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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 20th 2010
    • (edited Jun 20th 2010)

    I posted this as a question over at MO, but I figured I’d post it here as well:

    Question:


    Let D:A(XC) be a κ-good S-tree rooted at X for a collection of morphisms S in C, where κ is a fixed uncountable regular cardinal. Then according to the proof of Lemma A.1.5.8 of Higher Topos Theory by Lurie, for any κ-small downward-closed BA, the colimit of the restricted diagram, varinjlimD|B is κ-compact in (XC).

    Why is this true? (It is stated without proof.)

    Definitions:


    For your convenience, here are the definitions:

    Recall that an object X in C is called κ-compact if hX():=Hom(X,) preserves all κ-filtered colimits (where κ-filtered means “< κ“-filtered, since the terminology is different depending on the source).

    Recall that an S-tree rooted at X for a collection of morphisms S in C consists of the following data:

    • An object X in C (the root)
    • A partially ordered set A whose order structure is well-founded (the index)
    • A diagram D:A(XC) such that given any element αA, the canonical map
    varinjlimD|{β:β<α}D(α)

    is the pushout of some map UαVαS.

    We say that an S-tree is κ-good if for all of the morphisms UαVα above, Uα and Vα are κ-compact, and such that for any αA, the subset {β:β < α}A is κ-small.

    Edit: It’s easy to reduce the proof to showing that D(α) is κ-compact, since projective limits of diagrams BSet are |Arr(B)|-accessible (and therefore κ-accessible since B is κ-small), we perform the computation for I a κ-filtered poset, and F:IC, assuming that D(α) is κ-compact for all αB:

    varinjlimIHomC(varinjlimBD,F)varinjlimIvarprojlimBopHomC(D,F)varprojlimBopvarinjlimIHomC(D,F)varprojlimBopHomC(D,varinjlimIF)HomC(varinjlimBD,varinjlimIF)

    Edit 2: I think the above reduction actually won’t work, since it doesn’t use the hypothesis that B is downward-closed.

    nForum Edit 1: For some reason, \varinjlim, \varprojlim, and \cdot don’t render.