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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2018
    • (edited Jun 28th 2018)

    I have spelled out statement and proof of:

    1. every reflection is a reflective localization,

    2. reflective localizations are given by full subcategories of local objects.

    I have also tried to produce decent cross-links to various entries that mention something related. But there will be more left to do.

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2018

    Conversely, when is reflection onto a full subcategory of local objects also a localization in the sense of universally inverting the given morphisms?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 29th 2018

    I have to puzzle through the terminology – you’re asking if, given a category with weak equivalences (C,W)(C,W) such that the full subcategory Loc WLoc_W of WW-local objects in reflective, whether the reflector CLoc WC \to Loc_W is equivalent to the localization functor CC[W 1]C\to C[W^{-1}] that universally inverts WW?

    By property (1), it seems that this is equivalent to asking whether every functor CDC\to D that inverts WW also inverts every WW-local equivalence (morphism that is seen as an isomorphism by every WW-local object)?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2018
    • (edited Jun 29th 2018)

    Yes, for WW any class of morphisms. In “Modalities in homotopy type theory” you write: “We will not be concerned here with the universal property of the localized category”. Suppose you were concerned, what could you say?

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJun 29th 2018
    • (edited Jun 29th 2018)

    Well, it is true in general that every left adjoint F:CDF:C\to D that inverts WW also inverts every WW-local equivalence. For if GG is its right adjoint, then FF inverting WW means that GG takes values in WW-local objects, which in turn implies that FF preserves WW-local equivalences. Since the reflector CLoc WC\to Loc_W is also a left adjoint, it follows that it is the coinverter of WW in the 2-category of categories and left adjoints (which contains, in particular, the 2-category of locally presentable categories and left adjoints).

    However, for non-adjoints I think it is false; here is a proposed counterexample. Let PP be a poset with binary meets but no top element, and let C=P C = P^\triangleright be PP with a new top element \top adjoined. Take W=MorPMorCW = Mor P \subsetneq Mor C, i.e. every inequality xyx\le y in CC where yy\neq \top. Then for any xPx\in P there is a yPy\in P with yxy\nleq x (since PP has no top element), but xyxx\wedge y \le x, so xx does not invert the weak equivalence xyyx\wedge y \le y; thus no element of PP is WW-local. But of course \top is WW-local, so Loc W={}Loc_W = \{\top\} is contractible and reflective. However, C[W 1]=P[W 1] C[W^{-1}] = P[W^{-1}]^\triangleright, which is not contractible unless P=P = \emptyset; thus they are inequivalent.

    • CommentRowNumber6.
    • CommentAuthorRichard Williamson
    • CommentTimeJun 30th 2018
    • (edited Jun 30th 2018)

    Have not read the thread or page closely, so might be missing something, but is this not just a question of whether WW is saturated? If I recall correctly, it is an if and only if.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 30th 2018
    • (edited Jun 30th 2018)

    Hi Mike, thanks for the bit about left adjoints. I was just wondering about this being the right condition, since this is how Hovey’s book defines localization.

    Richard, yes, every reflector is the localization at the morphisms it inverts, but I was hoping for a tighter condition. I suppose localization by left adjoins is useful here.

  1. Yes, perhaps one could say that one is asking for conditions to ensure saturation. The one at saturated class of morphisms is no doubt already familiar to you.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 30th 2018

    Okay, I have recorded the statement about left-adjoint localization, here.

  2. arrow direction

    Tim Richter

    diff, v8, current

  3. typo

    Tim Richter

    diff, v8, current

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