Spahn axiomatic cohesion (Rev #5, changes)

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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

Definiion 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_!~~ $q_! h=p_!$

\begin{matrix} E & \to \overset{p_{!}}{\to} & S \\ ↓ ↓^{h} ↗ & ↗^{q_{!}} \\ F \end{matrix}

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

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

Revision on January 7, 2013 at 20:35:44 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.