The New York City
Category Theory Seminar
Fall 2009

October 19, 2009. Fred E.J. Linton, Wesleyan University
Title: Something Old, Something New, Something Borrowed, ...
Abstract: I talk about a peculiar topological space that Peter Johnstone
and Bob Paré and I used some 30 years ago to solve a problem in
topos
theory.
Remarks on selected words or phrases above:
"Peculiar"? it has a discrete, dense, open subset, whose (closed)
complement is a discrete subspace as well,
yet ...
"Problem in topos theory"? ... the whole space admits a double cover with
no continuous global section;
and
"Some 30 years ago"? offered in the Proceedings of a 1978 Berlin
categories gettogether.
Remarks on the choice of title:
"old"? as said, the work cited dates back to 1978.
"new"? the main proof offered here is a new cardinalitycounting
argument,
rather than the Baire category reasoning of 30odd years ago.
"borrowed"? it's the base space that is borrowed:
that's Heath's Vspace, and is rather much older than 30 years.
Seemed like the sort of mix of areas seminar participants might
appreciate.
And the pictures may be a pleasant diversion, too.

October 26, 2009. Louis Thrall, The Graduate Center, CUNY
Title: Category theory, Combinatorial Homotopy Theory and Coverings of
"Spaces".
Abstract: I will begin by briefly reviewing the definition of Cech
coverings of a space and the associated nerve. There is then a theorem
that states that the geometric realization of the nerve and the original
space are weakly equivalent. This may then be generalized to Grothendieck
topologies, if one uses hypercovers. Some immediate applications to
algebraic geometry can then be stated. In the course of this talk, we will
flesh out what all of this means. We end with some concrete
examples of this in practice: models of the S^2 and CP^n will be shown.

November 2, 2009. Louis Thrall, The Graduate Center, CUNY
Title: Category theory, Combinatorial Homotopy Theory and Coverings of
"Spaces" Part II.
Abstract: I will begin by briefly reviewing the definition of Cech
coverings of a space and the associated nerve. There is then a theorem
that states that the geometric realization of the nerve and the original
space are weakly equivalent. This may then be generalized to Grothendieck
topologies, if one uses hypercovers. Some immediate applications to
algebraic geometry can then be stated. In the course of this talk, we will
flesh out what all of this means. We end with some concrete
examples of this in practice: models of the S^2 and CP^n will be shown.

November 23, 2009. Alex Hoffnung, University of California,
Riverside
Title: The categorified Hecke algebra and the Zamolodchikov
tetrahedron equation
Abstract: We will look at a categorification of the Hecke algebra and its
connections to incidence geometries and buildings. Further, we will see
how this construction gives rise to solutions of the Zamolodchikov
tetrahedron equation.

December 7, 2009. Dustin Mulcahey, The Graduate Center, CUNY
Title: Towards a Change of Comonads Theorem
Abstract: The issue of computing derived functors with respect to some
comonad is a central one in unstable homotopy theory. In particular, the
E2 term of any variant of the unstable Adams spectral sequence comes about
this way. Here, I present the following question, "When do two comonads
(over possibly different categories) yield the same derived functors?" I
will go over some progress in answering this question. The discussion
will be mostly categorical in nature, though the ultimate application is
in unstable BP theory.

December 14, 2009. Rick Jardine, The University of Western Ontario
Title: Cocycle categories
Abstract: The set of maps between objects in a homotopy category can be
identified with path components of a category of cocycles, in great
generality. The applications to date have been predominantly consisted of
homotopy classification results in nonabelian cohomology theory.
Cocycle category methods lead to an expanded (and interesting) version of
the Verdier hypercovering theorem. Cocycle categories might also be used
to give a new description of etale homotopy theory, which would involve
neither hypercovers nor proobjects.
Rick will also be speaking on Concurrency on Thursday December 10th, 2009
(4:155:30) at the
The Computer Science Colloquium