25 April 2006

1:00 - 2:30  S Niefield 

Exponentiability in Lax Slices of Top

Abstract:

Suppose \le is a partial order on a topological space B. The lax slice
Top//B is the category whose objects are continuous maps p:X-->B and
morphisms are triangles which commute up to \le. It is easy to see that
Top//B has binary products which are preserved by the forgetful functor to
Top if and only if (B,\le) is a topological meet-semilattice.

When \le is the specializataion order, Top//B can be thought of as a
category of B-indexed diagrams of open sets. For example, if 2 is the
Sierpinski, then Top//2 is isomorphic to the category whose objects are
pairs (X,U) with U open in X, and morphisms (X,U)-->(Y,V) are continuous
maps f:X-->Y such that f(U) is a subset of V. For the opposite of the
specialization order, we get B-indexed diagrams of closed sets.

In this talk, we present characterizations of exponentiable objects in
Top//B for these two natural orders on B.