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Stream: learning: questions

Topic: Oppositization functor for Prof

Asad Saeeduddin (Jul 16 2020 at 09:42):

In $\mathbf{Cat}$, we have a mapping of 0-cells given by taking the opposite of categories, a mapping of 1-cells given by taking the opposite of functors (the action on objects and morphisms of the resulting functor is identical), and a mapping of 2-cells given by taking the opposite natural transformation (the families of morphisms selected by the resulting natural transformation are identical). Taken together, this gives us a 2-functor $(-)^{\mathbf{op}} : \mathbf{Cat^{co}} \to \mathbf{Cat}$, where $\mathbf{Cat^{co}}$ is a 2-category similar to $\mathbf{Cat}$, except with a 2-cell $\eta : F \to G$ taken to mean a natural transformation from $G$ to $F$.

Is it possible to extend this concept to 2-categories besides $\mathbf{Cat}$? E.g. there should be a similar pseudofunctor for the bicategory $\mathbf{Prof}$, no?

Nathanael Arkor (Jul 16 2020 at 10:36):

Yes, this is called a duality involution. Mike Shulman's Contravariance through enrichment could be a good place to start if you want to learn more.

Morgan Rogers (he/him) (Jul 16 2020 at 10:38):

I have a related question about toposes, since we can apply the presheaf construction to the "op" 2-functor to get a bifunctor from the bicategory of presheaf topos and essential geometric morphisms to the same bicategory with 2-cells reversed. I tried for a short while with @Riccardo Zanfa to extend this to a broader duality for Grothendieck toposes and geometric morphisms, but this work is far from complete; I wonder if anyone has attempted this before with any success?

Asad Saeeduddin (Jul 20 2020 at 12:45):

@Nathanael Arkor Thank you. Does this mean there is indeed an analogous pseudofunctor for Prof?

Asad Saeeduddin (Jul 20 2020 at 12:45):

Having not yet understood the contents of the paper you linked I can't tell right off the bat whether this follows

Asad Saeeduddin (Jul 20 2020 at 12:52):

My overall goal is to figure out what the correct $\mathbf{Prof}^{\mathbf{co}}$ would be in order to have such an involution, since I have other business with that bicategory once I know what it is

Nathanael Arkor (Jul 20 2020 at 12:52):

Map(Prof) has a duality involution because Prof is a compact closed bicategory (see the second paragraph of page 5 of ibid.), which suggests Prof itself may not (at least as a bicategory).

Nathanael Arkor (Jul 20 2020 at 12:54):

But it seems plausible that Prof could have a duality involution as a pseudo-double category/proarrow equipment. I'm not sure if this has been worked out. Weber's Yoneda structures from 2-toposes has some suggestive formulations (e.g. see Definition 2.14).

Reid Barton (Jul 20 2020 at 13:09):

I think there's a different kind of duality $(-)^{\mathbf{op}} : \mathbf{Prof}^\mathbf{op} \to \mathbf{Prof}$. (If you represent a profunctor from $C$ to $D$ as a functor $C \times D^\mathbf{op} \to \mathrm{Set}$, then you can also regard it as a functor $D^\mathbf{op} \times C \to \mathrm{Set}$, that is, a profunctor from $D^\mathbf{op}$ to $C^\mathbf{op}$.)

Reid Barton (Jul 20 2020 at 13:12):

Or a fancier perspective is that $C$ and $C^\mathbf{op}$ are actually dual objects in $\mathbf{Prof}$, and this operation is the duality functor.

Asad Saeeduddin (Jul 20 2020 at 13:14):

@Reid Barton Does that mean you don't turn around Set to get Set^op though?

Reid Barton (Jul 20 2020 at 13:14):

All I did was switch the order of the arguments

Reid Barton (Jul 20 2020 at 13:15):

In general $C \times D^\mathbf{op}$ and $D^\mathbf{op} \times C$ are equivalent categories (of course), while $C \times D^\mathbf{op}$ and $C^\mathbf{op} \times D$ are not

Reid Barton (Jul 20 2020 at 13:17):

So you can have an identification $\mathbf{Prof}(C, D) = \mathbf{Prof}(D^\mathbf{op}, C^\mathbf{op})$ but not an identification $\mathbf{Prof}(C, D) = \mathbf{Prof}(C^\mathbf{op}, D^\mathbf{op})$

Asad Saeeduddin (Jul 20 2020 at 13:18):

I see. I think this is the mistake I was making. I was taking the opposite functor of the original profunctor

Asad Saeeduddin (Jul 20 2020 at 13:18):

as we do in $\mathbf{Cat}$

Asad Saeeduddin (Jul 20 2020 at 13:49):

Ok, so we can't get a functor $\mathbf{Prof}^\mathbf{co} \to \mathbf{Prof}$, because the hom-mapping doesn't work without flipping around the 1-cells, as @Reid Barton mentioned. But can we get $\mathbf{Prof}^\mathbf{coop} \to \mathbf{Prof}$, where the 2-cells are flipped around as well?

Reid Barton (Jul 20 2020 at 15:55):

The Hom-categories of Prof are presheaf categories, and so their opposites won't be presheaf categories (except in degenerate situations)