Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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.

  1. more elementary language, also fixed an error: C and Spec should be contravariant functors.

    Yuxi Liu

    diff, v36, current

    • CommentRowNumber12.
    • CommentAuthorGuest
    • CommentTimeApr 21st 2022
    What happens if the category of test spaces is Set?
    • CommentRowNumber13.
    • CommentAuthormattecapu
    • CommentTimeMay 23rd 2022

    add Isbell duality as related entry

    diff, v37, current