Spahn axiomatic cohesion (Rev #6)

II. Cohesion versus non-cohesion; quality types

He interprets geometric morphisms as “Contrasts” between cohesion and non-cohesion and between variation and non-variation. There is also a distinction between cohesion and variation.

Definition

A full and faithful functor f *:SFf^*: S\to F between extensive categories which is a Frobenius functor in that it is reflective and coreflective by the same functor by definition exhibits FF as a quality type over SS.

Definition (category of cohesion)

III. Extensive quality; intensive quality in its rarefied and condensed aspects; the canonical qualities form and substance

Definition

An extensive quality quality on a category of cohesion p:ESp:E\to S is defined to be a functor hh such that

  • hh preserves finite coproducts

  • the codomain of hh is a quality type q:FSq:F\to S

  • q !h=p !q_! h=p_!

E p ! S h q ! F\array{ E&\stackrel{p_!}{\to}&S\\ \downarrow^h&\nearrow^{q_!}\\ F }

i.e. an extensive quality of XX has the same number of connected pieces as XX.

Theorem (Hurewicz)

Any category of cohesion has a canonical extensive quality defined by F(X,Y)=p !(Y,X)F(X,Y)=p_!(Y,X) where hhis the identity on objects and preserves finite products and exponentiation.

Revision on January 7, 2013 at 21:22:11 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.