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 nforum 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).

nLab > Latest Changes: sliced adjoint functors -- section

Bottom of Page

1 to 6 of 6

  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2021

    this is a bare little section, to be !include-ed as a Properties-subsection at slice category and adjoint (∞,1)-functor (where two copies of this same section used to be all along) and also at slice (∞,1)-category and adjoint functor (where, for completeness, the same should be recorded, too, but wasn’t until now)

    It would also be good to expand a little here, for instance by adding a pointer to a 1-category textbook account (this is probably in Borceux, but I haven’t checked yet), or, of course, by adding some indication of the proof.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2021
    • (edited Jul 15th 2021)

    I have added the proof of the slice adjunction

    𝒟 /L(c)⊥⟶R /c⟵L /c𝒞 /c \mathcal{D}_{/L(c)} \underoverset {\underset{\;\;R_{/c}\;\;}{\longrightarrow}} {\overset{\;\;L_{/c}\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/c}

    in 1-category theory – and in tikzcd diagrams.

    The typesetting looks quite neat in my local pdf rendering. On the nnLab it currently comes out not quite as neat since:

    (1) the font size of the tikzcd diagrams does not match that of the ambient text,

    (2) tikzcd diagrams can’t be put inline here, or even just next to each others in a line.

    But I guess it’s okay.

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2021

    added statement and proof also of the other slicing

    𝒟 /b⊥⟶R /b⟵L /b𝒞 /R(b) \mathcal{D}_{/b} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/R(b)}

    What’s a canonical reference for this? This ought to be in Borceux somewhere, but I haven’t looked for it yet.

    diff, v5, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2021

    I have added the remark (here) that the left adjoint of the “2nd form” of the sliced adjunction sends slice morphisms to their adjuncts.

    diff, v6, current

  5. Previous edit added the joint generalisation of both ways of slicing, which is proven in HTT. This edit fixes some typos.

    Jan Steinebrunner

    diff, v9, current

  6. I’m very confused about how to typeset the adjunctions, sorry… Should be fixed now

    Jan Steinebrunner

    diff, v9, current

1 to 6 of 6

  1.