The New York City Category Theory Seminar This semester there is a special three-part lecture series on TQFT and Computation. These lectures take place on February 8, 15 and 22 in The Graduate Center. See below. Department of Computer Science <http://cs.gc.cuny.edu> Department of Mathematics <http://math.gc.cuny.edu/> The Graduate Center of The City University of New York <http://www.gc.cuny.edu/> THIS SEMESTER, SOME TALKS WILL BE IN-PERSON AND SOME WILL BE ON ZOOM. Time: Wednesdays 07:00 PM Eastern Time (US and Canada) IN-PERSON INFORMATION: 365 Fifth Avenue (at 34th Street) map <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> (Diagonally across from the Empire State Building) New York, NY 10016-4309 Room 5417 (not the usual Room 6417) The videos of the lectures will be put up on YouTube a few hours after the lecture. ZOOM INFORMATION: https://brooklyn-cuny-edu.zoom.us/j/89472980386?pwd=Z3g3Q3h3V1dQUmg2ZlVGU1Rw SEhMZz09 Meeting ID: 894 7298 0386 Passcode: NYCCTS Seminar web page. <http://www.sci.brooklyn.cuny.edu/~noson/Seminar/index.html> Videoed talks. <https://www.youtube.com/channel/UCNOfhimbNwZwJO2ltv1AZOw/videos> Previous semesters. <http://www.sci.brooklyn.cuny.edu/~noson/Seminar/Previous%20Semesters.html> researchseminars.org page. <https://researchseminars.org/seminar/Category_Theory> Contact N. Yanofsky <mailto:noson@sci.brooklyn.cuny.edu> to schedule a speaker or to add a name to the seminar mailing list. _____ _____ Spring 2023 _____ _____ * Speaker: Igor Baković, University of Osijek, Croatia. * Date and Time: Wednesday February 1, 2023, 7:00 - 8:30 PM. IN PERSON TALK. * Title: Enhanced 2-adjunctions. * Abstract: Whenever one has a class of objects possessing certain structure and a hierarchy of morphisms that preserve structure more or less tightly, we are in an enhanced context. Enhanced 2-categories were introduced by Lack and Shulman in 2012 with a paradigmatic example of an enhanced 2-category T-alg of strict algebras for a 2-monad and whose tight and loose 1-cells are pseudo- and lax morphisms of algebras, respectively. They can be defined in two equivalent ways: either as 2-functors, which are the identity on objects, faithful, and locally fully faithful, or as categories enriched over the cartesian closed category F, whose objects are functors that are fully faithful and injective on objects. Lack and Shulman called objects of F full embeddings, but we will call them "enhanced categories" because they are nothing else but categories with a distinguished class of objects, which we call tight.The 2-category F has a much richer structure besides being cartesian closed; there are additional closed (but not monoidal) structures, and we show how 2-categories with a right ideal of 1-cells as in 2-categories with Yoneda structure on them can be presented as categories enriched in F in the sense of Eilenberg and Kelly. Since Lack and Shulman were mainly motivated by limits in enhanced 2-categories, they didn't further develop the theory of enhanced (co)lax functors and their enhanced lax adjunctions. The purpose of this talk is to lay the foundations of the theory of enhanced 2-adjunctions and give their examples throughout mathematics and theoretical computer science. _____ * Special Topic: TQFT and Computation, First Lecture. * Speaker: Mikhail Khovanov, Columbia University. * Date and Time: Wednesday February 8, 2023, 7:00 - 8:30 PM. IN PERSON TALK. * Title: Universal construction and its applications. * Abstract: Universal construction starts with an evaluation of closed n-manifolds and builds a topological theory (a lax TQFT) for n-cobordisms. A version of it has been used for years as an intermediate step in constructing link homology theories, by evaluating foams embedded in 3-space. More recently, universal construction in low dimensions has been used to find interesting structures related to Deligne categories, formal languages and automata. In the talk we will describe the universal construction and review these developments. _____ * Special Topic: TQFT and Computation, Second Lecture. * Speaker: Mee Seong Im, United States Naval Academy, Annapolis. * Date and Time: Wednesday February 15, 2023, 7:00 - 8:30 PM. IN PERSON TALK. * Title: Automata and topological theories. * Abstract: Theory of regular languages and finite state automata is part of the foundations of computer science. Topological quantum field theories (TQFT) are a key structure in modern mathematical physics. We will interpret a nondeterministic automaton as a Boolean-valued one-dimensional TQFT with defects labelled by letters of the alphabet for the automaton. We will also describe how a pair of a regular language and a circular regular language gives rise to a lax one-dimensional TQFT. _____ * Special Topic: TQFT and Computation, Third Lecture. * Speaker: Joshua Sussan, CUNY. * Date and Time: Wednesday February 22, 2023, 7:00 - 8:30 PM. IN PERSON TALK. * Title: Non-semisimple Hermitian TQFTs. * Abstract: Topological quantum field theories coming from semisimple categories build upon interesting structures in representation theory and have important applications in low dimensional topology and physics. The construction of non-semisimple TQFTs is more recent and they shed new light on questions that seem to be inaccessible using their semisimple relatives. In order to have potential applications to physics, these non-semisimple categories and TQFTs should possess Hermitian structures. We will define these structures and give some applications. _____ * Speaker: Jens Hemelaer, University of Antwerp. * Date and Time: Wednesday March 15, 2023, 7:00 - 8:30 PM. IN PERSON TALK. * Title: EILC toposes. * Abstract: In topos theory, local connectedness of a geometric morphism is a very geometric property, in the sense that it is stable under base change, can be checked locally, and so on. In some situations however, the weaker property of being essential is easier to verify. In this talk, we will discuss EILC toposes: toposes E such that any essential geometric morphism with codomain E is automatically locally connected. It turns out that many toposes of interest are EILC, including toposes of sheaves on Hausdorff spaces and classifying toposes of compact groups. _____ * Speaker: Jim Otto. * Date and Time: Wednesday March 29, 2023, 7:00 - 8:30 PM. IN PERSON TALK * Title: P Time, A Bounded Numeric Arrow Category, and Entailments. * Abstract:We revisit the characterization of the P Time functions from our McGill thesis. 