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 2019

The NYC Category Theory Seminar and NY Haskell Meetup are organizing a

Reading Group on Categorical Logic.

The reading group is run by Gershom Bazerman and Raymond Puzio.
We will be reading:
Introduction to Higher-Order Categorical Logic by J. Lambek and P.J. Scott

We meet on at least two Wednesdays (exact schedule to be determined) of each month from 7:00 to 8:30 PM in Room 6417, CUNY Grad Center, 365 Fifth Avenue (at 34th Street) The first meeting will be on Wednesday, September 18, 2019. This book introduces many fundamental concepts in category theory, beginning with the character of cartesian-closedness.
From there, it develops the connection between category theory and logic via the lambda calculus and higher order type theories.
As such, this material should be of interest to students of computer science and programming language semantics, as well as categorists and logicians.


These reading groups proceed slowly and methodically, and we make sure to cover the basic material as we go, so even those with little-to-no background in category theory should be able to follow along and acquire the necessary tools as we go.

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



  • Speaker:     Gershom Bazerman.

  • Date and Time:     Wednesday September 18, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:    Orientation meeting and starting Introduction to Higher-Order Categorical Logic.

  • Abstract: We will get oriented and start the book from the beginning.



  • Speaker:     Callan McGill.

  • Date and Time:     Wednesday September 25, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:    Higher-Order Categorical Logic: Natural Transformations.

  • Abstract: We will be starting at Section 0.2 and going forward.



  • Speaker:     Jonathan Weinberger, TU Darmstadt.

  • Date and Time:     Thursday October 10, 2019, 7:00 - 8:30 PM., Room 6417. (NOTICE SPECIAL DAY)

  • Title:    Modalities and fibrations for synthetic (∞,1)-categories.

  • Abstract: Higher-dimensional categories play an increasing role in many areas of Mathematics, such as topology, geometry, number theory, and logic. Higher categories are often defined analytically, i.e. entirely based on sets and ordinary categories (endowed with some additional structure), even though these objects themselves do not carry any higher-dimensional information. Riehl and Shulman have introduced a synthetic framework for (∞,1)-category theory where, in contrast, the basic objects intrinsically are of homotopical flavor. One motivation in developing such a synthetic theory is to find a language that is closer to ordinary 1-category theory, and hide much of the (mathematical) implementation of higher-dimensional and homotopical objects. We present a variation of this synthetic theory that allows for additional constructions such as opposite categories and twisted arrow spaces. Our approach is crucially based on Licata--Riley--Shulman's fibrational framework for modal type theories. Furthermore, we suggest a notion of synthetic co-/cartesian fibrations, based on Riehl--Shulman's work on synthetic discrete co-/contravariant fibrations. The topics presented are joint work in progress with Ulrik Buchholtz.



  • Speaker:     Jaime Piedra.

  • Date and Time:     Wednesday October 16, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:    Higher-Order Categorical Logic: Adjoint functors.

  • Abstract: We will be starting at Section 0.3 and going forward.



  • Speaker:     Alex Martsinkovsky, Northeastern University.

  • Date and Time:     Wednesday October 23, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:    Stabilization of additive functors II.

  • Abstract: Over the last couple of years we have seen a growing number of unexpected applications of stabilized functors. This term refers to the passage from an additive functor between abelian categories to a functor vanishing on injectives (or projectives). This concept has been around since M. Auslander’s pioneering work in the mid-1960s, but apparently not much work has been done on it later on. In my previous talk (December 2018) I gave precise definitions and outlined two applications: an extension of the Auslander-Reiten formula and a generalization of Tate homology, both applicable to arbitrary modules over arbitrary rings.

    In the present talk, I will review the definitions and will present yet another application of stabilization. First, we shall redefine and extend the classical torsion over commutative domains. This algebraic concept goes back to Poincaré, who described (and named) it in a topological setting around 1900. In 1959, Bass observed that the kernel of the canonical bidualization map, which we call the Bass torsion, from a finitely generated module over a commutative domain coincides with the classical torsion of the module. Using the injective stabilization of the tensor product, Jeremy Russell and I defined, for arbitrary modules over arbitrary rings, a new torsion radical, which agrees with both the classical torsion over commutative domains and the Bass torsion for finitely presented modules over arbitrary rings. The functorial nature of the new torsion makes it amenable to dualization, yielding, for the first time, a notion of cotorsion, also applicable to arbitrary modules over arbitrary rings. The informal, metamathematical dualization process used to define the cotorsion can be effected by a purely mathematical tool, known as the Auslander-Gruson-Jensen functor.



  • Speaker:     James Myer.

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

  • Title:    Higher-Order Categorical Logic: Equivalence of Categories.

  • Abstract: We will be starting at Section 0.4 and going forward.



  • Speaker:     Dan Shiebler, Oxford University.

  • Date and Time:     Wednesday November 6, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:    Incremental Monoidal Categories for Speech.

