composite qualities (extensive x intensive)

composite qualities (extensive x intensive)

[Posina Venkata Rayudu]

Dear All,

I hope and pray that you and your families are all well.

If I may, space, according to Kant, is the form of intuition, i.e. the
canonical extensive quality of intuition.

Along similar lines, time, according to Proust, is the shape of
memory, i.e. the canonical extensive quality of memory.

Now, given a category of objects, we can find out its qualities.  For
example, the category of irreflexive graphs satisfies one of the
axioms of cohesion (see F. William Lawvere, 2007, Axiomatic Cohesion;
http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf), which is: subobject
classifier is 1 piece (see Lawvere and Schanuel, Conceptual
Mathematics, p. 345), but doesn't satisfy another cohesion axiom:
pieces of product equals the product of pieces of the factors (p(A x
A) = 3, while p(A) x p(A) = 1 x 1 = 1, where A = * ---> *, one of the
two basic shapes of the category of irreflexive graphs).  However,
reflexive graphs, wherein arrows have distinguished loops as sources
and targets, satisfy the product preservation of the pieces functor
(for illustration see Fig. 6 in
https://zenodo.org/record/4048096#.YW4e2RpBzIU).  Simply put, given a
category, we can see if it is cohesive; its qualities: both extensive
and intensive qualities.

Now, according to Poincaré (see p. 45 in F. William Lawvere, Axiomatic
Cohesion; http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf):

Extensive E and intensive I qualities jointly reflect isomorphisms.
For example, with (E1 x I1) and (E2 x I2) as products of the canonical
extensive AND intensive qualities of two cohesive objects C1 and C2,
respectively, if composite quality E1 x I1 is isomorphic to E2 x I2,
then the cohesive object C1 is isomorphic to C2 (see pp. 173-174 in
Lawvere and Schanuel's Conceptual Mathematics).

In light of the above, I was wondering if we can characterize the
nature of a category whose extensive quality is space and/or that of a
category whose extensive quality is time.  Isn't it a problem of
knowing only one of the two variables of a 2-variable function (e.g.
instead of a surface on a plane we have a curve).

Independent of Kant and Proust, isn't it mathematically interesting to
know if the calculation of qualities of a category can be inverted
(not necessarily isomorphisms, but, say, inverse images or fibers of a
map from the category C of categories to a product category E x I of
extensive and intensive quality types)?

Returning to Poincaré, is the product category (E x I) of categories
of extensive (E) and intensive (I) quality types coadequate for the
category C of categories?

Now, given a functor

C ---> E x I

we get a pair of functors

C ---> E AND C ---> I

by aft composition with corresponding projections to factor categories E and I.

With extensive quality

E = S

(category of spaces/modes of cohesion), we get to know how the
codomain intensive qualities I structure the domain category C of
categories.  Similarly, with extensive quality

E = T

(category of time(s?)/types of change), we get to know how intensive
qualities structure the category of categories.

These calculations can then be brought to bear on the nature of and
relations between intuition and memory (since they have space and time
as their respective extensive qualities; for worked out examples of
cohesive categories (reflexive graphs) and quality types
(idempotents), see: https://zenodo.org/record/3924474).

I look forward to your corrections of any mistakes I might have made
in my description of extensive and intensive qualities.

Thanking you,
posina


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