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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 20th 2015
   • (edited Oct 20th 2015)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexFrom that [discussion](http://nforum.ncatlab.org/discussion/6773/adjoint-logic/) on temporal logic, I was thinking about the comparison of 'henceforth' with necessarily, and why doing base change and dependent product seemed different for $Time_1 \rightrightarrows Time_0$ and $Worlds \to \ast$. I concluded that we can reformulate the latter as pullback $PairsAccessibleWorlds \rightrightarrows Worlds$ 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, $Time_0 \to ??$. Equivalence relations may be treated either way, but not in general. But then when it comes to representation theory, wouldn't the analogy with $Worlds$ go: * take the epimorphism $Worlds \to \ast$, then pullback to $PairsAccessibleWorlds \rightrightarrows Worlds$ * take the epimorphism $\ast \to B G$, then pullback to $G \rightrightarrows \ast$ 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 $Worlds \to \ast$ it's not $\ast \to B G$, but $B G \to \ast$ that I care about.

   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 $Time_1 \rightrightarrows Time_0$ and $Worlds \to \ast$. I concluded that we can reformulate the latter as pullback $PairsAccessibleWorlds \rightrightarrows Worlds$ 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, $Time_0 \to ??$. Equivalence relations may be treated either way, but not in general.

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

   • take the epimorphism $Worlds \to \ast$, then pullback to $PairsAccessibleWorlds \rightrightarrows Worlds$
   • take the epimorphism $\ast \to B G$, then pullback to $G \rightrightarrows \ast$

   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 $Worlds \to \ast$ it’s not $\ast \to B G$, but $B G \to \ast$ that I care about.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 20th 2015
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexCan we not have an epi $Worlds \to Time_0$, and then induce the groupoid with objects $Worlds$ from $Time_1\rightrightarrows Time_0$?

   Can we not have an epi $Worlds \to Time_0$, and then induce the groupoid with objects $Worlds$ from $Time_1\rightrightarrows Time_0$?