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 complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration 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 object 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).

nLab > Latest Changes: representable functor theorem

Bottom of Page

1 to 8 of 8

  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 22nd 2023

    Added:

    Specifically, a continuous functor C→SetC\to Set is a right adjoint functor if and only if it is representable, in which case the left adjoint functor Set→CSet\to C sends the singleton set to the representing object

    Related concepts

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2023

    I see now that, originating with the creation of the entry back in 2011, it has a line starting with the words

    As representable functors are ubiquitous,…

    This does not make sense to me, neither the claim itself nor the suggestion that it implies the statement that follows. I wonder what was really meant here. But it looks like just deleting these words would not take anything away from the paragraphs that follow. (?)

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 22nd 2023

    I changed the opening of the first sentence under “Related facts” a little so that it ties in better with the phrase that follows (the earlier version, about the ubiquity of representable functors, seemingly echoes similar statements about the ubiquity of (the concept of) adjoint functors – see the quotations of Mac Lane given in Categories for the Working Mathematician).

    diff, v5, current

    • CommentRowNumber4.
    • CommentAuthorncfavier
    • CommentTimeOct 31st 2023

    Added some details and corrections. Please double-check! I don’t have a reference, sadly, I’m just going off what I could convince Agda of.

    diff, v6, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2023

    I have added two references (here)

    You changed “complete” to “cocomplete” in the first sentence. But it looks to me like the original version was correct.

    diff, v7, current

    • CommentRowNumber6.
    • CommentAuthorvarkor
    • CommentTimeOct 31st 2023

    Reformulated the statement of the representable functor theorem to make explicit the distinction between representability and corepresentability, which was leading to confusion.

    diff, v8, current

    • CommentRowNumber7.
    • CommentAuthorncfavier
    • CommentTimeNov 1st 2023
    • (edited Nov 1st 2023)

    Okay, I think my confusion came from the fact that you need CC to be cocomplete in order to be able to say that F:C→SetF : C \to Set has a left adjoint iff it is representable, and I assumed that the representable functor theorem followed from the adjoint functor theorem and this equivalence; but the representable functor theorem actually stands on its own (without the assumption that CC is cocomplete), and then in order to deduce the adjoint functor theorem from that you only need D(x,Fy)D(x, Fy) to be representable, so there’s no copowering nonsense.

    I think the “specifically” paragraph should be rewritten now, but I am not sure how.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 3rd 2023

    I don’t think there’s any “nonsense”.

1 to 8 of 8

  1.