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 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 sheaves 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: category object in an (infinity,1)-category

Bottom of Page

1 to 12 of 12

  1.  
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeNov 28th 2012

    It was pointed out to me today that in the very special case of internal (0,1)-category objects in Set, what we are calling a “pre-category” reduces to a preordered set, while adding the “univalence/Rezk-completeness” condition to make it a “category” promotes it to a partially ordered set. I feel like surely I knew that once, but if so, I had forgotten. It provides some extra weight behind this term “precategory”, especially since some category theorists like to say merely “ordered set” to mean “partially ordered set”.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeNov 28th 2012

    Interesting!

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2012
    • (edited Nov 28th 2012)

    In general, the sSetsSet-nerve of a category is complete Segal precisely if the only isomorphisms are identities (what’s the name again for such a category?).

    I have added a paragraph on this to Segal space – Examples – In Set

    (this could alternatively go to various other entries, but now I happen to have it there, and linked to from elsewhere).

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeNov 28th 2012

    Sometimes it is called “rigid”.

  5. Sometimes it is called “gaunt”.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2012
    • (edited Nov 28th 2012)

    Ah, of course. I knew that term of yours, I wrote all those notes on your article, after all. But I forgot. Thanks for reminding me. There is now an entry gaunt category, so that I shall never forget again.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeNov 29th 2012

    Right, thanks. That’s better than “rigid”.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2012
    • (edited Dec 3rd 2012)

    I have been further expanding the Definition-section at category object in an (infinity,1)-category, adding a tad more details concerning proofs of some of the statements. But didn’t really get very far yet.

    Also reorganized again slightly. I am afraid that in parts the notation is now slightly out of sync. I’ll get back to this later today.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2012

    I have edited still a bit further. This will probably be it for a while, unless I spot some urgent mistakes or omissions.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 2nd 2021

    Added reference to

    • Louis Martini, Yoneda’s lemma for internal higher categories, (arXiv:2103.17141)

    diff, v56, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2022

    added these references on development of ∞\infty-category theory internal to any (∞,1)-topos:

    internal (∞,1)-Yoneda lemma:

    internal (infinity,1)-limits and (infinity,1)-colimits:

    internal cocartesian fibrations and straightening functor:

    internal presentable (∞,1)-categories:

    diff, v58, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 8th 2023

    added pointer to the recent:

    diff, v61, current

1 to 12 of 12

  1.