CAUTION: The Sender of this email is not from within Dalhousie. The New York City Category Theory Seminar YouTube Channel: here. <https://www.youtube.com/channel/UCNOfhimbNwZwJO2ltv1AZOw/videos> Webpage: http://www.sci.brooklyn.cuny.edu/~noson/CTseminar.html Information also here: https://researchseminars.org/seminar/Category_Theory <http://cs.gc.cuny.edu> Department of Computer Science <http://math.gc.cuny.edu/> Department of Mathematics <http://www.gc.cuny.edu/> The Graduate Center of The City University of New York THE TALKS WILL ALL BE DONE ON ZOOM THIS SEMESTER. THIS WEEKS ZOOM INFORMATION: Topic: New York City Category Theory Seminar Time: Wednesdays 07:00 PM Eastern Time (US and Canada) Join Zoom Meeting <https://us02web.zoom.us/j/81139396554> https://us02web.zoom.us/j/81139396554 Meeting ID: 811 3939 6554 Passcode: NYCCTS Usually our talks take place at 365 Fifth Avenue (at 34th Street) <http://maps.google.com/maps?sourceid=navclient&ie=UTF-8&rlz=1T4GFRC_enUS206 US206&q=365+Fifth+Avenue,+ny&um=1&sa=X&oi=geocode_result&resnum=1&ct=title> map (Diagonally across from the Empire State Building) New York, NY 10016-4309 _____ _____ Spring 2021 _____ _____ * Speaker: Jason Parker, Brandon University in Manitoba. * Date and Time: Wednesday February 3, 2021, 7:00 - 8:30 PM., on Zoom. * Title: Isotropy Groups of Quasi-Equational Theories. * Abstract: In [2], my PhD supervisors (Pieter Hofstra and Philip Scott) and I studied the new topos-theoretic phenomenon of isotropy (as introduced in [1]) in the context of single-sorted algebraic theories, and we gave a logical/syntactic characterization of the isotropy group of any such theory, thereby showing that it encodes a notion of inner automorphism or conjugation for the theory. In the present talk, I will summarize the results of my recent PhD thesis, in which I build on this earlier work by studying the isotropy groups of (multi-sorted) quasi-equational theories (also known as essentially algebraic, cartesian, or finite limit theories). In particular, I will show how to give a logical/syntactic characterization of the isotropy group of any such theory, and that it encodes a notion of inner automorphism or conjugation for the theory. I will also describe how I have used this characterization to exactly characterize the 'inner automorphisms' for several different examples of quasi-equational theories, most notably the theory of strict monoidal categories and the theory of presheaves valued in a category of models. In particular, the latter example provides a characterization of the (covariant) isotropy group of a category of set-valued presheaves, which had been an open question in the theory of categorical isotropy. [1] J. Funk, P. Hofstra, B. Steinberg. Isotropy and crossed toposes. Theory and Applications of Categories 26, 660-709, 2012. [2] P. Hofstra, J. Parker, P.J. Scott. Isotropy of algebraic theories. Electronic Notes in Theoretical Computer Science 341, 201-217, 2018. _____ * Speaker: Peter Hines, University of York. * Date and Time: Wednesday February 10, 2021, 7:00 - 8:30 PM., on Zoom. * Title: Shuffling cards as an operad. * Abstract: The theory of how two packs of cards may be shuffled together to form a single pack has been remarkably well-studied in combinatorics, group theory, statistics, and other areas of mathematics. This talk aims to study natural extensions where 1/ We may have infinitely many cards in a deck, 2/ We may take the result of a previous shuffle as one of our decks of cards (i.e. shuffles are hierarchical), and 3/ There may even be an infinite number of decks of cards. Far from being 'generalisation for generalisation's sake', the original motivation came from theoretical & practical computer science. The mathematics of card shuffles is commonly used to describe processing in multi-threaded computations. Moving to the infinite case gives a language in which one may talk about potentially non-terminating processes, or servers with an unbounded number of clients, etc. However, this talk is entirely about algebra & category theory -- just as in the finite case, the mathematics is of interest in its own right, and should be studied as such. We model shuffles using operads. The intuition behind them of allowing for arbitrary n-ary operations that compose in a hierarchical manner makes them a natural, inevitable choice for describing such processes such as merging multiple packs of cards. We use very concrete examples, based on endomorphism operads in groupoids of arithmetic operations. The resulting structures are at the same time both simple (i.e. elementary arithmetic operations), and related to deep structures in mathematics and category theory (topologies, tensors, coherence, associahedra, etc.) We treat this as a feature, not a bug, and use it to describe complex structures in elementary terms. We also aim to give previously unobserved connections between distinct areas of mathematics. _____ * Speaker: Richard Blute, University of