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 internal-categories k-theory 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 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).

nLab > Latest Changes: transfinite composition

Bottom of Page

1 to 14 of 14

  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 22nd 2010
    • (edited Jan 22nd 2010)

    I should know this, but now I am getting mixed up, so I'll ask, at the risk of just making a fool of myself:

    how do I see whether the transfinite composition of some weak equivalences is again a weak equivalence?

    I came across the statement by somebody that SSet has the advantage over Top that weak equivalences are closed under transfinite composition. Then I tried to think about this and found that I got myself mixed up....

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJan 22nd 2010

    I don't know any universal answer.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 22nd 2010
    • (edited Jan 22nd 2010)

    thanks, that makes me feel better ;-)

    but is that right about WEs in SSet being and in Top not being closed under transcomp? what would be a reference?

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJan 23rd 2010

    I don't recall right now, but it seems plausible, since WEs in sSet are determined by a "finite amount of data".

    • CommentRowNumber5.
    • CommentAuthorHarry Gindi
    • CommentTimeJan 23rd 2010
    I'm pretty sure it's not true in general for WEs, since they don't form a weakly saturated class. (Most weakly saturated classes are classes with lifting properties, which the WEs do not in general have.)
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 23rd 2010

    Thanks, Mike, this is already helpful.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJan 23rd 2010

    Yes, it's certainly not true that WEs in a general model category are closed under transfinite composition. But it is true sometimes, for instance it's certainly true for WEs in the canonical model structure on Cat.

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 19th 2015

    What are some interesting examples of model categories whose weak equivalences are not closed under transfinite compositions?

    • CommentRowNumber9.
    • CommentAuthorKarol Szumiło
    • CommentTimeOct 19th 2015

    Is the standard model structure on spaces interesting enough?

    If so, let U iU_is be contractible neighbourhoods of 11 in S 1S^1 whose intersection is just {1}\{1\} and let f i:S 1→S 1f_i \colon S^1 \to S^1 be a weak equivalence collapsing the complement of U iU_i to −1-1 and fixing 11. Then the colimit of f if_is is a connected two-point space so it is contractible.

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 19th 2015

    Indeed; one more reason to use simplicial sets instead of topological spaces…

    Now I wonder if there are any interesting combinatorial examples…

    • CommentRowNumber11.
    • CommentAuthorZhen Lin
    • CommentTimeOct 19th 2015
    • (edited Oct 19th 2015)

    If your class of morphisms is closed under filtered colimits, then it is also closed under transfinite composition. In the case of combinatorial model categories, a sufficient condition for the class of weak equivalences to be closed under filtered colimits is the existence of a set of generating cofibrations and a set of generating trivial cofibrations such that the domains and codomains of these are finitely presentable.

    So, if I had to guess, perhaps a combinatorial presentation of a non-hypercomplete ∞\infty-topos (on a non-finitary site) would be an example of what you are looking for.

    • CommentRowNumber12.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 20th 2015

    @ZhenLin: I’m confused now, I thought that any left Bousfield localization of simplicial presheaves automatically has weak equivalences closed under transfinite compositions.

    Indeed, consider a transfinite sequence X of weak equivalences in a left Bousfield localization of sPSh(C). Choose a cofibrant replacement q: QX→X in the projective model structure on transfinite sequences. The transition maps of QX are acyclic cofibrations in the left Bousfield localization. Acyclic cofibrations are weakly saturated, so the transfinite composition of QX is a weak equivalence.

    It remains to see that the transfinite composition of QX is weakly equivalent to the transfinite composition of X. This amounts to showing that the filtered colimit of q_n is a weak equivalence. However, the individual components q_n are acyclic fibrations in the left Bousfield localization, and acyclic fibrations don’t change under left Bousfield localizations, so q_n are componentwise acyclic fibrations of simplicial presheaves, and weak equivalences of simplicial sets are closed under filtered colimits.

    • CommentRowNumber13.
    • CommentAuthorZhen Lin
    • CommentTimeOct 20th 2015
    • (edited Oct 20th 2015)

    Huh. Interesting. That argument works in any combinatorial model category in which the trivial fibrations are closed under filtered colimits. So that rules out a lot of Cisinski model categories.

    • CommentRowNumber14.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 20th 2015

    @ZhenLin: Yes, and more generally, using the same argument, weak equivalences are closed under transfinite compositions whenever maps with a right lifting property with respect to all cofibrations between compact objects are weak equivalences. (Note that this is weaker than saying that such cofibrations generate all cofibrations.)

1 to 14 of 14

  1.