The New York City
Category Theory Seminar
Spring 2009
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February 9, 2009. Noson S. Yanofsky, Brooklyn College, CUNY
Title: On the algorithmic informational content of categories.
Abstract: With Kolmogorov complexity theory, researchers define the
algorithmic informational content of a string as the length of the
shortest program/Turing machine that describes the string. After a brief
review of some features of Kolmogorov complexity theory we present the
rudimentary beginnings of a programming language that can be used to
describe categories, functors and natural transformations. With this in
hand we define the informational content of these categorical notions as
the shortest such program. Some basic consequences of our definition are
presented and we show that our definition is a generalization of
Kolmogorov complexity theory of strings.
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February 16, 2009. President's Day. No Seminar.
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February 23, 2009. Alex Hoffnung, University of California,
Riverside
Title:A categorification of Hecke algebras.
Abstract: Given a Dynkin diagram and the finite field F_q, where
q is a prime power, we get a finite algebraic group G_q.
We will show how to construct a categorification of the Hecke
algebra H(G_q) associated to this data. This is an example
of the Baez/Dolan program of ``Groupoidification'', a method of
promoting vector spaces to groupoids and linear operators to
spans of groupoids. For example, given the A_2 Dynkin diagram,
for which G_q = SL(3,q), the spans over the G_q-set of
complete flags in F_q^3 encode the relations of the Hecke
algebra associated to SL(3,q). Further, we will see how the
categorified Yang-Baxter equation is derived from incidence
relations in projective plane geometry.
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March 2, 2009. Dustin Mulcahey, The Graduate Center, CUNY
Title: Spectral sequences for a working category theorist.
Abstract: The relation between cosimplicial objects, completions, homotopy
inverse limits, and homotopy spectral sequences was first touched upon in
Bousfield and Kan's 'Yellow Monster' ("Homotopy limits, completions and
localizations") and subsequently generalized to arbitrary simplicial model
categories satisfying certain assumptions by Bousfield in his 2003 paper,
"Cosimplicial resolutions and homotopy spectral sequences in model
categories".
This machinery is part of a growing toolbox for tackling problems in
unstable homotopy theory. I will attempt to give an overview, only
supposing some basic knowledge about model categories. Given time, I
would also like to talk about various applications, including Haynes
Miller's proof of the Sullivan conjecture, Bendersky-Curtis-Miller's
Unstable Novikov Spectral Sequence, and Bendersky-Thompson's unstable
K-theory completion of the spheres.
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March 16, 2009. Louis Thrall, The Graduate Center, CUNY
Title: Applications of Braided Tensor Categories (Part 1)
Abstract: In this talk, we will briefly discuss the definition of Braided
Tensor Categories (BTC). We will then talk about the relationship between
BTC's and Braids. After this we will discuss two equivalent definitions of
Quasi Triangular Quasi Hopf Algebras. One is an algebra with a coalgebra
structure satisfying some equations. The other definition by demanding
that
the category of representations of this algebra forms a BTC. Then we will
talk about groups of automorphisms of BTC's called Grothendieck
Teichmuller groups. Lastly we will relate these Grothendieck Teichmuller
groups to Galois Theory.
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March 23, 2009. Florian Lengyel, The Graduate Center, CUNY
Title: The Curry-Howard-Lambek isomorphism for CCCs with LP
structure.
Abstract: A generalization of the Curry-Howard-Lambek isomorphism for
Cartesian
closed categories and typed lambda calculi is given for the LP
categories, which correspond to the positive conjunction fragment of the
intuitionistic Logic of Proofs (LP) of Artemov, and LP-typed lambda
calculi. LP categories are obtained from LP deductive systems, which are
deductive systems with new types called proof terms, along with arrows
that internalize operations on arrows. Such systems can internalize
their own proofs. LP-typed lambda calculi are defined in the expected
way. The categorical equivalence of CCCs and typed lambda calculi
strictly generalizes to the new setting.
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March 30, 2009. Louis Thrall, The Graduate Center, CUNY
Title: Applications of Braided Tensor Categories (Part 2)
Abstract: In this talk, we will briefly discuss the definition of Braided
Tensor Categories (BTC). We will then talk about the relationship between
BTC's and Braids. After this we will discuss two equivalent definitions of
Quasi Triangular Quasi Hopf Algebras. One is an algebra with a coalgebra
structure satisfying some equations. The other definition by demanding
that
the category of representations of this algebra forms a BTC. Then we will
talk about groups of automorphisms of BTC's called Grothendieck
Teichmuller groups. Lastly we will relate these Grothendieck Teichmuller
groups to Galois Theory.
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May 11, 2009. Andre Rodin, Paris, ENS
Title: Categories without Structures
Abstract: MacLane, Awodey and some others have argued that category theory
(CT) provides a strong support for Mathematical Structuralism, while
Hellman has argued that CT is unable to provide an autonomous
structuralist framework adequate to the needs of mathematics. In my talk I
shall briefly overview arguments of both parties of this controversy and
then argue that CT suggests a new understanding of the subject-matter of
mathematics, which doesn't reduce to Structuralism or to any other earlier
established view. While Structuralism, by Awodey's word, conceives of the
subject-matter of mathematics as an "invariant form" Category theory
studies mutual relative variations of mathematical constructions, which
generally have no absolute invariants. Using a physical language one can
say that in the new framework the notion of invariance is replaced by a
more general notion of covariance (i.e. functoriality). Like Structuralism
this new categorical view has a bearing on further philosophical issues
about mathematics, some of which I shall try to explore. I shall try to
make clear the shift from Structuralism to the new
viewpoint through a
thorough analysis of Lawvere's paper of 1965 The Category of Categories as
a Foundation for Mathematics. I shall particularly stress the changing
notion of axiomatic method in this context.
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