Spahn axiomatic cohesion (Rev #11, 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 *:SFf^*: 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 gg s.t. (gf *g)(g\dashv f^*\dashv g) ) by definition exhibits FF as a quality type over SS.

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

Definition (category of cohesion)

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

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

is an adjoint quadruple of functors such that

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

(b) p !p_! preserves SS-parametrized powers in that

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

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

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

Comments
  1. The functors directed “downwards” (i.e. p !p_! and p *p_*) express the opposition between “points” and “pieces”.

  2. The functors directed “upwards” (i.e. p *p^* and p !p^!) express the opposition between “pure anti-cohesion” (discreteness) and “pure cohesion” (codiscreteness).

  3. (c) implies that p !(X×Y)p !(X)×p !(Y)p_!(X\times Y)\to p_!(X)\times p_!(Y) “mapping pieces of a product to the product of pieces” is an epimorphism.

  4. 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 qualityextensive 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&\mathrlap{\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 pp has a canonical extensive quality h:(p:ES)(q:FS)h:(p:E\to S)\to (q:F\to S) such that

  • hh is the identity on objects. hh preserves finite products and exponentiation.

  • the hom objects in FF are defined by F(X,Y)=p ![Y,X]F(X,Y)=p_![Y,X].

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

Since FF is cartesian closed the internal homhom-object [Y,X][Y,X] is defined.

Definition

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

  • s *s_* preserves finite products and finite coproducts.

  • q *s *=p *q_*s_*=p_*

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

Revision on January 9, 2013 at 23:14:19 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.