a semi-monad. In this talk\, I will improve t his 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/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Anders Kock (Aarhus\, Denmark) DTSTART;VALUE=DATE-TIME:20210618T150000Z DTEND;VALUE=DATE-TIME:20210618T152000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca/live\n\nAbstrac t\nIn terms of synthetic differential geometry\, it makes sense to compare the infinitesimal structure of a space and of its tangent bundle. This hi nges of the possibility to form certain affine combinations (barycentic ca lculus) of the algebra maps from A to B\, where A and B are arbitrary comm utative rings.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/27/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Kadri Ilker Berktav (Middle East Technical University\, Turkey) DTSTART;VALUE=DATE-TIME:20210618T153000Z DTEND;VALUE=DATE-TIME:20210618T155000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca/li ve\n\nAbstract\nThis is a talk on higher structures in geometry and physic s. We\, indeed\, intend to overview the basics of derived algebraic geomet ry and its essential role in encoding the formal geometric aspects of cert ain moduli problems in physics. Throughout the talk\, we always study obje cts with higher structures in a functorial perspective\, and we shall focu s on algebraic local models for those structures. With this spirit\, we wi ll investigate higher spaces and structures in a variety of scenarios. In that respect\, we shall also mention some of our works in this research di rection.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/28/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Rowan Poklewski-Koziell (University of Cape Town) DTSTART;VALUE=DATE-TIME:20210618T160000Z DTEND;VALUE=DATE-TIME:20210618T162000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http:// www.birs.ca/live\n\nAbstract\nA Frobenius monad on a category is a monad-c omonad pair whose multiplication and comultiplication are related via the Frobenius law. Street has given several equivalent definitions of Frobeniu s monads. In particular\, they are those monads induced from ambidextrous adjunctions. On a dagger category\, much of this comes for free: every mon ad on a dagger category is equivalently a comonad\, and all adjunctions ar e ambidextrous. Heunen and Karvonen call a monad on a dagger category whic h satisfies the Frobenius law a dagger Frobenius monad. They also define t he appropriate notion of an algebra for such a monad\, and show that it ca ptures quantum measurements and aspects of reversible computing. In this t alk\, we will show that these definitions are exactly what is needed for a formal theory of dagger Frobenius monads\, with the usual elements of Eil enberg-Moore object and completion of a 2-category under such objects havi ng dagger counterparts. This may pave the way for characterisations of cat egories of Frobenius objects in dagger monoidal categories and generalisat ions of distributive laws of monads on dagger categories.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/29/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Tarmo Uustalu (Reykjavik University) DTSTART;VALUE=DATE-TIME:20210618T163000Z DTEND;VALUE=DATE-TIME:20210618T165000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z UID:BIRS_21w5251/30 DESCRIPTION:by Tarmo Uustalu (Reykjavik University) as part of BIRS worksh op : Tangent Categories and their Applications\n\nView-only livestream: ht tp://www.birs.ca/live\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/30/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Nicolas Blanco (University of Birmingham) DTSTART;VALUE=DATE-TIME:20210618T170000Z DTEND;VALUE=DATE-TIME:20210618T172000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca/live\n\nAbstra ct\nPolycategories are structures generalising categories and multicategor ies by letting both the domain and codomain of the morphisms to be lists o f objects. This provides an interesting framework to study models of class ical multiplicative linear logic. In particular the interpretation of the connectives ise given by objects defined by universal properties in contra st to their interpretation in a *-autonomous category.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/31/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Simona Paoli (Leicester University) DTSTART;VALUE=DATE-TIME:20210618T173000Z DTEND;VALUE=DATE-TIME:20210618T175000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca/live\n\nAb stract\nWeakly globular double categories are a model of weak 2-categories based on the notion of weak globularity\, and they are known to be suitab ly equivalent to Tamsamani 2-categories. Fair 2-categories\, introduced by J. Kock\, model weak 2-categories with strictly associative compositions and weak unit laws. In this talk I will illustrate how to establish a dire ct comparison between weakly globular double categories and fair 2-categor ies and prove they are equivalent after localisation with respect to the 2 -equivalences. This comparison sheds new light on weakly globular double c ategories as encoding a strictly associative\, though not strictly unital\ , composition\, as well as the category of weak units via the weak globula rity condition. \n\nReference: S. Paoli\, Weakly globular double categorie s and weak units\, arXiv:2008.11180v1\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/32/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Chad Nester (Union College) DTSTART;VALUE=DATE-TIME:20210618T190000Z DTEND;VALUE=DATE-TIME:20210618T192000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView -only livestream: http://www.birs.ca/live\n\nAbstract\nThe resource-theore tic interpretation of symmetric monoidal categories allows us to express p ieces of material history as morphisms. In this talk we will see how to ex tend this to capture concurrent interaction. \nSpecifically\, we will see that the resource-theoretic interpretation extends to single object double categories with companion and conjoint structure\, and that in this setti ng material history may be decomposed into interacting concurrent componen ts. \nAs an example\, we will show how transition systems with boundary (s pans of reflexive graphs) can be equipped to generate 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/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Cole Comfort (University of Oxford) DTSTART;VALUE=DATE-TIME:20210618T193000Z DTEND;VALUE=DATE-TIME:20210618T195000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca/live\nAbstrac t: TBA\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/34/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Nuiok Dicaire (University of Edinburgh) DTSTART;VALUE=DATE-TIME:20210618T200000Z DTEND;VALUE=DATE-TIME:20210618T202000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca/live\n\nAbstract\nG iven a “global” monad\, one wishes to obtain “local” monads such t hat these locally behave like the global monad. In this talk\, I will prov ide an overview of how subunits can be used to provide a notion of localis ation on monads. I will start by introducing subunits\, a special kind of subobject of the unit in a monoidal category. Afterwards\, I will provide two equivalent ways of understanding the localisation of monads. The first involves a strength on subunits\, while the second relies on the formal t heory of graded monads. I will also explain how to construct one from the other.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/35/ URL:http://www.birs.ca/live END:VEVENT BEGIN:VEVENT SUMMARY:Sacha Ikonicoff (University of Calgary) DTSTART;VALUE=DATE-TIME:20210618T203000Z DTEND;VALUE=DATE-TIME:20210618T205000Z DTSTAMP;VALUE=DATE-TIME:20210613T001145Z 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\nView-only livestream: http://www.birs.ca/live\n\nAbstract \nClassical divided power algebras are commutative associative algebras en dowed with `divided power' monomial operations. They were introduced by Ca rtan in the 1950's in the study of the homology of Eilenberg-MacLane space s\, and appear in several branches of mathematics\, such as crystalline co homology and deformation theory.\n \nIn this talk\, we will investigate di vided power algebras with derivation\, and identify the most natural compa tibility relation between a derivation and the divided power operations. T he work of Keigher and Pritchard on formal divided power series (also call ed Hurwitz series) suggests a certain `power rule'. We will prove\, using the framework of operads\, that this power rule gives a reasonable definit ion for a divided power algebra with derivation. We will extend this resul t to a more general notion of divided power algebras\, such as restricted Lie algebras\, with derivation.\n LOCATION:https://researchseminars.org/talk/BIRS_21w5251/36/ URL:http://www.birs.ca/live END:VEVENT END:VCALENDAR