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Stream: learning: questions

Topic: on the relation between twisted and untwisted arrows

fosco (May 06 2021 at 16:06):

There is something I have always found remarkable: consider the following construction.

Let $\mathcal C$ be a category, and the functor $K : \mathcal C \to {\sf Cat}$ sending $C$ into the slice category $\mathcal C/C$. More formally, this can be regarded as the functor sending $C$ to the category of elements of the representable presheaf on $C$.

Now, building the category of elements of $K$ one gets the category of arrows over $\mathcal C$, and the associated fibration is just the codomain fibration.

Very good! Now let's tweak this construction a little bit, and let's postcompose $K$ with the opping endofunctor of $\sf Cat$, sending $C \in \mathcal C$ to the opposite of the slice over C, i.e. $(\mathcal C/C)^\text{op}$. What we get is if we take the Grothendieck construction now is... the twisted arrow category of $\mathcal C$! Something so different, changing so little.

The most intuitive explanation I can give to myself about what's going on is that the morphisms in the total category of the Grothendieck construction of a prestack $F : \mathcal C \to \sf Cat$, and the morphisms in each fiber subcategory $F_C\subseteq {\cal E}(F)$ are of different kinds. Opping the latter does not result in some meaningful opping on the morphisms of the former.

As elementary and neat as this explanation is, I can't call it satsifying: I'd like to state a more general relation between the Grothendieck construction of a Grothendieck construction "performed fiberwise", and the Grothendieck construction "done just once".

What I have in mind is a theorem on the following lines:

Theorem Let $P : {\cal C}\times{\cal X}^\text{op} \to \sf Set$ be a profunctor; let $K_P : {\cal C} \to \sf Cat$ be the functor sending $C$ to the category of elements of the presheaf $P(C,-) : {\cal X}^\text{op}\to \sf Set$. Consider the category $\int K_P$, obtained taking the Grothendieck construction of $K_P$: each fiber of this category over C will be $\int P(C,-)$. On the other hand, consider the category of elements of $P$, done all at once. This is also called the "collage" of $\cal A,X$ along $P$. Then, $\int P$ and $\int K_P$ stand in the following relation: [...fill in the dots...]

In case $P=\hom$ what we get is that $\int K_{\hom}$ is the arrow category, and $\int\hom$ is the twisted arrow category.