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

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 $p:E\to S$ is defined to be a functor $h$ such that

$h$ preserves finite coproducts
the codomain of $h$ is a quality type $q:F\to S$
$q_! h=p_!$

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

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

Theorem (Hurewicz)

Any category of cohesion has a canonical extensive quality defined by $F(X,Y)=p_!(Y,X)$ where $h$ is 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.