Ottawa. * Date and Time: Wednesday February 17, 2021, 7:00 - 8:30 PM., on Zoom. * Title: Finiteness Spaces, Generalized Polynomial Rings and Topological Groupoids. * Abstract: The category of finiteness spaces was introduced by Thomas Ehrhard as a model of classical linear logic, where a set is equipped with a class of subsets to be thought of as finitary. Morphisms are relations preserving the finitary structure. The notion of finitary subset allows for a sharp analysis of computational structure. Working with finiteness spaces forces the number of summands in certain calculations to be finite and thus avoid convergence questions. An excellent example of this is how Ribenboim's theory of generalized power series rings can be naturally interpreted by assigning finiteness monoid structure to his partially ordered monoids. After Ehrhard's linearization construction is applied, the resulting structures are the rings of Ribenboim's construction. There are several possible choices of morphism between finiteness spaces. If one takes structure-preserving partial functions, the resulting category is complete, cocomplete and symmetric monoidal closed. Using partial functions, we are able to model topological groupoids, when we consider composition as a partial function. We can associate to any hemicompact etale Hausdorff groupoid a complete convolution ring. This is in particular the case for the infinite paths groupoid associated to any countable row-finite directed graph. _____ * Speaker: Joshua Sussan, Medgar Evers, CUNY. * Date and Time: Wednesday March 3, 2021, 7:00 - 8:30 PM., on Zoom. * Title: Categorification and quantum topology. * Abstract: The Jones polynomial of a link could be defined through the representation theory of quantum sl(2). It leads to a 3-manifold invariant and 2+1 dimensional TQFT. In the mid 1990s, Crane and Frenkel outlined the categorification program with the aim of constructing a 3+1 dimensional TQFT by upgrading the representation theory of quantum sl(2) to some categorical structures. We will review these ideas and give examples of various categorifications of quantum sl(2) constructions. _____ * Speaker: *** ***. * Date and Time: Wednesday March 10, 2021, 7:00 - 8:30 PM., on Zoom. * Title: ****. * Abstract: *** _____ * Speaker: Tobias Fritz, University of Innsbruck. * Date and Time: Wednesday March 17, 2021, 7:00 - 8:30 PM., on Zoom. * Title: Categorical Probability and the de Finetti Theorem. * Abstract: I will give an introduction to categorical probability in terms of Markov categories, followed by a discussion of the classical de Finetti theorem within that framework. Depending on whether current ideas work out or not, I may (or may not) also present a sketch of a purely categorical proof of the de Finetti theorem based on the law of large numbers. Joint work with Tomáš Gonda, Paolo Perrone and Eigil Fjeldgren Rischel. _____ * Speaker: *** ***. * Date and Time: Wednesday March 24, 2021, 7:00 - 8:30 PM., on Zoom. * Title: ****. * Abstract: *** _____ * Speaker: *** ***. * Date and Time: Wednesday April 7, 2021, 7:00 - 8:30 PM., on Zoom. * Title: ****. * Abstract: *** _____ * Speaker: Ross Street, Macquarie University. * Date and Time: Wednesday April 14, 2021, 7:00 - 8:30 PM., on Zoom. * Title: Absolute colimits for differential graded categories. * Abstract: TBA _____ * Speaker: *** ***. * Date and Time: Wednesday April 21, 2021, 7:00 - 8:30 PM., on Zoom. * Title: ****. * Abstract: *** _____ * Speaker: *** ***. * Date and Time: Wednesday April 28, 2021, 7:00 - 8:30 PM., on Zoom. * Title: ****. * Abstract: *** _____ * Speaker: Juan Orendain, University of Mexico, UNAM. * Date and Time: Wednesday May 5, 2021, 7:00 - 8:30 PM., on Zoom. * Title: Categorical General Boundary Formulation. * Abstract: The General Boundary Formulation, or GBF for short, is an axiomatic framework for physical theories based on the principles of locality and operationalism. It extends the classical formulation for quantum mechanics in order to accomodate concepts from general relativity. In this talk I will argue that theories in the GBF should be thought of as 'object values' of more general theories defined in terms of involutive symmetric monoidal functors. We call such theories equivariant local field theories. I will further explain how part of the data of an equivariant local field theory can be expressed through a form of lax equivariant TQFT in the form of an involutive symmetric monoidal double functor defined on the double category of bordisms. I will not assume prior knowledge of quantum field theory or double categories. The following are references for the General Boundary Formulaion: [1] R. Oeckl, A "general boundary" formulation for quantum mechanics and quantum gravity, Physics Letters B, 575, 3-4,(2003) 318-324 [2] R. Oeckl, A local and operational framework for the foundations of physics, Adv. Theor. Math. Phys. 23 (2019) 437-592 _____ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]