1. We build on work of L. Roman (89) on primitive recursion and of A. Cobham (65) and Bellantoni-Cook(92) on P Time. 2. We use base 2 numbers with the digits 1 & 2. Let N be the set of these numbers. We split the tapes of a multi-tape Turing machine each into 2 stacks of digits 1 & 2. These are (modulo allowing an odd numberof stacks) the multi-stack machines we use to study P Time. 3. Let Num be the category with objects the finite products of N and arrows the functions between these. From its arrow category Num^2 we abstract the doctrine (here a category of small categories with chosen structure) PTime of categories with with finite products, base 2 numbers, 2-comprehensions, flat recursion, & safe recursion. Since PTime is a locally finitely presentable category, it has an initial category I. Our characterization is that the bottom of the image of I in Num^2 consists of the P Time functions. 4. We can use I (thinking of its arrows as programs) to run multi-stack machines long enough to get P Time.This is the completeness of the characterization. 5. We cut down the numeric arrow category Num^2, using Bellantoni-Cook growth & time bounds on the functions, to get a bounded numeric arrow category B. B is in the doctrine PTime. This yields the soundness of the characterization. 6. For example, the doctrine of toposes with base 1 numbers, choice, & precisely 2 truth values (which captures much of ZC set theory) likely lacks an initial category, much as there is an initial ring, but no initial field. 7. On the other hand, the L. Roman doctrine PR of categories with finite products, base 1 numbers, & recursion (that is, product stable natural numbers objects) does have an initial category as it consists of the strong models of a finite set of entailments. And is thus locally finitely presentable. We sketch the signature graph for these entailments. And some of these entailments. Similarly (but with more complexity) there are entaiments for the doctrine PTime. _____ * Speaker: Walter Tholen, York University. * Date and Time: Wednesday April 19, 2023, 7:00 - 8:30 PM. ZOOM TALK. * Title: What does “smallness” mean in categories of topological spaces? * Abstract: Quillen’s notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as the class of finite discrete spaces, or just the empty space , as the examples and remarks in the existing literature may suggest? In this talk we will demonstrate that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces, such as those of T_i-spaces (i= 0, 1, 2), can be quite challenging and may lead to unexpected surprises. In fact, we will show that there are significant differences in this regard even amongst the categories defined by the standard separation conditions, with the T1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of Top. (Based on joint work with J. Adamek, M. Husek, and J. Rosicky.) _____ * Speaker: Dusko Pavlovic, University of Hawai‘i at Mānoa. * Date and Time: Wednesday April 26, 2023, 7:00 - 8:30 PM. ZOOM TALK. * Title: Program-closed categories. * Abstract: > Let CC be a symmetric monoidal category with a comonoid on every object. Let CC* be the cartesian subcategory with the same objects and just the comonoid homomorphisms. A *programming language* is a well-ordered object P with a *program closure*: a family of X-natural surjections CC(XA,B) <<--run_X-- CC*(X,P) one for every pair A,B. In this talk, I will sketch a proof that program closure is a property: Any two programming languages are isomorphic along run-preserving morphisms. The result counters Kleene's interpretation of the Church-Turing Thesis, which has been formalized categorically as the suggestion that computability is a structure, like a group presentation, and not a property, like completeness. We prove that it is like completeness. The draft of a book on categorical computability is available from the web site dusko.org. _____ * Speaker: Gemma De las Cuevas, University of Innsbruck. * Date and Time: Wednesday May 3, 2023, 7:00 - 8:30 PM. ZOOM TALK. * Title: A framework for universality across disciplines. * Abstract: What is the scope of universality across disciplines? And what is its relation to undecidability? To address these questions, we build a categorical framework for universality. Its instances include Turing machines, spin models, and others. We introduce a hierarchy of universality and argue that it distinguishes universal Turing machines as a non-trivial form of universality. We also outline the relation to undecidability by drawing a connection to Lawvere’s Fixed Point Theorem. Joint work with Sebastian Stengele, Tobias Reinhart and Tomas Gonda. _____ * Speaker: Arthur Parzygnat, Nagoya University. * Date and Time: Wednesday May 17, 2023, 7:00 - 8:30 PM. IN PERSON TALK. * Title: Inferring the past and using category theory to define retrodiction. * Abstract: Classical retrodiction is the act of inferring the past based on knowledge of the present. The primary example is given by Bayes' rule P(y|x) P(x) = P(x|y) P(y), where we use prior information, conditional probabilities, and new evidence to update our belief of the state of some system. The question of how to extend this idea to quantum systems has been debated for many years. In this talk, I will lay down precise axioms for (classical and quantum) retrodiction using category theory. Among a variety of proposals for quantum retrodiction used in settings such as thermodynamics and the black hole information paradox, only one satisfies these categorical axioms. Towards the end of my talk, I will state what I believe is the main open question for retrodiction, formalized precisely for the first time. This work is based on the preprint https://arxiv.org/abs/2210.13531 and is joint work with Francesco Buscemi. _____ _____ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]