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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)D:A\to (X\downarrow C) be a κ\kappa-good SS-tree rooted at XX for a collection of morphisms SS in CC, where κ\kappa 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\kappa-small downward-closed BAB\subseteq A, the colimit of the restricted diagram, varinjlimD| B\varinjlim D|_B is κ\kappa-compact in (XC)(X\downarrow C).

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

    Definitions:


    For your convenience, here are the definitions:

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

    Recall that an SS-tree rooted at XX for a collection of morphisms SS in CC consists of the following data:

    • An object XX in C (the root)
    • A partially ordered set AA whose order structure is well-founded (the index)
    • A diagram D:A(XC)D:A\to (X\downarrow C) such that given any element αA\alpha\in A, the canonical map
    varinjlimD| {β:β<α}D(α)\varinjlim D|_{\{\beta:\beta &lt; \alpha\}}\to D(\alpha)

    is the pushout of some map U αV αSU_\alpha\to V_\alpha\in S.

    We say that an SS-tree is κ\kappa-good if for all of the morphisms U αV αU_\alpha\to V_\alpha above, U αU_\alpha and V αV_\alpha are κ\kappa-compact, and such that for any αA\alpha\in A, the subset {β:β\{\beta: \beta < α}A\alpha \}\subseteq A is κ\kappa-small.

    Edit: It’s easy to reduce the proof to showing that D(α)D(\alpha) is κ\kappa-compact, since projective limits of diagrams BSetB\to Set are |Arr(B)||Arr(B)|-accessible (and therefore κ\kappa-accessible since BB is κ\kappa-small), we perform the computation for II a κ\kappa-filtered poset, and F:ICF:I\to C, assuming that D(α)D(\alpha) is κ\kappa-compact for all αB\alpha\in B:

    varinjlim IHom C(varinjlim BD,F) varinjlim Ivarprojlim B opHom C(D,F) varprojlim B opvarinjlim IHom C(D,F) varprojlim B opHom C(D,varinjlim IF) Hom C(varinjlim BD,varinjlim IF)\begin{matrix} \varinjlim_I Hom_C(\varinjlim_B D, F)&\cong& \varinjlim_I\varprojlim_{B^{op}} Hom_C(D,F)\\ &\cong& \varprojlim_{B^{op}} \varinjlim_I Hom_C(D,F)\\ &\cong& \varprojlim_{B^{op}} Hom_C(D,\varinjlim_IF)\\ &\cong& Hom_C(\varinjlim_B D,\varinjlim_IF) \end{matrix}

    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.