Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: what are the main properties of monadic functors?

John Baez (May 04 2023 at 20:38):

I guess I have two questions: what are the main (simplest or most important) properties of monadic functors in general, and also monadic functors to $\mathsf{Set}$?

John Baez (May 04 2023 at 20:41):

The monadicity theorem is one answer to this question, but I'm more interested in necessary conditions for a functor to be monadic than sufficient ones.

John Baez (May 04 2023 at 20:56):

I've just rewritten the nLab article Monadic functors: basic properties to highlight some of the basic properties of monadic functors. Namely:

• they are [[faithful]]
• they are [[conservative]]
• they create all limits that exist in their codomain
• they create all colimits that exist in their codomain and are preserved by the corresponding monad.

John Baez (May 04 2023 at 20:56):

They are full on isomorphisms, too - right?

Kevin Arlin (May 04 2023 at 20:58):

A functor that's strictly monadic, i.e. isomorphic to the category of algebras for a monad, is an isofibration, which generalizes the fact that you can transfer a group structure along a bijection of sets, and that rhymes a bit with being full on isomorphisms. But it's not true that every bijection of groups is an isomorphism, so they aren't full on isos.

John Baez (May 04 2023 at 21:00):

Aargh, what a blunder. Indeed, I was trying to take "isofibration" and weaken it to get something that's true for all monadic functors, not just strictly monadic ones. But I failed completely.

Kevin Arlin (May 04 2023 at 21:01):

Isofibrations are weird! Every functor's equivalent to an isofibration, so it doesn't tell you anything about non-strict monadic guys.

John Baez (May 04 2023 at 21:02):

Here's what I was actually trying to say: given a monadic functor $U: D \to C$ and an isomorphism in $C$, there is some isomorphism in $D$ that maps to it. This is weaker than "full on isomorphisms". But now I think this is false, too.

John Baez (May 04 2023 at 21:05):

I now think if I finally say the true thing that resembles what I was thinking, I might say $U$ is equivalent to an isofibration. But this is vacuous, as you point out.

John Baez (May 04 2023 at 21:25):

What I was trying to say, all jargon aside, is that given a monad $T$ on $C$ and a $T$-algebra structure on $c \in C$, you can transport this $T$-algebra structure along any isomorphism $f: c \to c'$ to get a $T$-algebra structure on $c'$. But this just says that the forgetful functor $T \mathrm{Alg} \to C$ is an isofibration. And this says that any strict monadic functor is an isofibration. Apparently it says nothing at all about general monadic functors!

Kevin Arlin (May 04 2023 at 21:39):

Something I think is less well known about monadic categories over set is that they're exactly the Barr exact (i.e. internal equivalence relations work decently) cocomplete categories that have a "projective generator" $G$, which is like the free widget on one element in that maps out lift against epimorphisms and each widget admits an epi from some power of $G.$ Then the functor represented by the $G$ is monadic.

Kevin Arlin (May 04 2023 at 21:40):

...and thus $G$ really is a free widget on one element!

John Baez (May 04 2023 at 21:51):

Interesting! Hmm, the nLab doesn't seem to have that yet; apparently it only says any category monadic over a power of $\mathsf{Set}$ is Barr exact.

Kevin Arlin (May 04 2023 at 22:10):

Yeah, I learned it from this paper of Vitale: http://www.numdam.org/article/CTGDC_1994__35_4_351_0.pdf. I should add it to nLab sometime. I think you can actually characterize categories monadic over a power of set, which are very general multisorted algebraic theories, by just allowing a small family of projective generators in the characterization for Set.

John Baez (May 04 2023 at 23:16):

Yes, please add it, and that reference... it's a nice fact!

John Baez (May 05 2023 at 14:31):

Nobody really answered my question so I'll ask it again. Are there some interesting properties of monadic functors other than these:

• they are [[faithful]]
• they are [[conservative]]
• they create all limits that exist in their codomain
• they create all colimits that exist in their codomain and are preserved by the corresponding monad.

and those implicit in Beck's [[monadicity theorem]]?

Morgan Rogers (he/him) (May 05 2023 at 14:38):

A negative answer: I've thought about monadicity a bunch and haven't encountered any properties beyond the ones you just listed. On the other hand, I haven't spent so much time figuring out the properties of monads that weren't either monads on Set or idempotent, so I'm sure there could be some further properties lurking around which are hidden by the relative simplicity of those cases.

John Baez (May 05 2023 at 14:41):

A negative answer like yours is still helpful! I have always avoided thinking about properties of monadic functors until now, when it turns out that to understand the representation theory of Clifford algebras and its connection to symmetric spaces I need to understand essential fibers of monadic functors! (Who knew?)

Mike Shulman (May 05 2023 at 15:09):

Every monadic functor over Set is a [[solid functor]], but I don't think that's true over arbitrary base categories.

John Baez (May 05 2023 at 15:38):

John Baez said:

Yes, please add it, and that reference... it's a nice fact!

Too late, I added it:

• Monadic functor: monadic functors to Set.

Kevin Arlin (May 05 2023 at 15:48):

You're quick! I fixed the link to "solid in physics" to be to "solid functor".

John Baez (May 05 2023 at 15:50):

Quick? It was almost 24 hours! Any longer and I would have forgotten. :upside_down:

Thanks for fixing that.