The nature of the spatial background for classical analysis and for modern
theories of continuum physics requires more than the partial invariants of
locales and cohomology rings for its description. As Maxwell emphasized,
this description has various levels of precision depending on the needs of
investigation. These levels correspond to different categories of space,
all of which have intuitively the feature of cohesion. Our aim here is to
continue the axiomatic study of such categories, which involves the
following aspects:
I. Categories of space as cohesive backgrounds
II. Cohesion versus non-cohesion; quality types
III. Extensive quality; intensive quality in its rarefied and
condensed aspects; the canonical qualities form and substance
IV. Non-cohesion within cohesion via constancy on infinitesimals
V. The example of reflexive graphs and their atomic numbers
VI. Sufficient cohesion and the Grothendieck condition
VII. Weak generation of a subtopos by a quotient topos
I look forward to further work on each of these aspects, as well as
development of categories of dynamical laws, constitutive relations, and
other mathematical structures that naturally live in cohesive categories.
Keywords: Cohesion, qualities, graphs, nature of space
2000 MSC: 18A40, 18B25, 18B30, 74A60, 74A99
Theory and Applications of Categories,
Vol. 19, 2007,
No. 3, pp 41-49.
http://www.tac.mta.ca/tac/volumes/19/3/19-03.dvi
http://www.tac.mta.ca/tac/volumes/19/3/19-03.ps
http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/19/3/19-03.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/19/3/19-03.ps