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Stream: learning: questions

Topic: Copresheaves

Keith Elliott Peterson (Sep 05 2024 at 04:29):

Why is it that the category of copresheaves of some $\mathcal{C}$ is sometimes given as $[\mathcal{C},\mathrm{Set}]$ and other times as $[\mathcal{C},\mathrm{Set}]^\mathrm{op}$?

Joshua Meyers (Sep 05 2024 at 05:12):

Where is it given as the latter? I've only ever seen the former

Nathanael Arkor (Sep 05 2024 at 05:31):

The appropriate definition is typically the latter, because it has a universal property with respect to C: it is the free completion under small limits.

Nathanael Arkor (Sep 05 2024 at 05:36):

As to why some people use a different convention, I imagine many people aren't aware of the reason to use the latter convention, and the former seems more intuitive.

Mike Shulman (Sep 05 2024 at 08:32):

Interesting. I would refer to $[C,\rm Set]^{\rm op}$ as the "co-presheaf category" of $C$ for the reasons given by @Nathanael Arkor . But in my head this is bracketed as "co-(presheaf category)", and I would not call its objects "copresheaves" or refer to it as the "category of copresheaves" -- to me, those would refer to $[C,\rm Set]$.

Josselin Poiret (Sep 05 2024 at 09:42):

the latter is also the natural counterpoint of $Psh(C)$ for Isbell duality, precisely because you have the co-yoneda embedding $C \to [C, \mathrm{Set}]^{\mathrm{op}}$

Morgan Rogers (he/him) (Sep 05 2024 at 11:04):

Using "copresheaves on $C$" to mean $[C,\mathrm{Set}]$ can be justified by the convention that the "co-" prefix refers to what we get by applying a given construction to $C^{\mathrm{op}}$.

Nathanael Arkor (Sep 05 2024 at 11:13):

I would also tend to use "copresheaf category" rather than "category of copresheaves", but I do think it would be confusing to use both these terms in a single paper, but with different meanings.

Mike Shulman (Sep 05 2024 at 14:52):

Yes, I agree. But in separate papers it could be okay.

Mike Shulman (Sep 05 2024 at 14:54):

Actually I don't really like the word "copresheaf" at all -- I prefer "Set-valued functor". It annoys me sufficiently already that we have to have a separate word for "contravariant Set-valued functor" and that that word is derogated by a prefix like "pre-"; why do we then have to double-dualize to talk about covariant functors? (-:O

Nathanael Arkor (Sep 05 2024 at 14:56):

In the enriched context, presheaves and copresheaves are not necessarily functors, so I do think it is useful to have separate terms. (Though whether "presheaf" and "copresheaf" are good terms is a different question...)

Nathanael Arkor (Sep 05 2024 at 14:59):

Maybe one could use "left-distributor" and "right-distributor" in analogy with "left-module" and "right-module" :upside_down:

Mike Shulman (Sep 05 2024 at 15:00):

Or one could just say "left module" and "right module". (But then one has to remember which is "left" and which is "right"...)

Nathanael Arkor (Sep 05 2024 at 15:00):

That's true, but I personally do find it helpful to have a specific term for modules of categories.

Nathanael Arkor (Sep 05 2024 at 15:01):

But maybe that's simply a habit to be broken.

Josselin Poiret (Sep 05 2024 at 15:46):

covariant vs. contravariant rather than left vs. right?