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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 12th 2010

    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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 12th 2018

    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 XX by a test space UU turns out to coincide with the actual maps from UU to XX, when UU 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:

    diff, v32, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 12th 2018

    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.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 12th 2018

    Or rather the nucleus is the invariant part of a general construction which includes Isbell duals.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 12th 2018

    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?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 12th 2018

    added pointer that regarding presheaves as generalized spaces is really the perspective of functorial geometry. Also cross-linked back from that entry to here.

    diff, v32, current

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJun 12th 2018

    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”..

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 12th 2018

    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’.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJun 12th 2018

    Hmm… maybe, but I don’t immediately see it from the definitions. Not confidently enough to write anything myself about the relationship.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 12th 2018

    It seems like just variations on the theme of matrices in the end.

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