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worked on space and quantity a bit
tried to polish the introduction and the Examples-section a bit
added a section on the adjunction with a detailed end/coend computation of the fact that it is an adjunction.
did a little bit of polishing: fixed the formatting of the two citations and of the pointers to it, and made a pointer to these appear right at the beginning of the Idea-section. Added floating table-of-contents.
Then I expanded the first paragraphs of the Idea-section with the remark that from the point of view of “presheaves are generalized spaces”, the Yoneda lemma, Yoneda embedding and the sheaf condition consistute three consistency conditions on this interpretation:
One may view the Yoneda lemma and the resulting Yoneda embedding as expressing consistency conditions on this perspective: The Yoneda lemma says that the prescribed rule for how to test a generalized space $X$ by a test space $U$ turns out to coincide with the actual maps from $U$ to $X$, when $U$ is itself regarded as a generalized space, and the Yoneda embedding says that, as a result, the nature of maps between test spaces does not depend on whether we regard these as test spaces or as generalized spaces.
Beyond this automatic consistency condition, guaranteed by category theory itself, typically the admissible (co)presheaves that are regarded as generalized spaces and quantities are required to respect one more consistency condition, the sheaf condition:
I still think we should make Chu and Isbell talk to each other, as I said here.
Then there’s yet another page nucleus of a profunctor, which indicates that Isbell duality is a special case. So presumably Chu construction should be tied in with the former.
Or rather the nucleus is the invariant part of a general construction which includes Isbell duals.
Looking back at “Taking categories seriously”, p. 17, the discussion there is somewhat abrupt. I seem to remember that the same kind of idea is introduced in a more inviting way in other articles by Lawvere. Anyone remember a good alternative to cite?
added pointer that regarding presheaves as generalized spaces is really the perspective of functorial geometry. Also cross-linked back from that entry to here.
In the other thread we mentioned a couple general notions that seem at least jointly inspired by Chu-like and Isbell-like ideas, but I didn’t see any precise connection yet. I suppose we could just add links saying “duality is also relevant to X”..
But Chu construction and nucleus of a profunctor are more directly connected, no? Simon Willerton’s series on the latter went through - The Nucleus of a Profunctor: Some Categorified Linear Algebra; Formal Concept Analysis; Classical Dualities and Formal Concept Analysis; Classical Dualities and Formal Concept Analysis; Galois Correspondences and Enriched Adjunctions. Chu spaces speaks about ’formal concept analysis’.
Hmm… maybe, but I don’t immediately see it from the definitions. Not confidently enough to write anything myself about the relationship.
It seems like just variations on the theme of matrices in the end.
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