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* categories: The New York City Category Theory Seminar --- Spring 2023 Lineup of speakers.
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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 <> 
Department of Mathematics <>  
The Graduate Center of The City University of New York

Time: Wednesdays 07:00 PM Eastern Time (US and Canada) 

365 Fifth Avenue (at 34th Street) map

(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

Meeting ID: 894 7298 0386 
Passcode: NYCCTS 

Seminar web page.
Videoed talks.
Previous semesters.
<> page.

Contact N. Yanofsky <>  to schedule a
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

*  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

*  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

*  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

*  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

*  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

*  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

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

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

*  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



*  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

*  Title:     Inferring the past and using category theory to define

*  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 and is joint work with Francesco Buscemi. 




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