Generalized Homotopy Theories

**Abstract**

If you have a monodial closed category, a notion of "the object of all
path components" and an object you are willing to call a circle, you
can define the homotopy objects π_{n}(*X*). Mike
Barr's paper "Building Closed Categories" helps us construct useful
examples. In general, there will be a "generic path" and in one of our
examples the maps from the generic path to itself are all given by
polynomials. Mike Barr, Bob Raphael and I are just starting this work
and the talk will mainly explore some examples.