The New York City
Category Theory Seminar
Spring 2010

February 22, 2010. Noson S. Yanofsky, Brooklyn College, CUNY
Title: Galois Theory of Algorithms.
Abstract: Many different programs are the implementation of the same
algorithm. This gives us a surjective map from the collection of
programs
to the collection of algorithms. Similarly, there are many different
algorithms
that perform the same function. This gives us a surjective map from
the
collection of algorithms to the collection of computable functions.
Algorithms are intermediate between programs and functions:
Programs  > Algorithms   > Functions.
We investigate how rigid programs are by looking at the group of
automorphism of
programs that preserve functionality. The fundamental theorem of Galois
theory says
that the subgroup lattice of this group is isomorphic to the dual lattice
of intermediate types of algorithms. Only basic category theory is needed
to understand this talk.
March 1, 2010. Joseph Hirsh, The Graduate Center, CUNY
Title: Operads and Homotopy Theory for Algebraic Structures
(Part I).
Abstract: I will begin with a handson definition of Operads in Sets,
Topological Spaces, dgVect, and examples of operads. This will motivate a
handson definition of Operads in an arbitrary symmetricmonoidal category
C. In nice cases, one can construct a symmetric monoidal category of
"collections in C," whose monoids are exactly Operads. I will explain the
relationship between Collections(C) and End(C), the exact statement of
which will make precise the idea that Operads are just "representable"
monads.
March 8, 2010. Joseph Hirsh, The Graduate Center, CUNY
Title: Operads and Homotopy Theory for Algebraic Structures
(Part II).
Abstract: For an operad O, I will define a category of Oalgebra objects
in C. For an Oalgebra object A in C, we define a category of Amodule
objects in C. We will discuss some nice properties that these categories
have, and some nice properties that the forgetful functor OAlg(C) > C
has. I will then discuss, without too much technical detail, how a
"homotopy theory" on the underlying category C induces a homotopy theory
on the category of Oalgebra objects in C. I will try to illustrate that
the functor OAlg(C) > C is not particularly well behaved with respect
to this homotopy theory. This motivates us to develop a package of
theorems we wish were true about the functor but are not.
March 15, 2010. Joseph Hirsh, The Graduate Center, CUNY
Title: Operads and Homotopy Theory for Algebraic Structures
(Part III).
Abstract: Finally, I will choose a particular symmetric monoidal category
C with a homotopy theory: dgVect. I will describe a class of Operads
{O_{\infty} } for which the package of theorems about
O_{\infty}Alg(C)> C are true, and describe how one may pass from any
operad O to an operad O_{\infty}, so that O_{\infty} is an operad for
which our package of theorems are true, while OAlg(C) >
O_{\infty}Alg(C) is an equivalence of homotopy theories.
May 17, 2010. Emily Riehl,
University of Chicago
Title: Natural weak factorization systems in model structures
Abstract: Factorization systems abound in mathematics: orthogonal
factorization systems (ofs) have nice categorical properties but are too
strict to describe many homotopical situations, while weak factorization
systems (wfs) do figure prominently, especially in the definition of a
model structure on a category, but aren't as well behaved. Natural weak
factorization systems (nwfs) "algebraicize" the notion of wfs and sit
somewhere in between the two, and thanks to Richard Garner's small object
argument, it's possible to construct a plethora of examples.
My work explores the consequences of incorporating nwfs into model
categories: a natural model structure will consist of a pair of nwfs
together with a comparison map between them such that the underlying
classes of maps form a model structure in the ordinary sense. After giving
the basic definitions and exploring some of the properties of nwfs, I'll
devote the remainder of the talk to persuading my audience that this was a
good idea. The most significant results build toward an algebraization of
the notion of a Quillen adjunction, each of which include five adjunctions
of nwfs. In addition to supplying the definition, I am able to show that
this extra algebraic structure can be found for a large family of known
examples of Quillen adjunctions.