20 June 2006 2:30 - 4:00 M Barr Some topological *-autonomous categories Theorem: Suppose \Asc is an additive equational category and \Tsc is the category of topological \Asc objects. Suppose \Ssc_0 is a full subcategory closed under finite products and closed subobjects and contains an object K with the following properties: 1. K is injective in \Ssc_0; 2. K contains a neighbourhood U that a. contains no non-zero subobject; b. for any T in Ob(\Tsc) a map T --> K is continuous iff the inverse image of U is open; 3. K is complete in its natural uniformity. Let \Ssc be the closure of \Ssc_0 under products and subobjects. Then every object of \Ssc generates a new object with the same underlying object of \Asc and the same set of maps to K and that has the coarsest possible topology with that property as well as another with the finest possible topology. The two subcategories so defined are equivalent to each other as well as to chu(\Asc,K) and are thus *-autonomous. Here are some examples: |-------------------------------------------------| | \Asc | K | \Ssc_0 | |======================|====|=====================| |real vector spaces | R | normed spaces | |----------------------|----|---------------------| |complex vector spaces | C | normed spaces | |----------------------|----|---------------------| |K-vector spaces | K | linearly discrete | |(K locally compact) | | vector spaces | |----------------------|----|---------------------| |abelian groups |R/Z | locally compact | | | | spaces | |-------------------------------------------------| Linearly discrete vector spaces are just sums, in the category \Tsc, of copies of the field. Although the real and complex numbers are locally compact, the first two examples are distinct from the third because the choice of \Ssc_0 is different. It is known that the locally compact fields are: discrete fields, R, C, finite algebraic extensions of the p-adic completions of Q, and finite extensions of fields of the fields of Laurent series in one variable over a finite field.