The New York City

Category Theory Seminar

Department of Computer Science
Department of Mathematics
The Graduate Center of The City University of New York
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.
Room 6417 .

Some of the talks are videoed and available here.

Contact N. Yanofsky to schedule a speaker
or to add a name to the seminar mailing list.

Fall 2018

The New York Category Theory Seminar and the New York Haskell Meetup is sponsoring a

Topos Theory Reading Group

We will be reading

Sheaves in Geometry and Logic: A First Introduction to Topos Theory

by Saunders MacLane, Ieke Moerdijk

The list for announcements and scheduling regarding this group is also at the hott-nyc google group:!forum/hott-nyc

We meet the first and third Wednesdays of each month from 7:00 to 8:30 PM in Room 6417. The meetings are run by Gershom Bazerman.

  • Speaker:     Christoph Dorn, Oxford University.

  • Date and Time:     Wednesday September 5, 2018, 7:00 - 8:30 PM., Room 6417.

  • Title:    Associative n-categories.

  • Abstract: What is a ``natural setting'' for higher category theory? In the past decades, a guiding intuition to answer this question has been provided by analogy with the homotopy hypothesis, which says that spaces are a ``natural setting'' for higher groupoids. More precisely, the analogy suggests that we can think of the building blocks of higher categories as ``directed'' n-cells and their compositions. This intuition is reflected in most approaches to higher categories. In this talk we show that n-fold categories indicate a more general point of view. N-fold categories have a geometric calculus which specializes to a geometric calculus for n-categories called calculus of ``manifold diagrams''. By a combinatorial analysis of manifolds diagrams (of which the common string diagrams are a special case) we will built a new notion of higher categories: associative n-categories. These semi-strict higher categories have powerful features: (A) They have a geometric model in manifold diagrams (B) They are equivalent to a fully weak model of higher categories, giving evidence to a (strengthened!) version of Simpson's conjecture (C) They have a formulation as categories presented by generators. In particular the latter is a powerful and novel tool, for instance, allowing us to compute with finitely generated spaces or to implement higher categories on a computer as higher inductive types.

  • Speaker:     Phillip Bressie, Kansas State University.

  • Date and Time:     Wednesday October 24, 2018, 7:00 - 8:30 PM., Room 6417.

  • Title:    On Tautological Globular Operads.

  • Abstract: I will begin by giving a nonstandard description of the construction of the classical tautological, or endomorphism, operad taut(X) on a set X. Then I shall describe how globular operads are a strict generalization of classical operads. From this perspective a description will be given of the analogous construction for the tautological globular operad Taut(X) on a globular set X by way of describing the internal hom functor for the monoidal category Col, of collections and collection homomorphisms, with respect to the monoidal composition tensor product used to define globular operads.
  • Slides of the talk.

  • Speaker:     André Joyal, Université du Québec à Montréal

  • Date and Time:     Friday October 26, 2018, 7:00 - 8:30 PM., Room 4102 The Science Center. (NOTICE THE SPECIAL DAY AND ROOM)

  • Title:    Homotopy type theory: a new bridge between logic, category theory and topology.

  • Résumé: Martin-Lôf type theory is a language for constructive mathematics with applications to program verifications and proof assistants. The homotopy interpretation of Martin-Lôf type theory, discovered by Voevodsky and independantly by Awodey and Warren, has initiated a project for a new foundation of constructive mathematics and of homotopy theory. An elementary notion of higher topos is emerging in the process. We shall describe the various connections involved.
  • Slides of the talk.

  • EVENT:    Category Theory OctoberFest 2018.

  • DATE AND TIME:     Saturday October 27, 2018, 9:00 AM - 5:00 PM.

  • DATE AND TIME:     Sunday October 28, 2018, 9:00 AM - 1:00 PM.

  • PLACE:    304 Shepard Hall; City College, CUNY; 160 Convent Ave; New York, NY 10031.


  • Speaker:     Ieke Moerdijk, University of Utrecht.

  • Date and Time:     Wednesday October 31, 2018, 7:00 - 8:30 PM., Room 6417.

  • Title:    The topos of dendroidal sets.

  • Abstract: The topos of simplicial sets sits as an open subtopos inside the larger topos of dendroidal sets. These two toposes are quite similar in some respects, but completely different in others. For example, dendroidal sets carry a somewhat mysterious tensor product, which coincides with the ordinary cartesian product when applied to simplicial sets. While simplicial sets generalize small categories through the construction of the nerve, dendroidal sets generalize operads in a similar way. There are various ways in which this can be expressed, some of them quite refined in terms of Quillen model categories.

