[Alta-Logic] working seminar

[Pieter Hofstra]

Dear all,

During the fall semester, the peripatetic seminar has been filled with a
series of talks by Kristine Bauer and myself on Quillen Model Categories
and abstract homotopy theory. Most of the participants have indicated an
interest in pursuing this topic in some further detail, possibly by
concentrating on one or more modern lines of research. 

Since we do not wish to hold the peripatetic seminar hostage for another
semester, we will have this research seminar independently of the
peripatetic seminar. 

This leaves us with two concrete problems. First, we need to find a time
slot for the working seminar. I would like to ask everyone who is
planning to attend the seminar to indicate preferences and/or
obstructions.
Second, we need to decide on a topic for the seminar. Two possible
projects are described below, but we welcome suggestions for topics
and/or concrete problems. 

Hoping to hear your opinion on these matters,

Pieter Hofstra


Project 1: Quillen homology of operadic categories

The goal of this project would be to understand the Quillen homology of 
categories of algebras over operads.  I envision this as first being a 
discussion of operads and algebras over operads, as well as classical 
examples (like Stasheff's associahedron and loop spaces over the little 
cube operads).  From there, we would need to discover what the Quillen 
model category structure is for the category of algebras over a given 
operad (I believe this is known and at least in some cases is an easy 
extension of the model structure of the ground category).  We would
need 
to define and understand what Andre-Quillen homology is, and try to 
understand some of the classical examples (e.g. in the category of
dga's).  
Finally, we would need to compute the Andre-Quillen homology for
algebras 
over an operad (this may be new, I am checking references) and try to 
understand it's context - for example, does it recover anything
classical 
in special cases?

Project 2: Quillen model structures and cellular categories. (Program by
A. Joyal)

Cellular sets are a generalization of simplicial sets, and constitute
one of the many possible settings in which one can define higher
categories. Just like quasi-categories are simplicial sets satisfying a
restricted Kan condition, one can define higher categories as cellular
sets satisfying a similar horn-filler condition. 

The first step for us would be to cover the basic theory quasi-
categories. For every category X, the nerve of X is a quasi-category. A
large part of basic category theory can be generalized to the level of
quasi-categories. There are two important results to understand: first,
there is a model structure on simplicial sets for which the quasi-
categories are exactly the fibrant objects. Second, a quasi-category C
is a Kan complex if and only if the homotopy category of C is a
groupoid.

Next, we would go over cellular sets, which relate to simplicial sets in
the same way as disks relate to intervals. Then we study the definition
of higher (cellular) categories as cellular sets subject to a particular
horn-filler requirement. 

Some interesting problems are: can one find a Quillen Model Structure on
the category of cellular n-categories? Does the collection of cellular
n-categories form a cellular n+1-category? Is there an inductive
definition of cellular n-categories making use of enrichment?