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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 20th 2015
    • (edited Oct 20th 2015)

    From that discussion on temporal logic, I was thinking about the comparison of ’henceforth’ with necessarily, and why doing base change and dependent product seemed different for Time 1Time 0Time_1 \rightrightarrows Time_0 and Worlds*Worlds \to \ast. I concluded that we can reformulate the latter as pullback PairsAccessibleWorldsWorldsPairsAccessibleWorlds \rightrightarrows Worlds and the comparison is much closer.

    So then we seem to have greater flexibility with the latter in the sense that I can’t do the projection onto classes trick with time, Time 0??Time_0 \to ??. Equivalence relations may be treated either way, but not in general.

    But then when it comes to representation theory, wouldn’t the analogy with WorldsWorlds go:

    • take the epimorphism Worlds*Worlds \to \ast, then pullback to PairsAccessibleWorldsWorldsPairsAccessibleWorlds \rightrightarrows Worlds
    • take the epimorphism *BG\ast \to B G, then pullback to G*G \rightrightarrows \ast

    But I’m not going to want to base change from *\ast in the latter case, just because there is a disanalogy in that unlike Worlds*Worlds \to \ast it’s not *BG\ast \to B G, but BG*B G \to \ast that I care about.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 20th 2015

    Can we not have an epi WorldsTime 0Worlds \to Time_0, and then induce the groupoid with objects WorldsWorlds from Time 1Time 0Time_1\rightrightarrows Time_0?