  • Abstract: In some systems new information is incrementally introduced. For example, each new word in spoken speech modifies the structure and content of a sentence. Although monoidal categories are a popular foundation for linguistic modeling, they are not natively equipped with structure to model incrementality along the tensor-product dimension. In this work we present a characterization of formal grammars as monoidal categories, which we call monoidal grammars. We also characterize automata that parse formal grammars as F-coalgebras. We use these characterizations to derive a functor from the category of monoidal grammars to the category of F-coalgebras.



  • Speaker:     James Myer.

  • Date and Time:     Wednesday November 13, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:    Higher-Order Categorical Logic: Limits.

  • Abstract: We will be starting at Section 0.5 and go forward.



  • Speaker:     Raymond Puzio.

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

  • Title:    Posets, Lifting properties, and Completions.

  • Abstract: Ever since Dedekind, it has been known that special classes of posets can be characterized in terms of forbidden configurations. We note how these characterizations result from lifting properties which, in turn, correspond to propositions in regular logic. This leads us to consider subcategories of morphisms between posets and idempotent completions.



  • Speaker:     James Myer.

  • Date and Time:     Wednesday November 27, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:    Higher-Order Categorical Logic: Monads.

  • Abstract: We will be starting at 0.6 and go forward.



  • Speaker:     Philipp Rothmaler, The Graduate Center and BCC.

  • Date and Time:     Wednesday December 4, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:    Martsinkovsky-Russell torsion done definably.

  • Abstract: I will show that the torsion radical in question is, in every module, a (usually infinite) sum of first-order definable subgroups (of the additive group) of the module. Moreover, the collection of formulas involved is uniform: it is all the so-called pp (=positive primitive) formulas that vanish in flat modules. Among the consequences are many of the known features of that torsion theory, as well as new ones. Besides, this lays the foundation for other pp definable torsion radicals, which I will discuss time permitting. Note, the formulas in question constitute functors, namely subfunctors of the forgetful functor.



  • Speaker:     Callan McGill.

  • Date and Time:     Wednesday December 11, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:    Higher-Order Categorical Logic: Cartesian Closed Categories and Lambda-Calculus.

  • Abstract: We will be starting at Part 1 and go forward.


  • Speaker:     Gershom Bazerman.

  • Date and Time:     Wednesday January 8, 2020, 7:00 - 8:30 PM., Room 6417.

  • Title:    Higher-Order Categorical Logic: Sections 1.1 and 1.2.

  • Abstract: We will be looking at on deductive systems and the "deduction theorem.



  • Speaker:     David Ellerman, University of California at Riverside.

  • Date and Time:     Monday (NOTE SPECIAL DAY) January 13, 2019, 7:00 - 8:30 PM., Room 6417.

  • Title:    Four New Partition-related Theories.

  • Abstract:

    Category theory gives us the duality between subsets and quotient sets (= partitions = equivalence relations). Certain ‘classical’ theories are based on the subset or subobject side of the duality: Boolean logic of subsets (usually presented as “propositional logic”) and the Birkhoff-von-Neumann quantum logic of subspaces (of a separable Hilbert space). Hence there are two dual theories: 1) Since partitions are dual to subsets, there is a dual logic of partitions, and 2) for vector spaces, direct-sum decompositions are the partitional dual to subspaces, so there is a quantum logic of direct-sum decompositions.

    The quantitative version of Boolean subset logic is finite discrete probability theory, and the quantitative version of partition logic is 3) the logical theory of information—where the Shannon notions of simple, compound, conditional, and mutual entropies are derived by a uniform requantifying transformation from the corresponding natural logical notions of entropy. And the standard quantum information theory notion of von Neumann entropy can be arrived at from Shannon entropy by substituting density matrices for probability distributions and traces for sums. Similarly, the new 4) quantum logical entropy is arrived at from the notion of logical entropy by the same substitutions. Those are the four new partition-related theories that will be sketched in the talk.

     

    References (downloadable from www.ellerman.org) :

    1)      Ellerman, David. 2010. “The Logic of Partitions: Introduction to the Dual of the Logic of Subsets.” Review of Symbolic Logic 3 (2 June): 287–350., or Ellerman, David. 2014. “An Introduction to Partition Logic.” Logic Journal of the IGPL 22 (1): 94–125. https://doi.org/10.1093/jigpal/jzt036. See also Brendan Fong’s MIT Category Theory Seminar talk on partition logic at: https://www.youtube.com/watch?v=5I7v9mvOC2E

    2)      Ellerman, David. 2018. “The Quantum Logic of Direct-Sum Decompositions: The Dual to the Quantum Logic of Subspaces.” Logic Journal of the IGPL 26 (1 (January)): 1–13. https://doi.org/10.1093/jigpal/jzx026.

    3)      Ellerman, David. 2017. “Logical Information Theory: New Foundations for Information Theory.” Logic Journal of the IGPL 25 (5 Oct.): 806–35. https://doi.org/10.1093/jigpal/jzx022.

