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 beauty 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 deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality education 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 lie lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories 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 science 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.
    • CommentAuthorUrs
    • CommentTimeNov 10th 2009
    • (edited Nov 10th 2009)

    I added to Quillen bifunctor as a further "application" the discussion of Bousfield-Kan type homotopy colimits.

    At some point I want to collect the material on homotopy (co)limits currently scattered at Bousfield-Kan map at weighted limit and now at Quillen bifunctor into one coherent entry.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 15th 2009

    After pointing to it from MathOverflow here, I polished a tiny little bit some parts of homotopy limit.

    More needs to be done here.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2010

    somehow I felt like expanding the section global definition at homotopy limit – even though I should really be doing something else.

    I amplified the relation to homotopy Kan extension a bit more, spent more words on the special case of simplicial model categories, on the enriched Quillen adjunctions that limit and colimit form there, and then at the end wrote some paragraphs on the derived weigheted colimits and how they can be used to computed unweighted derived colimits.

    This duplicates now some of the discussion that appears later in the Examples, section, but I felt this deserved to be the in the Theory-part, before explicit examples are discussed.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2010

    added now also a section with some pointers to derivators.

    The fact that homotopy limit is the first hit presented by Google is something that not so much makes me feel glorious, as that it makes me feel worried about the long distance this entry has from perfection… But I guess I should relax.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJun 1st 2010

    I think the page is pretty good. The main problem of organization I see is that there is some duplication in the discussion of projective/injective model structures between the section global definition and the section general formula. Also I’d like to see a mention of Reedy model structures as well; maybe I’ll add that if I get the chance.

    Is the section on simplicial presheaves really appropriate on the general homotopy limit page? Would it make more sense at simplicial presheaf or a separate page of its own?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2010

    I agree. Yes, the section on simplicial presheaves would be better kept at the entry on simplicial presheaves themselves.

    I can move it later this week. Unless you beat me to it.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeJun 2nd 2010

    Unless you beat me to it.

    Urs Schreiber starring in a horror movie: "Beaten by nlabizens".

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2010

    Added statement of proof of the standard fact that every simplicial set is the “hocolim over its cells” to homotopy colimit – examples – hocolims of simplicial diagrams

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeFeb 9th 2011

    The query at homotopy limit seems outdated. So I move it here as the archived version:

    Tim Mike, are you intending to treat the case of when the domain category, DD is the above, is enriched as well? This would handle the example of homotopy limts of homotopy coherent diagrams, both in Vogt’s sense and in the simplicially enriched case looked at by Bourn and Cordier. This would also allow the GG in one of the examples to be a simplicial or topological group, or to be (?) and A-infinity category. (Some of those examples may be already dealt with in others of the entries as different people classify things in different ways.)

    Perhaps some of the more classical referencs, Vogt, Bousfield-Kan etc. might be included for completeness.

    Mike: Yes, certainly; my paper referenced above deals with the case when DD is enriched as well. There are cofibrancy technicalities, of course. I’m not against including the classical references, although I find them fairly impenetrable myself.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 9th 2011
    • (edited Feb 9th 2011)

    Thanks, Zoran, for you effots of cleaning up old discussion in the entries and reviving them here. I think this is a very good service.

    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeFeb 9th 2011
    • (edited Feb 9th 2011)

    I added the link to Maltsiniotis’ lectures at Sevilla. Notice that in lecture II, in the setting of a localizer (C,W)(C,W), he defines a homotopy (co)limit as an adjoint to a localized version Δ¯ I:W 1CW I 1C I\bar{\Delta}^I: W^{-1} C \to W_I^{-1} C^I of the diagonal functor Δ I:CC I\Delta^I: C\to C^I on the localized category of II-diagrams; to start with one does not ask even for 2 out of 3 property for weak equivalences nor for any kind of enrichment. (He neglects the size questions, which are usually resolvable only under more specific assumptions.) See lecture I, localizers pdf pages 12-13.

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeFeb 9th 2011

    Added to homotopy limit:

    Alternative definitions can be formulated at the level of the homotopy category W 1CW^{-1} C one defines a localized version Δ¯ I:W 1CW I 1C I\bar{\Delta}^I : W^{-1} C\to W_I^{-1} C^I of the diagonal functor Δ I:CC I\Delta^I : C\to C^I and define the homotopy limits and colimits as the adjoints of Δ¯ I\bar{\Delta}^I (at least at the points where the adjoints are defined). Here W IMor(C I)W_I\subset Mor(C^I) are the morphisms of diagrams whose all components are in WMor(C)W\subset Mor(C). The above definitions via derived functors (Kan extensions) follow once one applies the general theorem that the derived functors of a pair of adjoint functors are also adjoint and noticing that (Δ I,Δ¯ I)(\Delta^I,\bar{\Delta}^I) is a morphism of localizers (and in particular that Δ¯ I\bar{\Delta}^I with the identity 2-cell is a Kan extension (simultaneously left and right)).

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 9th 2011

    The previous paragraphs seemed a bit confused about adjoints versus Kan extensions, so I fixed them.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 8th 2011

    added to homotopy colimit very briefly the statement that the inclusion Δ onlyfacesΔ\Delta_{only faces} \hookrightarrow \Delta is homotopy final

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)