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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 12th 2021

    Created:

    Idea

    Partial model categories are one of the many intermediate notions between relative categories and model categories.

    They axiomatize those properties of model categories that only involve weak equivalences.

    Definition

    A partial model category is a relative category such that its class of weak equivalences satisfies the 2-out-of-6 property (if sr and ts are weak equivalences, then so are r, s, t, tsr) and admits a 3-arrow calculus, i.e., there are subcategories U and V (which can be thought of as analogues of acyclic cofibrations and acyclic fibrations) such that U is closed under cobase changes (which are required to exist), V is closed under base changes, and any morphism can be functorially factored as the composition vu for some uU and vV.

    Properties

    If (C,W) is a partial model category, then any Reedy fibrant replacement of the Rezk nerve N(C,W) is a complete Segal space.

    Related concepts

    References

    v1, current

    • CommentRowNumber2.
    • CommentAuthorHurkyl
    • CommentTimeJul 12th 2021

    Functoral factorization into vu is for weak equivalences only.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorHurkyl
    • CommentTimeJul 12th 2021

    Clarified U,V are subcategories of the weak equivalences, and that (co)base changes are along arbitrary morphisms, not just weak equivlaences.

    diff, v2, current