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• CommentRowNumber1.
• CommentAuthorDavidRoberts
• CommentTimeJun 23rd 2018

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.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJun 24th 2018

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.

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeJun 24th 2018

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.