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.
    • CommentAuthorFinnLawler
    • CommentTimeDec 17th 2011

    Stephan Alexander Spahn has created descent object, with some definitions from Street’s Categorical and combinatorial aspects of descent theory.

    If I get the opportunity this weekend I’ll add details from Street’s Correction to ’Fibrations in bicategories’ and Lack’s Codescent objects and coherence. Anyone know of any other references?

    Looking at Street’s paper again, what he describes as the ’n=0n=0’ case of codescent objects looks to be just the notion of a coequalizer. I would have expected reflexive coequalizers, though, because the higher-nn case uses n+2n+2-truncated simplicial objects. Is there a reason for this?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeDec 17th 2011

    I think a page on descent objects should emphasize on the 2 and higher dimensional case; Street’s starting with 1-dimension is a bit idiosyncratic. By all means we should emphasize that it’s a generalization of coequalizers, but we shouldn’t give people the impression that “descent object” is just a fancy word for “coequalizer”. Perhaps this is what you were saying too.

    I don’t know what Street would say if you asked him the question about reflexive coequalizers, but one answer I can think of is that 2-dimensional descent objects are often/usually defined using two levels of face maps, but only one level of degeneracies. Going down a level, you’d have one level of face maps but no degeneracies at all. I think the reason for this is that for defining the limit notion, at least, the top level of degeneracy is unnecessary. A coequalizer of a reflexive pair is the same as a coequalizer of the same pair with the common splitting ignored. Same for whether or not you include the second level of degeneracies in a descent object. But you can’t (in general) leave out the first level of degeneracies in a descent object, or you get a different kind of limit (I think).

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 17th 2011
    • (edited Dec 17th 2011)

    Notice that Street’s definition of descent is lacking a condition to ensure that it gives the right answer. This is discussed at Verity on descent for strict omega-groupoid valued presheaves.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 17th 2011

    I have added to descent object an Idea-section, and a section on descent objects for ordinary presheaves (aka matching families).

    There is much more to be said about examples, but much of it is already said at descent.

    I think I’ll create a floating TOC for descent now, in order to provide a better overview of the topic.

    • CommentRowNumber5.
    • CommentAuthorFinnLawler
    • CommentTimeJan 13th 2012
    • (edited Jan 13th 2012)

    Is it definitely true that

    for defining the limit notion, at least, the top level of degeneracy is unnecessary

    where (co)descent objects are concerned? I ask because I think I have a characterization of ’2-final’ 2-functors F:IJF \colon I \to J as those for which the lax slice j//Fj // F is non-empty, connected and locally connected for each j (where in a zigzag of triangles witnessing connectedness the backwards ones must contain an invertible 2-cell). It follows that if I and J are 1-categories then F is 2-final if and only if each ordinary slice j/Fj/F is non-empty and codiscrete, i.e. a connected preorder.

    If that’s right, then (fully weak) coequalizers are not the same thing as reflexive ones, because, taking F to be the inclusion of the free parallel pair into the free reflexive pair (a subcategory of Δ\Delta), the morphisms δ 0σ\delta_0 \sigma and δ 1σ\delta_1 \sigma are idempotent and so are non-trivial endomorphisms of themselves in [1]/F[1]/F. That was surprising, but not shocking.

    But the same thing would seem to happen for the inclusion into Δ 2\Delta_{\leq 2} of itself without the top level of degeneracies, so it appears that colimits over the two categories need not coincide. Have I missed something, or does this show that my characterization is wrong?

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJan 14th 2012

    Hmm… given the characterization of homotopy final functors as those whose slices j/Fj/F all have (weakly) contractible nerves, I would expect a 2-final functor to be one where all the slices j/Fj/F have nerves with trivial π 0\pi_0 and π 1\pi_1.

    • CommentRowNumber7.
    • CommentAuthorFinnLawler
    • CommentTimeJan 15th 2012

    There are definitely a couple of mistakes in what I wrote – for a start, σ\sigma doesn’t exist in the free parallel pair, so can’t be part of any morphism in the slice categories. Also, we’re talking about colimits over the opposites of these categories, so we want the slices F/jF/j. But even with these errors corrected there are still unequal parallel morphisms in F/[1]F/[1], and the same problem as before arises for codescent objects, so there’s still something wrong.

