The New York City
Category Theory Seminar
Department of Computer Science
Department of Mathematics
The Graduate Center of The City University of New York
THE TALKS WILL ALL BE DONE ON ZOOM THIS SEMESTER.
Time: Wednesdays 07:00 PM Eastern Time (US and Canada)
Meeting ID: 841 3433 1639
Usually our talks take place at
365 Fifth Avenue (at 34th Street) map
(Diagonally across from the Empire State Building)
New York, NY 10016-4309
Wednesdays 7:00 - 8:30 PM
Research seminars page.
Contact N. Yanofsky to
schedule a speaker
or to add a name to the
seminar mailing list.
Speaker: Gemma De las Cuevas, University of Innsbruck.
Date and Time:   Wednesday October 6, 2021, 7:00 - 8:30 PM., on Zoom.
Title: From simplicity to universality and undecidability.
Abstract: Why is it so easy to generate complexity? I will suggest that this is due to the phenomenon of universality — essentially every non-trivial system is universal, and thus able to explore all complexity in its domain. We understand universality in spin models, automata and neural networks. I will present the first step toward rigorously linking the first two, where we cast classical spin Hamiltonians as formal languages and classify the latter in the Chomsky hierarchy. We prove that the language of (effectively) zero-dimensional spin Hamiltonians is regular, one-dimensional spin Hamiltonians is deterministic context-free, and higher-dimensional and all-to-all spin Hamiltonians is context-sensitive. I will also talk about the other side of the coin of universality, namely undecidability, and will raise the question of whether universality is
visible in Lawvere’s Theorem.
Speaker: Dan Shiebler, Oxford University.
Date and Time:   Wednesday October 20, 2021, 7:00 - 8:30 PM., on Zoom.
Title: Out of Sample Generalization with Kan Extensions.
Abstract: A common problem in data science is
use this function defined over this small set to generate predictions over that larger set. Extrapolation, interpolation, statistical inference and forecasting all reduce to this problem. The Kan extension is a powerful tool in category theory that generalizes this notion. In this work we explore several applications of Kan extensions to data science.
Speaker: Dusko Pavlovic, University of Hawai‘i at Mānoa.
Date and Time:   Wednesday November 3, 2021, 7:00 - 8:30 PM., on Zoom.
Title: Geometry of computation and string-diagram programming in a monoidal computer.
Abstract: A monoidal computer is a monoidal category with a distinguished type carrying the structure of a single-instruction programming language. The instruction would be written as "run", but it is usually drawn as a string diagram. Equivalently, the monoidal computer structure can be viewed as a typed lambda-calculus without lambda abstraction, even implicit. Any Turing complete programming language, including Turing machines and partial recursive functions, gives rise to a monoidal computer. We have thus added yet another item to the Church-Turing list of models of computation. It differs from other models by its categoricity. While the other Church-Turing models can be programmed to simulate each other in many different ways, and each interprets even itself in infinitely many non-isomorphic ways, the structure of a monoidal computer is unique up to isomorphism. A monoidal category can be a monoidal computer in at most one way, just like it can be closed in at most one way, up to isomorphism. In other words, being a monoidal computer is a property, not structure. Computability is thus a categorical property, like completeness. This opens an alley towards an abstract treatment of parametrized complexity, one-way and trapdoor functions on one hand, and of algorithmic learning in the other.
Speaker: Marco Schorlemmer, Spanish National Research Council.
Date and Time:   Wednesday November 17, 2021, 7:00 - 8:30 PM., on Zoom.
Title: A Uniform Model of Computational Conceptual Blending.
Abstract: We present a mathematical model for the cognitive operation of conceptual blending that aims at being uniform across different representation formalisms, while capturing the relevant structure of this operation. The model takes its inspiration from amalgams as applied in case-based reasoning, but lifts them into category theory so as to follow Joseph Goguen’s intuition for a mathematically precise characterisation of conceptual blending at a representation-independent level of abstraction. We prove that our amalgam-based category-theoretical model of conceptual blending is essentially equivalent to the pushout model in the ordered category of partial maps as put forward by Goguen. But unlike Goguen’s approach, our model is more suitable to capture computational realisations of conceptual blending, and we exemplify this by concretising our model to computational conceptual blends for various representation formalisms and application domains.
