**The New York City**

**Category Theory Seminar**

**
Department of Computer Science**

Department of Mathematics

The Graduate Center of The City University of New York

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

(Diagonally across from the Empire State Building)

New York, NY 10016-4309

Room 6417

ZOOM INFORMATION:

https://us02web.zoom.us/j/82271122572?pwd=NFRhYmswWEdGaWFiazNBSUFiQVJnUT09

Meeting ID: 822 7112 2572

Passcode: NYCCTS

Seminar web page.

Videoed talks.

Previous semesters.

researchseminars.org page.

Contact N. Yanofsky to
schedule a speaker

or to add a name to the
seminar mailing list.

**Fall 2022 **

Speaker: ** Sergei Burkin, University of Tokyo.**

Date and Time: ** Wednesday September 7, 2022, 7:00 - 8:30 PM.**

Title:** Segal conditions and twisted arrow categories of operads. **

Abstract: Several categories, including the simplex category Delta and Moerdijk-Weiss dendroidal category Omega, allow to encode structures (in this case categories and operads reprectively) as Segal presheaves. There are other examples of such categories, which were defined intuitively, by analogy with Delta. We will describe a general construction of categories from operads that produces categories that admit Segal presheaves. This construction explains why these categories appear in homotopy theory, why these allow to encode homotopy coherent structures as simplicial presheaves that satisfy weak Segal condition. Further generalization of this construction to clones shows that these categories are not as canonical as one might have hoped.

Speaker: ** Prakash Panangaden, McGill University.**

Date and Time: ** Wednesday September 14, 2022, 7:00 - 8:30 PM.**

Title:** Quantitative Equational Logic. **

Abstract:

Speaker: ** James Torre CANCELLED!!! **

Date and Time: ** Wednesday September 28, 2022, 7:00 - 8:30 PM.**

Title:** Diagonalization, and the Limits of Limitative Theorems. **

Abstract:

Speaker: ** David Ellerman, University of Ljubljana.**

Date and Time: ** Wednesday October 19, 2022, 7:00 - 8:30 PM.**

Title:** To Interpret Quantum Mechanics:``Follow the Math'': The math of QM as the linearization of the math of partitions. **

Abstract: Set partitions are dual to subsets, so there is a logic of partitions dual to the Boolean logic of subsets. Partitions are the mathematical tool to describe definiteness and indefiniteness, distinctions and distinctions, as well as distinguishability and indistinguishability. There is a semi-algorithmic process or ``Yoga'' of linearization to transform the concepts of partition math into the corresponding vector space concepts. Then it is seen that those vector space concepts, particularly in Hilbert spaces, are the mathematical framework of quantum mechanics. (QM). This shows that those concepts, e.g., distinguishability versus indistinguishability, are the central organizing concepts in QM to describe an underlying reality of objective indefiniteness--as opposed to the classical physics and common sense view of reality as ``definite all the way down'' This approach thus supports what Abner Shimony called the ``Literal Interpretation'' of QM which interprets the formalism literally as describing objective indefiniteness and objective probabilities--as well as being complete in contrast to the other realistic interpretations such as the Bohmian, spontaneous localization, and many world interpretations which embody other variables, other equations, or other worldly ideas.

The underlying paper is forthcoming in the *Foundations of Physics*, and the preprint is in the ArXiv here.

Speaker: ** Ross Street, Macquarie University.**

Date and Time: ** Wednesday October 26, 2022, 7:00 - 8:30 PM.**

Title:** The core groupoid can suffice. **

Abstract: Let V be the monoidal category of modules over a commuative ring R.
I am interested in categories A for which there is a groupoid G such that the
functor categories [A,V] and [G,V] are equivalent.
In particular, G could be the core groupoid of A;
that is, the subcategory with the same objects and with only the invertible morphisms.
Every category A can be regarded as a V-category (that is, an R-linear category),
denoted RA, with the same objects and with hom R-module RA(a,b) free on the
homset A(a,b). Indeed, RA is the free V-category on A so that the V-functor category
[RA,V] is the ordinary functor category [A,V] with the pointwise R-linear structure.
In these terms, we are interested in when RA and RG are Morita equivalent V-categories.
In my joint work with Steve Lack on Dold-Kan-type equivalences, we had many examples
of this phenomenon. However, the example of Nick Kuhn, where A is the category of
finite vector spaces over a fixed finite field F with all F-linear functions and G is the
general linear groupoid over F, does not fit our theory. Yet the ``kernel'' of the equivalence
is of the same type. The present work shows that the category theory behind the Kuhn
result also covers our Dold-Kan-type setting. I plan to start with a baby example which
highlights the ideas.

I am grateful to Nick Kuhn and Ben Steinberg for their patient email correspondence
with me on this topic.

Slides.

Speaker: ** Astra Kolomatskaia, Stony Brook.**

Date and Time: ** Wednesday November 2, 2022, 7:00 - 8:30 PM. IN PERSON TALK. **

Title:** The Objective Metatheory of Simply Typed Lambda Calculus. **

Abstract: Lambda calculus is the language of functions. One reduces the application of a function to an argument by substituting the argument for the function's formal parameter inside of the function's body. The result of such a reduction may have further instances of function application. We can write down expressions, such as ((λ f. f f) (λ f. f f)), in which this process does not terminate. In the presence of types, however, one has a normalisation theorem, which effectively states that "programs can be run". One proof of this theorem, which only works for the most elementary of type theories, is to assign some monotone well-founded invariant to a given reduction algorithm. A much more surprising proof proceeds by constructing the normal form of a term by structural recursion on the term's syntactic representation, without ever performing reduction. Such normalisation algorithms fall under the class of Normalisation by Evaluation. Since the accidental discovery of the first such algorithm, it was clear that NbE had some underlying categorical content, and, in 1995, Altenkirch, Hofmann, and Streicher published the first categorical normalisation proof. Discovering this content requires first asking the question “What is STLC?”, perhaps preceded by the question “What is a type theory?”. In this talk we will lay out the details of Altenkirch's seminal paper and explore conceptual refinements discovered in the process of its formalisation in Cubical Agda.

