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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 16th 2018

    I have spelled out an argument (here) for the statement that in a local topos the full subcategory of concrete objects provides a factorization of ΓcoDisc\Gamma \dashv coDisc as

    ΓcoDisc:HAAconcAA AAι concAAH concAAA AAASet \Gamma \;\dashv\; coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } \mathbf{H}_{conc} \array{ \overset{\phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AAA}}{\hookleftarrow} } Set

    diff, v5, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2018
    • (edited Jun 22nd 2018)

    the proof of that Proposition (still here) was lacking the argument that im(η X )im( \eta^{\sharp}_X ) is indeed concrete. Have added that now.

    diff, v7, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 6th 2021

    I have added (here) an observation that in a cohesive \infty-topos concrete objects are such that cohesive maps to them “glue”, in that every cohesive map out of a cover whose map of underlying \infty-groupoids descends down the cover also descends as a cohesive map.

    This is an immediate consequence of the definitions and the (-1)-connected/(-1)-truncated orthogonality. The point is just to observe that this lifting problem does have this interpretation for concrete objects.

    diff, v11, current