Speaker: Robert Geroch, University of Chicago.
Date and Time:   Wednesday December 1, 2021, 7:00 - 8:30 PM., on Zoom.
Title: An Alien's Perspective on Mathematics (and Physics).
Abstract: We describe what might be called a "point of view" toward mathematics. This view touches on such issues as how Godel's theorem might be interpreted, the relevance to physics of mathematical axioms such as the axiom of choice, and the possibility of using physics to "solve" unsolvable mathematical problems.
Speaker: Jens Hemelaer, University of Antwerp.
Date and Time:   Wednesday December 8, 2021, 7:00 - 8:30 PM., on Zoom.
Title: Toposes of presheaves on a monoid and their points.
Abstract: In 2014, Connes and Consani constructed their Arithmetic Site, with as underlying topos the topos of presheaves on the monoid of nonzero natural numbers under multiplication. One of their surprising results is that the points of this topos are classified by a double quotient of the finite adeles, leading immediately to a link with number theory. Inspired by this, we will consider toposes of presheaves on various monoids, and discuss strategies of calculating their points. The most recent strategies (involving for example étale geometric morphisms and complete spreads) are based on joint work with Morgan Rogers.
Speaker: Samantha Jarvis, The CUNY Graduate Center.
Date and Time:   Wednesday December 15, 2021, 7:00 - 8:30 PM., on Zoom.
Title: Language as an Enriched Category.
Abstract: We review enriched category theory, with particular focus on enriching over posets such as [0,1]. We then apply this to natural language, making a language category into an enriched language category as in Bradley-Vlassopoulos-Terilla (our advisor!) [2106.07890.pdf (arxiv.org)]. The statements of enriched category theory have concrete (and interesting!) interpretations when applied to this enriched language category.
Speaker: Todd Trimble, Western Connecticut State University.
Date and Time:   Wednesday December 22, 2021, 7:00 - 8:30 PM., on Zoom.
Title: Categorifying negatives: roadblocks and detours.
Abstract: The challenge of finding meaningful categorified interpretations of "reciprocals" of objects and "negatives" of objects poses some intriguing problems. In this talk, we consider a few responses to this challenge, with particular attention to extending the substitution product on species to "negative species" and "virtual species".
Speaker: Ralph Wojtowicz, Shenandoah University.
Date and Time:   Wednesday February 2, 2022, 7:00 - 8:30 PM., on Zoom.
Title: On Logic-Based Artificial Intelligence and Categorical Logic.
Abstract: The objective of this talk is to reformulate the logic-based artificial intelligence algorithms and examples from the text of Russell and Norvig using the syntax and categorical semantics of Johnstone’s Sketches of an Elephant in order to: (1) identify the fragments of first-order logic required; (2) enable symbolic reasoning about richly-structured semantic objects (e.g., graphs, dynamic systems and objects in categories other than Set); (3) clarify the separation between syntax and semantics; and (4) support use of other category-theoretic infrastructure such as Morita equivalence and transformations between theories and sketches.
Speaker: Emilio Minichiello, CUNY Graduate Center.
Date and Time:   Wednesday February 16, 2022, 7:00 - 8:30 PM., on Zoom.
Title: Category Theory ∩ Differential Geometry.
Abstract: In this talk we will take a tour through some areas of math at the intersection of category theory and differential geometry. We will talk about how the use of category theory works towards solving 2 problems: 1) to give rigorous definitions and techniques to study increasingly complicated objects in differential geometry that are coming from physics, like orbifolds and bundle gerbes, and 2) to find good categories in which to embed the category of finite dimensional smooth manifolds, without losing too much geometric intuition. This involves the study of Lie groupoids, sheaves, diffeological spaces, stacks, and infinity stacks. I will try to motivate the use of these mathematical objects and how they help mathematicians understand differential geometry and expand its scope.
Speaker: David Roberts.
Date and Time:   Wednesday February 23, 2022, 7:00 - 8:30 PM., on Zoom.
Title: Do you have what it takes to use the diagonal argument?