Slides.

Speaker: ** Andrei Rodin, University of Lorraine (Nancy, France).**

Date and Time: ** Wednesday November 9, 2022, 7:00 - 8:30 PM.**

Title:** Kolmogorov's Calculus of Problems and Homotopy Type theory. **

Abstract: A. N. Kolmogorov in 1932 proposed an original version of mathematical intuitionism where the concept of problem plays a central role, and which differs in its content from the versions of intuitionism developed by A. Heyting and other followers of L. Brouwer. The popular BHK-semantics of Intuitionistic logic follows Heyting's line and conceals the original features of Kolmogorov's logical ideas. Homotopy Type theory (HoTT) implies a formal distinction between sentences and higher-order constructions and thus provides a mathematical argument in favour of Kolmogorov's approach and against Heyting's approach. At the same time HoTT does not support the constructive notion of negation applicable to general problems, which is informally discussed by Kolmogorov in the same context. Formalisation of Kolmogorov-style constructive negation remains an interesting open problem.

Speaker: ** Saeed Salehi, University of Tabriz.**

Date and Time: ** Wednesday November 23, 2022, Zoom Talk SPECIAL TIME 9:30AM-11:00AM.**

Title:** Self-Reference and Diagonalization: their difference and a short history. **

Abstract: What is now called the Diagonal (or the Self-Reference) Lemma, is the statement that for every formula *F*(*x*), with the only free variable *x*, there exists a sentence *σ* such that *σ* is equivalent to the *F* of the Gödel code of *σ*, i.e., *σ* **≡** *F*(**#***σ*); and this equivalence is provable in certain weak arithmetics. This lemma is credited to Gödel (1931), in the special case when *F* is the *un*provability predicate, and to Carnap (1934) in the more general case.

In this talk, we will argue that Gödel-Carnap's original Diagonal Lemma is not the modern formulation and was more similar to, but not exactly identical with, the Strong Diagonal (or Direct Self-Reference) Lemma. This lemma, so-called recently, says that for every formula *F*(*x*), in a sufficiently expressive language, there exists a sentence *σ* such that *σ* is equal to the *F* of the Gödel code of *σ*, i.e., *σ* **=** *F*(**#***σ*); and this equality is provable in sufficiently strong theories. We will attempt at tracking down the first appearance of the modern formulation of the Diagonal Lemma in the equivalent form, also in the strong direct form of equality.

Speaker: ** **

Date and Time: ** Wednesday November 30, 2022, 7:00 - 8:30 PM.**

Title:** TBA. **

Abstract:

Speaker: ** Robert Pare, Dalhousie University.**

Date and Time: ** Wednesday December 7, 2022, 7:00 - 8:30 PM.**

Title:** The horizontal/vertical synergy of double categories. **

Abstract: A double category is a category with two types of arrows, horizontal and vertical, related by double cells. Think of sets with functions and relations as arrows and implications as double cells. The theory is 2-dimensional just like for 2-categories. In fact 2-categories were originally defined as double categories in which all vertical arrows were identities.
Most of the theory of 2-categories extends to double categories resulting in a deeper understanding. This is one aspect of double categories: they’re “new and improved”
2-categories.

From a purely formal point of view, a double category is a category object in CAT. Once a familiarity with double categories has developed, it is amusing and instructive to see how the various constructs of formal category theory play out in this setting.

But these two aspects of double categories, fancy 2-categories or internal categories, are only part of the picture. Perhaps the most important thing is the interplay between the horizontal and the vertical.

I will start with some examples of double categories to give a feeling for the objects I will be discussing, and then look at several concepts indicative of the rich interplay between the horizontal and the vertical.

Speaker: ** Gemma De las Cuevas, University of Innsbruck.**

Date and Time: ** Wednesday December 14, 2022, 7:00 - 8:30 PM.**

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.

**Spring 2023 **

Speaker: ** TBA, TBA.**

Date and Time: ** Wednesday February 8, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

Abstract:

Speaker: ** TBA, TBA.**

Date and Time: ** Wednesday February 22, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

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Speaker: ** TBA, TBA.**

Date and Time: ** Wednesday March 1, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

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Speaker: ** TBA, TBA.**

Date and Time: ** Wednesday March 15, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

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Speaker: ** TBA, TBA.**

Date and Time: ** Wednesday March 29, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

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Speaker: ** TBA, TBA.**

Date and Time: ** Wednesday April 19, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

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Date and Time: ** Wednesday April 26, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

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Date and Time: ** Wednesday May 3, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

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Speaker: ** TBA, TBA.**

Date and Time: ** Wednesday May 10, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

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Speaker: ** TBA, TBA.**

Date and Time: ** Wednesday May 17, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

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Speaker: ** TBA, TBA.**

Date and Time: ** Wednesday May ??, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

Abstract: