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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 , where the vertical arrows are “functions” (i.e. maps = left adjoints) in , horizontal arrows are “relations” or arbitrary morphisms in , and the 2-cells are inequalities of type (viewed as morphisms in the horizontal category). He says that the horizontal category is again an allegory, where the opposite of is . 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 where is a category. Apparently the objects are functors and morphisms are families 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 . 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 treated as a horizontally discrete double category.