adequate and coadequate subcategories

[Posina Venkata Rayudu <posinavrayudu@gmail.com>]

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Dear All,

For some time, I have been staring at the basic relation of algebraic
geometry (https://zenodo.org/records/7079058<https://url.au.m.mimecastprotect.com/s/wvP_CP7L1NfLvjO6u6TO9V?domain=zenodo.org>, p. 2), which is an
adjoint relation between geometry and algebra:

V^Yop <===> (V^Y)op

which Professor F. William Lawvere was kind enough to suggest as a
framework to abstract the mathematical content of the fundamental
dialectic of philosophy:

([epistemology vs. ontology] vs. reality)

I think compounding epistemology and ontology into which reality is
resolved is a major outstanding scientific program. Surely, Newton
would be happy, having emphasized synthesis after analysis.

Professor F. William Lawvere referred to the above adjointness as
Isbell conjugacy
(http://www.tac.mta.ca/tac/reprints/articles/8/tr8.pdf<https://url.au.m.mimecastprotect.com/s/zq5yCQnM1WfwB28RSAZ2f-?domain=tac.mta.ca>, p. 17). Simply
put, Isbell conjugacy is about getting to know a category V in terms
of geometric Y-shaped figures vs. algebraic Y-valued properties
(https://zenodo.org/records/7059109<https://url.au.m.mimecastprotect.com/s/zK-HCRONg6s3nMJOSQX2Fw?domain=zenodo.org>, p. 49).

Going by our experience with sets, a single-element set 1 = {*} is
adequate enough to completely characterize every set and to test for
the equality of functions, but we need a two-element coadequate set 2
= {false, true} to tell apart elements of any domain set.

Before long I can't help but wonder if the relationship between

V^Aop <=?=> (V^C)op

(where A and C are adequate and coadequate subcategories,
respectively, of a category V) would be relatively more informative
than the above Isbell conjugacy.

Furthermore, in a CatList post (06 March 2009), Professor F. William
Lawvere points out that Isbell conjugacy is a special case of the
construction of the total category with two descriptions which
objectify adjointness (unfortunately I couldn't find any mention of
'total category' in a quick search of his Functorial Semantics of
Algebraic Theories
(http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf<https://url.au.m.mimecastprotect.com/s/j5ybCVARmOHp0W7ZuESkx2?domain=tac.mta.ca>, which he
cites).

I look forward to your corrections and suggestions!

Happy Bakrid :)

Thanking you,
Yours truly,
posina


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