Tuesday, 26 March 2002
2:30 - 4:00  Susan Niefield (Union College)

Homotopy Pullbacks, Lax Pullbacks, and Exponentiability

We consider a unified approach to homotopy pullbacks, lax pullbacks, and
pseudo-pullbacks, introduced completely in the 1-dimensional realm of the
categories in question.   This is done via the notion of an E-pullback,
where E is an object of a category {\cal E} with finite limits.  It turns
out that E-pullbacks of objects over B exist in {\cal E} if and only if
the exponentials B^E exist in {\cal E}.

The goal of the talk is to discuss E-exponentiable morphisms of {\cal E},
i.e., morphisms for which a certain E-pullback functor has a right
adjoint.  In particular, we will consider the relationship between
E-exponentiability and ordinary exponentiability of morphisms of {\cal
E}, and apply these results to the categories of topological spaces, small
categories, and Grothendieck toposes.