Abstract: Lawvere's reformulation of the diagonal argument captured many instances from the literature in an elegant and abstract category-theoretic treatment. The original version used cartesian closed categories, but gave a nod to how the statement of the argument could be adjusted so as to make fewer demands on the category. In fact the argument, and the fixed-point theorem that Lawvere provided as the positive version of the argument, both require much less than Lawvere stated. This talk will give an outline of Lawvere's version of the diagonal argument, his corresponding fixed-point theorem, and then cover a few versions obtained recently that drop assumptions on the properties/structure of the category at hand.
Speaker: Jin-Cheng Guu, Stony Brook University.
Date and Time:   Wednesday March 16, 2022, 7:00 - 8:30 PM., on Zoom.
Title: Topological Quantum Field Theories from Monoidal Categories.
Abstract: We will introduce the notion of a topological quantum field theory (tqft) and a monoidal category. We will then construct a few (extended) tqfts from monoidal categories, and show how quantum invariants of knots and 3-manifolds were obtained. If time permits, I will discuss (higher) values in (higher) codimensions based on my recent work on categorical center of higher genera (joint with A. Kirillov and Y. H. Tham).
Speaker: Joseph Dimos.
Date and Time:   Wednesday March 23, 2022, 7:00 - 8:30 PM., on Zoom.
Title: Introduction to Fusion Categories and Some Applications.
Abstract: Tensor categories and multi-tensor categories have strong alignment with module categories. We can use the multi-tensor categories C in conjunction with classifying tensor algebras wrt C. From here, we can illustrate some examples of tensor categories: the category Vec of k-vector spaces that gives us a fusion category. This is defined as a category Rep(G) of some finite dimensional k-representations of a group G. From here, I will walk through the correspondence of tensor categories (Etingof) and fusion categories. Throughout, I will indicate a few unitary and non-unitary cases of fusion categories. Those unitary fusion categories are those that admit a uniquely monoidal structure. For example, this draws upon [Jones 1983] for finite index and finite depth that bridges a subfactor A-bimodule B to provide a full subcategory of some category A by its module structure. I will discuss some of these components throughout and explain the Morita equivalence of fusion categories.
Speaker: Morgan Rogers, Universit`a degli Studi dell’Insubria.
Date and Time:   Wednesday March 30, 2022, 7:00 - 8:30 PM., on Zoom.
Title: Toposes of Topological Monoid Actions.
Abstract: Anyone encountering topos theory for the first time will be familiar with the fact that the category of actions of a monoid on sets is a special case of a presheaf topos. It turns out that if we equip the monoid with a topology and consider the subcategory of continuous actions, the result is still a Grothendieck topos. It is possible to characterize such toposes in terms of their points, and along the way extract canonical representing topological monoids, the complete monoids. I'll sketch the trajectory of this story, present some positive and negative results about Morita-equivalence of topological monoids, and explain how one can extract a semi-Galois theory from this set-up.
Speaker: Jason Parker, Brandon University in Manitoba.
Date and Time:   Wednesday April 6, 2022, 7:00 - 8:30 PM., on Zoom.
Title: Enriched structure-semantics adjunctions and monad-theory equivalences for subcategories of arities.
Abstract: Several structure-semantics adjunctions and monad-theory equivalences have been established in category theory. Lawvere (1963) developed a structure-semantics adjunction between Lawvere theories and tractable Set-valued functors, which was subsequently generalized by Linton (1969), while Dubuc (1970) established a structure-semantics adjunction between V-theories and tractable V-valued V-functors for a symmetric monoidal closed category V. It is also well known (and due to Linton) that there is an equivalence between Lawvere theories and finitary monads on Set. Generalizing this result, Lucyshyn-Wright (2016) established a monad-theory equivalence for eleutheric systems of arities in arbitrary closed categories. Building on earlier work by Nishizawa and Power, Bourke and Garner (2019) subsequently proved a general monad-theory equivalence for arbitrary small subcategories of arities in locally presentable enriched categories. However, neither of these equivalences generalizes the other, and there has not yet been a general treatment of enriched structure-semantics adjunctions that specializes to those established by Lawvere, Linton, and Dubuc.