  • Speaker:     John Connor, The Graduate Center, CUNY.

  • Date and Time:     Wednesday November 14, 2018, 7:00 - 8:30 PM., Room 6417.

  • Title:    Intuitionistic Epistemic Logic and Propositional Truncation in the Type Theory.

  • Abstract: Intuitionistic Epistemic Logic (IEL) is an extension of Intuitionistic Propositional Logic introduced by Artemov and Protopopescu. IEL introduces a co-reflexive modal axiom P ⇒ (K P), where the intended interpretation of (K P) is that P is known, but a proof of P is not necessarily at hand.

    In this talk I introduce an extension of the simply typed λ-calculus which is equivalent to IEL via an extension of the Curry-Howard correspondence. The categorical semantics of the type theory are interesting in that the interpretation of the modality in a syntactic category forms part of a non-idempotent monad. I will then show that by introducing a judgmental equality we transform the monad into an idempotent monad in a quotient category, and that the new monad corresponds to propositional truncation in the new calculus.

  • Speaker:     Micah Miller, Borough of Manhattan Community College, CUNY.

  • Date and Time:     Wednesday November 28, 2018, 7:00 - 8:30 PM., Room 6417.

  • Title:    Primer on Homotopy Limits.

  • Abstract: We go over the definition of a homotopy limit of a diagram in a simplicial model category M and some elementary computations. When we have a cosimplicial object in M, we can define its totalization. The Dwyer-Kan map is a map from the totalization to the homotopy limit. Under certain conditions, this map is a natural weak equivalence. Since the totalization is easier to compute than the homotopy limit, this gives us another way to calculate homotopy limits.

  • Speaker:     Alex Martsinkovsky, Northeastern University.

  • Date and Time:     Wednesday December 5, 2018, 7:00 - 8:30 PM., Room 6417.

  • Title:    Stabilization of additive functors.

  • Abstract: In their classical treatise on homological algebra, Cartan and Eilenberg make a claim that ``... the right derived functors are of real interest only if [the functor being derived] is left exact. … Similarly the left derived functors are mainly interesting for functors which are right exact.'' The goal of this talk is to show that this may not necessarily be true, even for the tensor product and the Hom functors. The formalism of zeroth derived functors leads to instantaneous proofs of the theorems of Eilenberg and Watts. Examining the canonical transformations between an additive functor and its zeroth derived functors, one is lead to the notions of injective or projective stabilization of the functor, originally introduced by Auslander and Bridger. This provides an extension of the classical Auslander-Reiten formula to arbitrary modules over arbitrary rings. An appropriately defined iteration of injective stabilization leads to an extension of Tate homology, which is viewed as a homological counterpart of Buchweitz's construction of Tate cohomology. Time permitting, I will mention an intriguing parallelism between this construction and the comparison map from Steenrod-Sitnikov homology to Čech homology.

    This will be an expository talk. All terms will be defined and explained. This is joint work with Jeremy Russell.
  • Slides of the talk.

    Spring 2019

  • Speaker:     Noson S. Yanofsky, Brooklyn College, CUNY.

  • Date and Time:     Wednesday March 20, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:    Quantum Algorithms.

  • Abstract: Contrary to what you probably heard, quantum computers are not faster than classical computers. The power of quantum computing is that quantum algorithms demand fewer operations than classical algorithms. We will introduce some quantum mechanics, quantum computing, and then discuss several quantum algorithms. We move on to discuss the seeming paucity of quantum algorithms and conclude with some speculations about the future of quantum computing.

    This talk is given in conjunction with the Women in Quantum Meetup (NYC). No category theory or quantum mechanics is needed for this talk. The only prerequisite is the ability to multiply two matrices with complex entries.

  • Speaker:     Eoin Moore, The Graduate Center, CUNY.

  • Date and Time:     Wednesday April 3, 2019, 7:45 - 9:00 PM., Room 6417. (NOTICE SPECIAL TIME.)

  • Title:    The Arithmetical Completeness and Soundness of the Logic of Proofs.

  • Abstract: BHK semantics were introduced to provide a semantics of intuitionistic logic (Int) whereby the truth of a proposition was demonstrated by exhibiting a proof of it. This semi-rigorous approach had some serious difficulties with its exact formalization. The major hurdle of formalization was completed with Artemov's Logic of Proofs (LP), which provided the desired provability semantics. Int was embedded into LP, which was then embedded into Peano Arithmetic (PA). In this talk I will discuss the second inclusion. I will show that there is a complete and sound interpretation of LP into PA, in which LP proof terms are mapped to PA provability formulas.

  • Speaker:     Tibor Beke, University of Massachusetts, Lowell.

