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.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 6th 2020

    Created.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 6th 2020

    Added a definition.

    v1, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 6th 2020

    Relation to model categories.

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeMay 6th 2020

    I fixed some formatting.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 6th 2020

    So what are they for?

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeMay 6th 2020

    I quote the abstract of one of Simon Henry’s papers:

    We introduce a notion of “weak model category” which is a weakening of the notion of Quillen model category, still sufficient to define a homotopy category, Quillen adjunctions, Quillen equivalences and most of the usual construction of categorical homotopy theory. Both left and right semi-model categories are weak model categories, and the opposite of a weak model category is again a weak model category. The main advantages of weak model categories is that they are easier to construct than Quillen model categories. In particular we give some simple criteria on two weak factorization systems for them to form a weak model category. The theory is developed in a very weak constructive framework and we use it to produce, completely constructively (even predicatively), weak versions of various standard model categories, including the Kan-Quillen model structure, the variant of the Joyal model structure on marked simplicial sets, and the Verity model structure for weak complicial sets. We also construct semi-simplicial versions of all these.

    I have not looked at the paper at all, and have no opinion on the contents, although the abstract does look quite interesting.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2020

    Sure, there is the references cited. But what David C. probably meant, certainly what I would mean, is that a sentence or two on the purpose of the definition is missing in the entry.

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 6th 2020

    Added an idea section.

    diff, v6, current

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 7th 2020

    Thanks.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2020

    The second paragraph of the Idea section, as of now, may need adjustment. It reads:

    In particular, one can construct left and right Bousfield localizations in this setting without assuming the model category to be left proper respectively right proper.

    Since “this setting” must be that of weak model categories, should “assuming the model category…” rather be “assuming the weak model category…”?

    And even with that, I feel something is missing: Continuing from the previous sentence and following “In particular” we need to first state examples of constructions of model category theory that exists also in the weak case. Then as an addeddum to “in particular” we can list extra advantages gained, such as no need for properness.

    • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 8th 2020

    Attempted to clarify the idea section.

    diff, v9, current

    • CommentRowNumber12.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 8th 2020

    redirects

    diff, v9, current

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 8th 2020

    Split overly-long sentence in second paragraph in two.

    diff, v10, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2020
    • (edited May 8th 2020)

    Thanks, Dmitri, thanks David. But allow me to make a suggestion for how to phrase it:

    The concept of weak model categories is a relaxation of that of model categories, even weaker than the concept of semimodel categories, but such that it still allows for a rich theory largely analogous to that of actual model categories:

    The weak analogue of the construction of the homotopy category of a model category still exists, as do notions of Quillen adjunction and Quillen equivalence.

    Also, for example, an analogue of left or right Bousfield localization of model categories still makes sense for weak model categories; and, as a bonus in contrast to the usual case, it does not require the assumption of left or right properness.

    diff, v11, current

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 8th 2020

    Added a criterion for constructing weak model structures.

    diff, v12, current

    • CommentRowNumber16.
    • CommentAuthorHurkyl
    • CommentTimeMay 9th 2020

    Added related concepts: model category and premodel category.

    diff, v13, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJun 12th 2020

    added the publication data for

    • Simon Henry, Weak model categories in classical and constructive mathematics, Theory and Applications of Categories, Vol. 35, 2020, No. 24, pp 875-958. (arXiv:1807.02650)

    that just came in. Also fixed the hyperlinking of the references in the text.

    diff, v14, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2021

    added mentioning (here) of the example of semi-simplicial sets

    diff, v15, current