# BIRS workshop : Tangent Categories and their Applications

Mathematics

Banff International Research Station

 Audience: Researchers in the topic Conference dates: 14-Jun-2021 to 18-Jun-2021 Curator: BIRS Programme Coordinator* *contact for this listing

One of the most fundamental notions when studying functions of a real variable is the rate of change of a function, as measured by its derivative. Geometrically, the derivative is the slope of the tangent line. Both the notion of the derivative and the tangent line (or more generally, the tangent bundle of a smooth manifold) can be defined purely axiomatically because of the underlying structure of the category of smooth functions. The derivative, for example, is determined by its properties (such as the sum and product formulae for differentiation, the chain rule, etc.). This structural approach to the derivative leads to the notion of a differential category. <\p>

Similarly, the tangent bundle of a manifold is determined by what one normally thinks of as properties - for example, properties of its sections. The abstraction of these properties to their most general setting is the notion of a tangent category. <\p> When working with models of these categories such as smooth functions or manifolds, the properties seem like natural consequence of the model. Differential and tangent categories, however, suggest a rather different perspective: namely, that the properties above determine the structure of the derivative or the tangent bundle and that one should look more broadly for instances of these structures in mathematics.

Category theory has proven to be a powerful way to organize mathematical structures and to show how these structures relate. The goal of this workshop is to utilize the cross-disciplinary language of tangent categories to identify and delineate general phenomena related to tangent structures in a wide variety of disciplines, including algebraic and differential geometry, algebraic topology and theoretical computer science.

Upcoming talks
Past talks
FriJun 1820:30Sacha IkonicoffDivided power algebras with derivation
FriJun 1820:00Nuiok DicaireLocalization of monads via subunits
FriJun 1819:30Cole ComfortA graphical calculus for Lagrangian relations
FriJun 1817:30Simona PaoliWeakly globular double categories and weak units
FriJun 1817:00Nicolas BlancoBifibrations of polycategories and MLL
FriJun 1816:00Rowan Poklewski-KoziellFrobenius-Eilenberg-Moore objects in dagger 2-categories
FriJun 1815:30Kadri Ilker BerktavHigher structures in physics
FriJun 1815:00Anders KockBarycentric calculus, and the log-exp bijection
ThuJun 1722:30Bryce ClarkeLenses as algebras for a monad
ThuJun 1722:00Priyaa SrinivasanTBA
ThuJun 1721:30Susan NiefieldLinear Bicategories: Quantales and Quantaloids
ThuJun 1721:00Dorette PronkExponentials and Enrichment for Orbispaces
ThuJun 1717:00Mario Alvarez-PicalloSoundness for automatic differentiation via string diagrams
ThuJun 1716:00Bruno GavranovicCategorical Foundations of Gradient-Based Learning
ThuJun 1715:00Jonathan GallagherDifferential programming
WedJun 1623:00André JoyalThe (higher) topos classifying ∞-connected objects
WedJun 1622:00Michael ChingDual tangent structures for infinity-toposes
WedJun 1621:00Kristine BauerTangent Infinity Categories
WedJun 1617:00Eric FinsterThe Nilpotence Tower
WedJun 1616:00Brenda JohnsonAn example of a cartesian differential category from functor calculus
WedJun 1615:00Tom GoodwillieFunctor calculus
TueJun 1523:00Ben MacAdamNew tangent structures for Lie algebroids and Lie groupoids
TueJun 1522:00Richard GarnerThe free tangent category on an affine connection
TueJun 1521:00Rory Lucyshyn-WrightConnections in Tangent
TueJun 1517:30Lionel VauxA groupoid of permutation trees (with applications to the Taylor expansion of λ-terms)
TueJun 1516:30Marie KerjeanFrom categorical models of differentiation to topologies in vector spaces
TueJun 1515:45Michele PaganiAutomatic differentiation in PCF
TueJun 1515:00Thomas EhrhardDifferentiation in probabilistic coherence spaces
MonJun 1423:00Richard GarnerWeil spaces, and the embedding theorem for tangent categories
MonJun 1422:00Ben MacAdamAn introduction to differential bundles
MonJun 1421:00Geoffrey CruttwellIntroduction to tangent categories
MonJun 1417:15Robin CockettThe Faa Di Bruno Construction and Skew Enrichment
MonJun 1416:15Jean-Simon LemayThe World of Differential Categories: A Tutorial on Cartesian Differential Categories
MonJun 1415:15Rick BluteSyntax And Semantics Of Differentiation
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