Not signed in (Sign In)

Start a new discussion

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 bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics 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 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 kan lie-theory limit 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 newpage nlab nonassociative 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 string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory 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
    • CommentTimeJun 2nd 2010

    I think the definition of the Grothendieck construction was wrong. The explicit definition was right, but the description in terms of a generalized universal bundle didn’t work out to that, if by “the category of pointed categories” was meant for the functors to preserve the points, which is the usual meaning of a category of pointed objects. I corrected this by using the lax slice. Since while I was writing it I got confused with all the op’s, I decided that the reader might have similar trouble, so I changed it to do the covariant version first and then the contravariant.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 2nd 2010

    ’the category of pointed categories’ should have weakly pointed functors as 1-arrows, and as you say is the lax slice *Cat*\downarrow Cat. I think the definition was lost in translation somewhere between my brain and (probably) Urs’ fingers :) (and this was a few years ago since that discussion!)

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 2nd 2010
    • (edited Jun 2nd 2010)

    Redacted

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2010
    • (edited Jun 2nd 2010)

    As discussed at generalized universal bundle (I think) the “generalized universal bundle” over CC is the pullback

    P *C * C I d 1 C, \array{ P_* C &\to& * \\ \downarrow && \downarrow \\ C^I &\stackrel{d_1}{\to}& C } \,,

    for the given point *C* \to C and the relevant path object C IC^I.

    For C=CatC = Cat the point is the terminal category and C IC^I is the 1-categorical internal hom [Δ[1],Cat][\Delta[1], Cat], the category whose objects are functors and whose morphisms are filled commuting diagrams.

    This gives the correct construction, and this is what is meant.

    I am thinking that eventually there should be a way to make precise the sense in which this kind of construction produces “comma object-weak pullbacks” in analogy to how the analogous construction for undirected interval objects gives homotopy pullbacks.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJun 2nd 2010

    I don’t know what a “filled commuting diagram” is; if “filled” means that it contains a 2-cell, then it doesn’t commute. Squares containing a 2-cell is what you need as the morphisms in order to get the right thing (i.e. the lax slice) as the “generalized universal bundle,” and that version of “[Δ[1],Cat][\Delta[1],Cat]” is only the internal hom relative to the lax version of the Gray tensor product on 2Cat2Cat. It’s certainly not the internal hom for any closed structure on CatCat.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2010
    • (edited Jun 2nd 2010)

    Commuting up to a 2-isomorphism, yes. And I mean the 2-categorical internal hom: 2-functors, pseudonatural transformations and modifications.

    So this diagram

    * C x C y \array{ && * \\ & \swarrow &\swArrow& \searrow \\ C_x &&\to&& C_y }

    that describes what the Grothendieck construction does over a morphism is the (single) component of a pseudonatural transformation between two 2-functors Δ[1]Cat\Delta[1] \to Cat where after restriction along d 1:Δ[0]Δ[1]d_1 : \Delta[0] \to \Delta[1] everything is required to be constant on the terminal category.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJun 2nd 2010

    No, the point is that it can’t commute up to a 2-isomorphism; you have to allow the transformation sitting in the square to be non-invertible. In other words, you need to use lax natural transformations. If it were a pseudonatural transformation, then in that diagram you just drew, the 2-morphism would be invertible – but in the Grothendieck construction a morphism in the total category from aa to bb over f:xyf:x\to y is a map f !(a)bf_!(a)\to b which is not necessarily invertible.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2010

    No, the point is that it can’t commute up to a 2-isomorphism; you have to allow the transformation sitting in the square to be non-invertible.

    Right of course. Or else it works for the groupoid version.

    Okay, now looking back at the discussion I clearly didn’t give very concentrated replies here, but it seems to me that this notwithstanding, it is kind of noteworthy that the Grothendieck construction is forming this limit here

    F * [Δ[1],Cat op] Cat op C Cat op \array{ \int F &\to&&\to& * \\ \downarrow &&&& \downarrow \\ && [\Delta[1], Cat^{op}] &\to& Cat^{op} \\ \downarrow && \downarrow \\ C &\to& Cat^{op} }

    for a suitable hom-obect [Δ[1],Cat op][\Delta[1],Cat^{op}].

    Speaking just about the lax (co)slice for the top-right pullback hides a bit this nice structure.

    With the suitable notion of “weak pullback” we should just be saying that the Grothendieck construction is the “weak pullback” of the point.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJun 2nd 2010

    Maybe. I prefer thinking about comma objects (including slice categories) as weighted limits in their own right, rather than always constructing them as a double pullback, but I can see that that other perspective can also be helpful. It also makes me uneasy when we say “weak” to include “lax” because some people use “weak” to mean specifically “pseudo.”

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJun 3rd 2010

    …although perhaps that is being too persnickety. I do like the “generalized universal bundle” point of view, although as this discussion shows there are some subtleties to beware of when you start having noninvertible higher cells around.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2010

    as this discussion shows there are some subtleties to beware of when you start having noninvertible higher cells around.

    Yes, indeed, I took the message home here that the discussion of this on the Lab, for the special case at hand but also for the general situation at generalized universal bundle, needs to be improved. I’ll try to look into this when I have more leisure.

    This week and next one I am attending conferences and need to focus my energy a little bit elsewhere, though.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)