Total Projective Systems

✍ Scribed by Sten Bjon

• Publisher: John Wiley and Sons
• Year: 1985
• Tongue: English
• Weight: 529 KB
• Volume: 124
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19851240116

✦ Synopsis

In trod uc t,ioii

With an object A in a category 31 is often associated a cone (pi : A +Ai) of morphisms to objects Ai in some subcategory di of &. For instance, with a HAUSDORFF locally convex space is associated a projective system of BANACH spaces, such that the locally convex space "approximately" is a limit of the system.

In this note we are interested in a factorization property (totality), which guarantees that there are "enough" inorphisms in the cone. Its simplest form (absolute totality) is equivalent to the statement : '(each morphism A -A in 31 with A i~ 0b(3tl) factors through some p j " . If F : 31 +8 is a left adjoint functor and if certain conditions are fulfilled, then P A = Lim FAi for any total cone ( A -Ai).

The general results are applied to categories of limit spaces and limit vector spaces and so we obtain for instance for an absolutely total cone (X -Xi) from a