Spahn axiomatic cohesion (Rev #9)

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 (i.e. there is a $g$ s.t. $(g\dashv f^*\dashv g)$ ) by definition exhibits $F$ as a quality type over $S$ .

Short: A quality type is a full and faithful Frobenius pair/triple.

Definition (category of cohesion)

Let $E,S$ be cartesian-closed extensive categories. $E$ is called a category of cohesion relative to $S$ if

(p_!\dashv p^*\dashv p_*\dashv p^!):E\stackrel{p_*}{\to} S

is an adjoint quadruple of functors such that

(a) $p_!$ preserves finite products and $p^!$ is full and faithful. Thus for toposes this would be phrased as $p$ is “connected surjective” and “local”, $p^!$ is a subtopos, and $p^*$ is an exponential ideal.

(b) $p_!$ preserves $S$ -parametrized powers in that

p_![p^* W,X]\simeq [W,p_! X]

is a natural isomorphism for all $X\in E$ and all $W\in S$ . This “continuity” property follows from (a) if all hom sets in $S$ are finite; it also holds if the contrast with $S$ is determined as in IV below ba an infinitely divisible interval in $E$ .

(c) The canonical map $p_*\to p_!$ in $S$ is epimorphic (Schreiber calls this “pieces have points”). This holds iff the other canonical map $p^*\to p^!$ in $E$ is monomorphic (Schreiber calls this “discrete objects are concrete). Lawvere calls this property the ”Nullstellensatz“.

Comments

The functors directed “downwards” (i.e. $p_!$ and $p_*$ ) express the opposition between “points” and “pieces”.
The functors directed “upwards” (i.e. $p^*$ and $p^!$ ) express the opposition between “pure anti-cohesion” (discreteness) and “pure cohesion” (codiscreteness).
(c) implies that $p_!(X\times Y)\to p_!(X)\times p_!(Y)$ “mapping pieces of a product to the product of pieces” is an epimorphism.
If (c) is an isomorphism, this implies (a) and (b). In particular a cartesian closed quality type is a category of cohesion (in an extreme sense).

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&\mathrlap{\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 $p$ has a canonical extensive quality $h:(p:E\to S)\to (q:F\to S)$ such that

$h$ is the identity on objects. $h$ preserves finite products and exponentiation.
the hom objects in $F$ are defined by $F(X,Y)=p_![Y,X]$ .

defined by $F(X,Y)=p_![Y,X]$ where $h$ is the identity on objects and preserves finite products and exponentiation.

Definition

An intensive quality on a category of cohesion $p:E\to S$ is a functor $s_*$ from $p$ to a quality type $q:L\to S$ such that

$s_*$ preserves finite products and finite coproducts.
$q_*s_*=p_*$

\array{ E&\stackrel{p}{\to}&S\\ \downarrow^{s_*}&\mathrlap{\nearrow}^{q}\\ L }

Revision on January 9, 2013 at 01:29:35 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.