Spahn axiomatic cohesion (Rev #4)

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

Definiion

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_!

E S F\array{ E&\to&S\\ \downarrow\nearrow\\ F }

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

Revision on January 7, 2013 at 19:35:47 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.