Motivated by these considerations, we develop a general axiomatic framework for studying enriched structure-semantics adjunctions and monad-theory equivalences for subcategories of arities, which generalizes all of the aforementioned results and also provides substantial new examples of relevance for topology and differential geometry. For a subcategory of arities J in a V-category C over a symmetric monoidal closed category V, Linton’s notion of clone generalizes to provide enriched notions of J-theory and J-pretheory, which were also employed by Bourke and Garner (2019). We say that J is amenable if every J-theory admits free algebras, and is strongly amenable if every J-pretheory admits free algebras. If J is amenable, then we obtain an idempotent structure-semantics adjunction between certain J-pretheories and J-tractable V-categories over C, which yields an equivalence between J-theories and J-nervous V-monads on C. If J is strongly amenable, then we also obtain a rich theory of presentations for J-theories and J-nervous V-monads. We show that many previously studied subcategories of arities are (strongly) amenable, from which we recover the aforementioned structure-semantics adjunctions and monad-theory equivalences. We conclude with the result that any small subcategory of arities in a locally bounded closed category is strongly amenable, from which we obtain structure-semantics adjunctions and monad-theory equivalences in (e.g.) many convenient categories of spaces.
Joint work with Rory Lucyshyn-Wright.
Speaker: Alex Martsinkovsky, Northeastern University.
Date and Time:   Wednesday April 13, 2022, 7:00 - 8:30 PM., on Zoom.
Title: A Reflector in Search of a Category.
Abstract: The last several months have seen an explosive growth of
activities centered around the defect of a finitely presented functor.
This notion made its first appearance in M. Auslander's fundamental
work on coherent functors in the mid-1960s, although at that time it
was mostly used just as a technical tool. A phenomenological study of
that concept was initiated by Jeremy Russell in 2016. What transpired
in the recent months is the ubiquitous nature of the defect, explained
in part by the fact that it is adjoint to the Yoneda embedding. Thus any
branch of mathematics, computer science, physics, or any applied
science that references the Yoneda embedding automatically becomes
a candidate for applications of the defect.
In this expository talk I will first give a streamlined introduction to the
original notion of defect of a finitely presented functor defined on a
module category and then show how to generalize it to arbitrary additive
functors. Along the way I will give a dozen or so examples illustrating
various use cases for the defect. The ultimate goal of this lecture is to
provide a background for the upcoming talk of Alex Sorokin, who will
report on his vast generalization of the defect to arbitrary profunctors
enriched in a cosmos.
This presentation is based on joint work in progress with Jeremy Russell.
Speaker: Alex Sorokin, Northeastern University.
Date and Time:   Wednesday April 27, 2022, 7:00 - 8:30 PM., on Zoom.
Title: The defect of a profunctor .
Abstract: In the mid 1960s Auslander introduced a notion of the defect of a finitely presented functor on a module category. In 2021 Martsinkovsky generalized it to arbitrary additive functors. In this talk I will show how to define a defect of any enriched functor with a codomain a cosmos. Under mild assumptions, the covariant (contravariant) defect functor turns out to be a left covariant (right contravariant) adjoint to the covariant (contravariant) Yoneda embedding. Both defects can be defined for any profunctor enriched in a cosmos V. They happen to be adjoints to the embeddings of V-Cat in V-Prof. Moreover, the Isbell duals of a profunctor are completely determined by the profunctor's covariant and contravariant defects. These results are based on applications of the Tensor-Hom-Cotensor adjunctions and the (co)end calculus.
Speaker: Gershom Bazerman, Arista Networks.
Date and Time:   Wednesday May 4, 2022, 7:00 - 8:30 PM., on Zoom.
Title: Classes of Closed Monoidal Functors which Admit Infinite Traversals.
Abstract: In functional programming, functors that are equipped with a traverse
operation can be thought of as data structures which permit an
in-order traversal of their elements. This has been made precise by
the correspondence between traversable functors and finitary
containers (aka polynomial functors). This correspondence was
established in the context of total, necessarily terminating,
functions. However, the Haskell language is non-strict and permits
functions that do not terminate. It has long been observed that
traversals can at times, in practice, operate over infinite lists, for
example in distributing the Reader applicative. We present work in
progress that characterizes when this situation occurs, making use of
the toolkit of guarded recursion.
Speaker: TBA, TBA.
Date and Time:   Wednesday May 11, 2022, 7:00 - 8:30 PM., on Zoom.
Speaker: TBA, TBA.
Date and Time:   Wednesday May 18, 2022, 7:00 - 8:30 PM., on Zoom.