    4)      Ellerman, David. 2018. “Logical Entropy: Introduction to Classical and Quantum Logical Information Theory.” Entropy 20 (9): Article ID 679. https://doi.org/10.3390/e20090679.

     

     

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

  • Date and Time:     Wednesday January 22, 2020, 7:00 - 8:30 PM., Room 6417.

  • Title:    Higher-Order Categorical Logic: From Section 1.5 and on.

  • Abstract: We will be talking about Polynomial Categories and Functional Completeness of Cartesian Closed Categories.

    Spring 2020



  • Speaker:     LINCOLN'S BIRTHDAY --- NO SEMINAR.

  • Date and Time:     Wednesday February 12, 2020, 7:00 - 8:30 PM., Room 6417.

  • Title:    

  • Abstract:



  • Speaker:     Todd Trimble, Western Connecticut State University.

  • Date and Time:     Wednesday February 19, 2020, 7:00 - 8:30 PM., Room 6417.

  • Title:    The Universal Property of the Bar Construction.

  • Abstract: The bar construction is a fundamental construction used throughout homological algebra and algebraic topology, including for example the construction of classifying bundles, deloopings of suitable H-spaces, and free resolutions of general algebras and the cohomology thereof. The underlying theme is that the bar construction produces canonical contractible or acyclic simplicial algebras, as usually explained by the acyclic models theorem. In this talk we sharpen this result, giving a precise sense in which the bar construction is a universal acyclic simplicial algebra, here recasting "acyclic" not as a property but as an algebraic structure, whereby acyclic structures are coalgebras over the decalage comonad.



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

  • Date and Time:     Wednesday February 26, 2020, 7:00 - 8:30 PM., Room 6417.

  • Title:    Higher-Order Categorical Logic: From Section 1.7 and on.

  • Abstract: We will be talking about Polynomial categories and Kleisli categories of cotriples. We will also talk about coproducts in Cartesian closed categories and natural number objects.



  • Speaker:     Noah Chrein, University of Maryland.

  • Date and Time:     Wednesday March 4, 2020, 7:00 - 8:30 PM., Room 6417.

  • Title:    Hierarchy and Anisotropy in Categorical Ontology.

  • Abstract: The theory of sheaves on a site allows us to break down objects into local pieces and recover data about the global object. We wish to treat systems outside of mathematics in the same way: by breaking down objects into local pieces, analyzing the local pieces, and recombining to get an analysis of the whole. When running a simulation, it's not always relevant to understand the atoms of every object, sometimes it is enough to understand objects abstractly, this is the concept of "anisotropy". We propose a modeling scheme that follows the development of sheaf theory, and adds a notion of hierarchical anisotropy. Namely, instead of a covering in a site, {U_i -> X}, we will treat the U_i and X in two different categories, with "Ontological Expansions" O(X) = {U_i}. In this way, we can decide to treat objects globally, or if we need more specific information, we can expand into local pieces. To this end we define a Hierarchical Ontology.



  • Speaker:     Tai-Danae Bradley, The Graduate Center, CUNY.

  • Date and Time:     Wednesday March 11, 2020, CANCELED!!!

  • Title:    Modeling Probability Distributions as Quantum States.

  • Abstract: This talk features a passage from classical probability to quantum probability. The quantum version of a classical probability distribution is a density operator on a Hilbert space. The quantum version of a marginal probability distribution is a reduced density operator, and the operation that plays the role of marginalization is the partial trace. In particular, every joint probability distribution on a finite set can be modeling as a rank 1 density operator—a pure quantum state. With the partial trace, we recover the classical marginal probabilities, but we also uncover additional information. This extra information can be understood explicitly from the spectral information of the reduced density operators. I’ll describe these ideas and share how they contribute to understanding mathematical structure within natural language.



  • Speaker:     Dan Bornside.

  • Date and Time:     Wednesday March 18, 2020, CANCELED!!!

  • Title:    Higher-Order Categorical Logic: From Section 1. and on.

  • Abstract: We will be talking about natural number objects and typed lambda-calculus.



  • Speaker:     Nicholas Meadows, Haifa University.

  • Date and Time:     Wednesday April 22, 2020, 7:00 - 8:30 PM., Zoom Meeting. Details to follow.

  • Title:    Higher Homotopy Operations in (\infty, 1)-categories.

  • Abstract: Traditionally, higher homotopy operations have three primary applications in homotopy theory: generating elements in the higher homotopy groups of spheres, as an obstruction theory to rectifying homotopy commutative diagrams and describing differentials in the spectral sequence of a (co)simplicial space. In this talk, we will define the spiral spectral sequence (which recovers the classical Bousfield-Friedlander spectral sequence from the E^{2} page on) in an arbitrary simplicial model category and describe the differentials in terms of higher homotopy operations. We will also explain how to represent elements in the filtration of the spiral spectral sequence as higher homotopy operations. Finally, we will sketch how one can define analogous higher homotopy operations in quasi-categories and simplicially enriched categories. .






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