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* subobject as monomorphism
@ 2025-01-25  8:51 Posina Venkata Rayudu
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From: Posina Venkata Rayudu @ 2025-01-25  8:51 UTC (permalink / raw)
  To: Michael Barr, Prof., categories, Andree Ehresmann

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Dear Professor Michael Barr,

I hope all is well.

If I may, Grothendieck redefines subobject as:

a subobject of A is not an object, but an object B, together with a
monomorphism u: B --> A

(https://www.math.mcgill.ca/barr/papers/gk.pdf<https://url.au.m.mimecastprotect.com/s/zB6vCXLW6DiXqQ7VMsmtASWjF5_?domain=math.mcgill.ca>, p. 2).

Your time permitting, would you be kind enough to shed some light on
Grothendieck's conceptual trajectory, along with whatever it is that
might have made it indispensable, from the then subset is a set to his
subset is a 1-1 function, a conceptual refinement, whose mathematical
value was more than adequately explained in Professor F. William
Lawvere's geometry of figures (Lawvere & Schanuel, Conceptual
Mathematics, pp. 370-371,
https://conceptualmathematics.wordpress.com/wp-content/uploads/2013/02/conceptual-mathematics-2.pdf)<https://url.au.m.mimecastprotect.com/s/ig9VCYW86EsLN7oE6tZu2SxsdUd?domain=conceptualmathematics.wordpress.com>,
which can be traced all the way back to Cayley's representation, not
to forget Yoneda embedding or Professor's F. William Lawvere's
functorial semantics, along the way.

If I am not mistaken, the above Tohoku paper you translated is the
first instance of a subobject reincarnating as monomorphism. When
viewed from the perspective of: a part of a whole is both itself and
its relationship to the whole (Lawvere, Objective Logic, p. 53,
https://zenodo.org/records/7059109)<https://url.au.m.mimecastprotect.com/s/cqzhCZY146s5D6nrofXCPSBcQbv?domain=zenodo.org>, shouldn't Grothendieck's
'subobject is monomorphism' have been readily incorporated into math
(without having to go through the trials and tribulations of having to
fight against the accusations of circumlocution, etc.).

I look forward to your corrections and clarifications!

Thanking you,
Yours truly,
posina
P.S. Rational passage between concepts
(https://conceptualmathematics.wordpress.com/2023/02/17/rational-passage/<https://url.au.m.mimecastprotect.com/s/vkuaC1WLjwsMk4w2EUOFJSV7Pes?domain=conceptualmathematics.wordpress.com>)
is my Year 2025 project :) So, I'd appreciate any pointers to go from
one concept to another. I could be wrong, but broadly it seems,
speaking of the passage between concepts (GRAPH, MONOID, FUNCTION,
etc.), since concepts are objectified as categories (the theory of
irreflexive directed graphs as a category consisting of two parallel
maps, i.e., source, target: Arrows --> Dots), it seems a passage
between concepts must necessarily be functorial (mapping of objects
must be respectful of their mutual relations, composition, etc).


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