    The actual condition for a general 2-functor F to be final is that each colimJ(j,F)colim J(j,F) should be equivalent to the terminal category, i.e. non-empty and codiscrete. But I’ve clearly made some kind of mistake in translating this through the prescription for 2-colimits that I put at 2-limit recently. More thinking required…

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJan 15th 2012

    Does colimJ(j,F)colim J(j,F) means a 2-categorical (pseudo) colimit of the diagram of hom-categories? That still doesn’t seem right to me; maybe if it were the lax colimit (whose nerve represents the homotopy colimit, I think).

    • CommentRowNumber9.
    • CommentAuthorFinnLawler
    • CommentTimeJan 15th 2012

    Yes, I mean colim iJ(j,Fi)\colim_i J(j,F i).

    I’m following Kelly’s book, section 4.5. Suppose you have F:IJF \colon I \to J as before and weights W:JCatW \colon J \to Cat and V:ICatV \colon I \to Cat. Then you can follow Kelly’s arguments (replacing isomorphisms in V with equivalences in Cat) to show that WLan FVW \simeq Lan_F V if and only if {W,D}{V,DF}\{W, D\} \simeq \{V, D F\} for any functor D (where the Kan extension is defined pointwise as usual: (Lan FV)j=J(F,j)V(Lan_F V) j = J(F-,j) \star V). Setting both weights equal to the constant functor Δ1\Delta \mathbf{1} at the terminal category you get the condition for F to be inital – that the identity transformation should exhibit Δ1:JCat\Delta\mathbf{1} \colon J \to Cat as Lan FΔ1Lan_F \Delta \mathbf{1} (where that Δ1:ICat\Delta\mathbf{1} \colon I \to Cat, of course). Then F is final if F opF^{op} is initial, which is where the earlier condition comes from.

    I’m pretty sure this all goes through OK in the weak/bicategorical setting. Unless I’ve missed something obvious here (which is not at all impossible), then the problem is with the translation of the condition on colim iJ(j,Fi)\colim_i J(j,F i) into one on the bicategory of elements J(j,F)=j//F\int J(j,F) = j//F. Any thoughts?

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJan 15th 2012

    I need to think about this some more, but just as a note: you can write \sslash to get \sslash (although apparently some people can’t view that character).

    • CommentRowNumber11.
    • CommentAuthorFinnLawler
    • CommentTimeJan 16th 2012

    Argh, I had forgotten that descent objects are not conical limits – according to Street’s Correction to Fibrations in bicategories they are weighted by the cosimplicial object in Cat whose objects are the terminal category, the free isomorphism and the free composable pair of isomorphisms. The finality/Kan-extension condition in that case reduces to checking that any descent datum automatically satisfies the extra conditions wrt the coface maps, which I’m pretty sure is true.

    With that in mind, the only slightly puzzling thing is that, in this fully weak setting at least, the coequalizer of a pair that has a common pseudo-section apparently need not be the colimit of the diagram with the section included. But I don’t think the condition I initially gave is quite equivalent to the one that says colimJ(j,F)1colim J(j,F) \simeq \mathbf{1}, so that could still be wrong.

    Thanks for your responses so far. I’ll keep thinking.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJan 16th 2012

    Descent objects are not conical strict 2-limits, but unless I’m mistaken they are conical as bicategorical limits, which I thought was the setting you were working in. Specifically, the free isomorphism and the free composable pair of isomorphisms are both equivalent to the terminal category, so bicategorical limits weighted by that weight should be equivalent to conical ones.

    • CommentRowNumber13.
    • CommentAuthorFinnLawler
    • CommentTimeJan 16th 2012

    Yes, I see. I hadn’t noticed that.

    • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 13th 2017

    We don’t actually have a description/construction of a descent object in a 2-category at descent object, even though at 2-limit#2limits_over_diagrams_of_special_shape it points there in lieu of given a description, in the point about equalisers.

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeAug 13th 2017

    Well, someone should fix that. (-: