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Stream: theory: category theory

Topic: monoid in a pseudomonoid

Joe Moeller (May 05 2020 at 18:38):

Monoids are pretty common things. A monoidal category is the perfect setting to define monoid objects. Groups are fancy monoids. The internal definition for groups requires some notion of duplication and deletion, which you naturally get in a cartesian monoidal category. Pseudomonoids in a monoidal 2-category are a categorification of this story. A pseudomonoid in Cat is a monoidal category, just as a monoid is Set is an ordinary monoid. So a natural question I think is what exactly is a monoid in a pseudomonoid? Here's the beginning of my attempt to figure out what it should be. But I'm disturbed by duplication appearing without deletion. So I have two questions:
1) has someone already worked out what a monoid in a pseudomonoid is?
2) for some reason, duplication appears in my definition, but not deletion yet. is this indicating that I made some mistake in my unit, or is this lopsidedness the right thing?
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Mike Shulman (May 05 2020 at 18:43):

I would define a monoid in a pseudomonoid $X$ to be a lax morphism of pseudomonoids $U\to X$. This doesn't require duplication or deletion.

Reid Barton (May 05 2020 at 18:44):

I wrote out my guess before I looked at your diagram and I think the source of the multiplication should be $U = U \boxtimes U \xrightarrow{x \boxtimes x} X \boxtimes X \xrightarrow{\otimes} X$.

Reid Barton (May 05 2020 at 18:44):

Unfortunately my notation on paper is therefore completely different than yours, so hopefully I transcribed it correctly.

Reid Barton (May 05 2020 at 18:49):

I guess the point is that, while I can't "duplicate $X$", I can "duplicate $U$".

Joe Moeller (May 05 2020 at 18:50):

Mike Shulman said:

I would define a monoid in a pseudomonoid $X$ to be a lax morphism of pseudomonoids $U\to X$. This doesn't require duplication or deletion.

Right. I think of this as the "top down" way of looking at it, which is probably the best way. I was trying to get at it "bottom up". Hopefully it would give the same thing.

Joe Moeller (May 05 2020 at 18:54):

Reid Barton said:

I wrote out my guess before I looked at your diagram and I think the source of the multiplication should be $U = U \boxtimes U \xrightarrow{x \boxtimes x} X \boxtimes X \xrightarrow{\otimes} X$.

Yeah, I like this. it doesn't require duplication as a component of the diagram.

Joe Moeller (May 05 2020 at 18:55):

It does seem that duplication is ultimately still present, but it's been pushed back to wherever K lives, which I said is monoidal 2-categories. Is there a way to talk about monoids that completely avoids duplication? It seems hopeless.

Nathanael Arkor (May 05 2020 at 18:58):

How do you talk about a binary operator (in any setting) without duplication?

Nathanael Arkor (May 05 2020 at 18:58):

If you want to restrict the two operands to the same value, you need to duplicate that value.

Joe Moeller (May 05 2020 at 19:01):

Yeah, you're right. I guess I was misinterpreting the way that the duplication kept retreating higher as you continue to internalize things.

Mike Shulman (May 05 2020 at 19:59):

Reid Barton said:

I wrote out my guess before I looked at your diagram and I think the source of the multiplication should be $U = U \boxtimes U \xrightarrow{x \boxtimes x} X \boxtimes X \xrightarrow{\otimes} X$.

This is equivalent to a lax pseudomonoid morphism $U\to X$; the "duplication" of $U$ is the inverse of its (unique, invertible) pseudomonoid structure.

Joachim Kock (May 11 2020 at 22:25):

Maybe you could be interested in the notion of internal algebra classifier. The basic example is the monoidal Delta, which is a classifier for monoids in monoidal categories: a monoid in an arbitrary monoidal category C is a monoidal functor Delta -> C. The notion makes sense in other 2-categories than Cat, and for other 2-monads; there is a whole theory about this. See [Batanin-Berger: Homotopy theory of polynomial monads] and [Weber: Internal algebra classifiers as codescent objects of crossed internal categories].