Series: Mathematics Colloquium

Date: Thursday, October 7, 2004

Time: 4:00 - 5:00 PM

Place: 201 Thomas Building

Host: Stephen Simpson

Refreshments: 3:15 - 4:00 PM, 321 Whitmore

Speaker: 

  Dana S. Scott, Carnegie Mellon University, 
  Mathematics - Philosophy - Computer Science

Title: 
  
  Topology, Categories, and lambda-Calculus

Abstract:

  Many categories have injective objects, but their properties depend
  on what families of subobjects are allowed.  In the case of
  topological T_0-spaces, for example, two alternatives suggest
  themselves: (1) arbitrary subspaces, and (2) dense subspaces.  Both
  notions are interesting.  Thus, a space D is injective in sense (1)
  iff for any space Y and subspace X and any continuous function
  f:X-->D there is a continuous extension f':Y-->D of f.  Injective
  spaces are easily proved to be closed under arbitrary products and
  continuous retracts, which facts provide many examples once a few
  such spaces are known.  Perhaps it is not so obvious, however, that
  injective spaces are also closed under the formation of function
  spaces, once the space of continuous functions is given the right
  topology; indeed the category of injective spaces and continuous
  functions is a cartesian closed category.  The talk will review old
  and new results about injective spaces, applications of the results,
  and a recent use of injectives to define a cartesian closed
  extension of the category of all T_0-spaces.  This topological point
  of view makes it obvious that (untyped) lambda-calculus models exist
  in many forms.