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 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 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).
    • 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


    • 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”.

  1. 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