  • Date and Time:     Wednesday April 17, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:    Schanuel functors and the Grothendieck (semi)ring of some theories.

  • Abstract: In a little known article, Schanuel defines a functor from semirings to idempotent semirings and a notion of dimension that is not linearly ordered. He uses it to give an elegant presentation of the Grothendieck semiring of semi-algebraic sets, from which the (much better known) structure of the Grothendieck ring of semi-algebraic sets easily follows. I will review his work and related results on the Grothendieck semiring of algebraically closed fields and similar geometric structures.

  • Speaker:     Andrei Rodin, Senior Researcher at the Institute of Philosophy of Russian Academy of Sciences.

  • Date and Time:     Wednesday April 24, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:     Directed Homotopy Type Theory and the (In)vertibility of Mathematics.

  • Abstract: Directed Homotopy Type theory (DHTT) is a generalization of Homotopy Type theory (HoTT) where fundamental groupoids of spaces are replaced by more general (higher) categories. Along with type formers for identity types which admit the standard HoTT interpretation in terms of invertible paths and their homotopies, DHTT comprises type formers for non-invertable homomorphisms of all levels which admit an interpreation in terms of non-invertable paths in appropriate spaces. The choice between DHTT and HoTT as foundational formal frameworks for building mathematical theories has an epistemological dimension, which concerns the epistemic significance of the invertibility condition. While HoTT and the related notion of Univalent Foundations support Mathematical Structuralism DHTT supports a more dynamic conception of Mathematics, which I shall outline in my talk.

    Related papers:

    Paige North, Towards a Directed Homotopy Type Theory, arXiv:1807.10566

    Andrei Rodin, Categories Without Structures, arXiv: 0907.5143 (published in Philosophia Mathematica 19/1 (2011), p. 20-46)

    Michael Warren, Directed Type Theory (video of talk in IAS Princeton)

  • Speaker:     Jonathan Funk, Queensborough Community College.

  • Date and Time:     Wednesday May 8, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:     Isotropy theory meets Galois theory.

  • Abstract: Isotropy theory for toposes is about internal symmetry of a topos. A topos may have trivial isotropy, said to be anisotropic. For example, a localic topos is anisotropic. The isotropy of a topos may be cancelled to yield what we call the isotropy quotient of a topos, although the quotient may itself have isotropy, or what we call higher isotropy of the given topos. (By analogy, the quotient of a group by its center may itself have non-trivial center.) Let us say that a topos is locally anisotropic if it has an etale cover by an anisotropic topos.

    THEOREM: A locally anisotropic topos has no higher isotropy. Equivalently, its isotropy quotient is anisotropic. Furthermore, a locally anisotropic topos is recovered as the topos of actions for a connected groupoid internal to its isotropy quotient.

    COROLLARY: An etendue, or locally localic topos, has no higher isotropy. An etendue may be recovered as the topos of actions for a connected groupoid internal to its isotropy quotient.

    Our argumentation of the theorem brings into focus how isotropy theory and Galois theory for toposes meet in a natural and evidently effective way.

    Joint work with Pieter Hofstra.

  • Speaker:     Sergei Artemov, The Graduate Center, CUNY.

  • Date and Time:     Wednesday May 15, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:     On the Provability of Consistency.

  • Abstract: We revisit the foundational question concerning Peano arithmetic PA:

    (1) can consistency of PA be established by means expressible in PA?

    The usual answer to (1) is “No, by Gödel’s Second Incompleteness Theorem.” In that theorem (G2), Gödel used an arithmetization of contentual mathematical reasoning and established that the arithmetical formula representing PA-consistency is not derivable in PA. Applying G2 to (1), one makes use of the formalization thesis (FT):

    FT: any proof by means expressible in PA admits Gödel’s arithmetization.

    Historically, there has been no consensus on FT; Gödel (1931) and Hilbert (1934) argued against an even weaker version of FT with respect to finitary proofs, whereas von Neumann accepted it.

    Note that the aforementioned negative answer to (1) is unwarranted: here is a counter-example to FT. Let Ind(F) denote the induction statement for an arithmetical formula F. The claim C, “for each formula F, Ind(F),” is directly provable by means of PA: given any F, argue by induction to establish Ind(F). However, C is not supported by any arithmetization as a single formula since PA is not finitely axiomatizable.

    We provide a positive answer to (1). We offer a mathematical proof of PA-consistency,

    No finite sequence of formulas is a PA-proof of 0=1,

    by means expressible in PA, namely, by partial truth definitions. Naturally, this proof does not admit Gödel’s arithmetization either.