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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 23rd 2018
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   Author: DavidRoberts
   Format: MarkdownItexJust to drop a mention of [this video](https://youtu.be/VpNXEbDJSnQ?t=3m5s) I just discovered, a talk on allegories, by Bénabou. I have not watched it yet, but will do so in case there is something worth linking for, and will add it to relevant entries at that point.

   Just to drop a mention of this video I just discovered, a talk on allegories, by Bénabou. I have not watched it yet, but will do so in case there is something worth linking for, and will add it to relevant entries at that point.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 24th 2018
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   Author: Todd_Trimble
   Format: MarkdownItexI watched it; I'll need either to watch parts of it again or work out on my own things I didn't catch the first time (there may be lacunae in the recording). But one of the questions he focuses on is how to promote the 1-categorical equivalence between regular categories and unital tabular allegories to a 2-categorical equivalence, because it is not immediately clear what notion of 2-cell on the allegorical side corresponds to a natural transformation between regular functors. One of the key constructions he introduces is a double category of "admissible squares" attached to an allegory $\mathbf{A}$, where the vertical arrows are "functions" (i.e. maps = left adjoints) $f, g$ in $\mathbf{A}$, horizontal arrows are "relations" or arbitrary morphisms $r, s$ in $\mathbf{A}$, and the 2-cells are inequalities of type $g r \leq s f$ (viewed as morphisms $(r, s): f \to g$ in the horizontal category). He says that the horizontal category is again an allegory, where the opposite of $(r, s): f \to g$ is $(r^\ast, s^\ast): g \to f$. He also says this construction when applied to a (unital) tabular category is again (unital) tabular. I couldn't quite catch all of it, but he generalizes this last construction to something he denotes $\mathbf{A}^X$ where $X$ is a category. Apparently the objects are functors $X \to Map(\mathbf{A})$ and morphisms are families $r x: F x \to G x$ for which the "naturality squares" are not commutative, but are admissible (so in other words, lax natural). The horizontal category of the last paragraph would be the case $X = \mathbf{2}$. It sounded like he was suggesting this construction was relevant to the problem of defining appropriate 2-cells between allegory maps, but as I say I didn't catch how.

   I watched it; I’ll need either to watch parts of it again or work out on my own things I didn’t catch the first time (there may be lacunae in the recording). But one of the questions he focuses on is how to promote the 1-categorical equivalence between regular categories and unital tabular allegories to a 2-categorical equivalence, because it is not immediately clear what notion of 2-cell on the allegorical side corresponds to a natural transformation between regular functors. One of the key constructions he introduces is a double category of “admissible squares” attached to an allegory $\mathbf{A}$, where the vertical arrows are “functions” (i.e. maps = left adjoints) $f, g$ in $\mathbf{A}$, horizontal arrows are “relations” or arbitrary morphisms $r, s$ in $\mathbf{A}$, and the 2-cells are inequalities of type $g r \leq s f$ (viewed as morphisms $(r, s): f \to g$ in the horizontal category). He says that the horizontal category is again an allegory, where the opposite of $(r, s): f \to g$ is $(r^\ast, s^\ast): g \to f$. He also says this construction when applied to a (unital) tabular category is again (unital) tabular.

   I couldn’t quite catch all of it, but he generalizes this last construction to something he denotes $\mathbf{A}^X$ where $X$ is a category. Apparently the objects are functors $X \to Map(\mathbf{A})$ and morphisms are families $r x: F x \to G x$ for which the “naturality squares” are not commutative, but are admissible (so in other words, lax natural). The horizontal category of the last paragraph would be the case $X = \mathbf{2}$. It sounded like he was suggesting this construction was relevant to the problem of defining appropriate 2-cells between allegory maps, but as I say I didn’t catch how.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJun 24th 2018
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   Author: Mike Shulman
   Format: MarkdownItexExcellent evidence that allegories should really be considered as double categories, not 2-categories. I rediscovered this notion of allegory transformation in [exact completions and small sheaves](http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html) (Definition 6.7). I think that "functor allegory" is also naturally a functor double category, with $X$ treated as a horizontally discrete double category.

   Excellent evidence that allegories should really be considered as double categories, not 2-categories. I rediscovered this notion of allegory transformation in exact completions and small sheaves (Definition 6.7). I think that “functor allegory” is also naturally a functor double category, with $X$ treated as a horizontally discrete double category.