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Stream: learning: questions

Topic: Tannakian categories

John Baez (Apr 16 2023 at 22:24):

Can we describe Tannakian categories without ever mentioning limits (e.g. kernels)? Usually a Tannakian category is defined to be something like a symmetric monoidal abelian $k$-linear category $C$ with duals for objects such that there exists a faithful exact functor $C \to \mathsf{Vect}_K$ where $K$ is some field extension of $k$.

John Baez (Apr 16 2023 at 22:25):

But I would like to get rid of "abelianness" and "exactness" here and replace them with concepts that only involve colimits.

John Baez (Apr 16 2023 at 22:26):

I think we can at least do this for semisimple Tannakian categories, though I might be confused about even that.

John Baez (Apr 16 2023 at 22:27):

The reason I want to do this is that Jim Dolan finds it unaesthetic to talk about doctrines that involve both limits and colimits, in cases where colimits are enough - and I know examples where this philosophy works well.

Morgan Rogers (he/him) (Apr 17 2023 at 08:10):

Is this a situation where the categories are nice enough to use a monadicity theorem to deduce the existence of limits from corresponding colimits, dually to what one does in topos theory? I don't know how you would deduce the coincidence or exactness properties from such a result even if it is possible, though, it will require some work.

John Baez (Apr 18 2023 at 02:33):

I don't know enough about using a monadicity theorem to deduce the existence of limits from colimits to answer you.

Morgan Rogers (he/him) (Apr 18 2023 at 12:05):

In topos theory, the power object functor (exponentiation by the subobject classifier $\Omega$) gives a functor from a topos $\mathcal{E}$ to $\mathcal{E}^{\mathrm{op}}$. This functor is monadic, as is the functor obtained by dualizing both categories (which happens to be its adjoint). Monadic functor create limits, which ensures that $\mathcal{E}^{\mathrm{op}}$ has limits as soon as $\mathcal{E}$ does, but limits in $\mathcal{E}^{\mathrm{op}}$ are colimits in $\mathcal{E}$. That's why we only need to assume the existence of finite limits, exponentials and a subobject classifier in an elementary topos, and we get finite colimits for free!

In your set-up, it might be possible to use exponentials (expressed as tensor with a dual object) to obtain a monadic functor in a similar way, but you might have to be more creative about it.

John Baez (Apr 18 2023 at 16:35):

Morgan Rogers (he/him) said:

In topos theory, the power object functor (exponentiation by the subobject classifier $\Omega$) gives a functor from a topos $\mathcal{E}$ to $\mathcal{E}^{\mathrm{op}}$. This functor is monadic, as is the functor obtained by dualizing both categories (which happens to be its adjoint). Monadic functors create limits, which ensures that $\mathcal{E}^{\mathrm{op}}$ has limits as soon as $\mathcal{E}$ does, but limits in $\mathcal{E}^{\mathrm{op}}$ are colimits in $\mathcal{E}$. That's why we only need to assume the existence of finite limits, exponentials and a subobject classifier in an elementary topos, and we get finite colimits for free!

Wow, thanks! I never understood this aspect of topos theory, probably because I bumped into it when I wasn't yet ready for it and gave up thinking "this is some fancy stuff I will ignore". But since then I've thought a lot about $2^- : \mathsf{Set} \to \mathsf{Set}^{\rm{op}}$, why it's monadic, and what that says about complete atomic boolean algebras. So now, even though I don't know why monadic functors create limits, I see the elegance of this argument.