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Stream: learning: questions

Topic: Irreducible monoids

John van de Wetering (Aug 02 2020 at 09:49):

I am wondering whether anyone could tell me whether the concept of an "irreducible monoid" has been considered before. Let me explain:
Let $C$ be a category with symmetric monoidal structure $\otimes$ and products $\times$, and let $A\otimes A\rightarrow A$ and $B\otimes B\rightarrow B$ be monoid objects in $C$. If the product distributes over $\otimes$ then you can show that $A\times B$ also forms a monoid object. Call a monoid object $C$ irreducible when $C=A\times B$ for monoid objects $A$ and $B$ implies that one of $A$ and $B$ is final, i.e. trivial (perhaps you need the additional condition here that the final object is also the monoidal unit).
In the category of vector spaces irreducible monoid objects correspond precisely to irreducible algebras, which is why I chose this name. But should I perhaps be using a different name for this?

John van de Wetering (Aug 07 2020 at 12:51):

Anyone?

Morgan Rogers (he/him) (Aug 07 2020 at 13:50):

I'm looking into this now. When you say that

If the product distributes over $\otimes$ then you can show that $A\times B$ also forms a monoid object,

I see two possible structure maps: $(A \times B) \otimes (A \times B) \cong A \times (B \otimes B) \to A \times B$, where the second morphism is $id_A \times \beta$, and another obtained by distributing on the other side. I see no reason they should be the same; is there an alternative I'm missing?

Morgan Rogers (he/him) (Aug 07 2020 at 13:53):

Ah, given the example, I gather you meant the distributivity to be the other way around.

Morgan Rogers (he/him) (Aug 07 2020 at 13:58):

So we have $(A \times B) \otimes (A \times B) \cong (A \otimes A) \times (A \otimes B) \times (B \otimes A) \times (B \otimes B) \to A \times B$ where the second map is $(\alpha \circ \pi_1) \times (\beta \circ \pi_4)$ as the induced monoid structure that you're referring to?

Morgan Rogers (he/him) (Aug 07 2020 at 14:07):

The literature I've been able to find on this stuff, cf for example this nLab page, seems to treat biproducts as colimits rather than limits. There are probably results about limits that you can deduce thanks to the monadicity results presented there.
As for irreducibility, this is also a concept more typically applied to coproducts, drawing intuition from situations where these behave as disjoint or orthogonal sums. The dual concept is typically called "prime", for reasons that you can probably guess.

Morgan Rogers (he/him) (Aug 07 2020 at 14:11):

I don't know if that helps you identify whether these have been studied before, but perhaps the alternative name will appeal to you.

Fabrizio Genovese (Aug 07 2020 at 14:25):

I suspect this kind of question is related to the problem of classifying algebraic structures. This is already monstrously difficult for groups (which are relatively simple as structures), and I am inclined to suspect there isn't a real set of techniques for this level of generality...

John van de Wetering (Aug 09 2020 at 09:15):

[Mod] Morgan Rogers said:

Ah, given the example, I gather you meant the distributivity to be the other way around.

Yes, that is indeed the type of distributivity I had in mind, and the monoid map I had in mind.

John van de Wetering (Aug 09 2020 at 09:30):

[Mod] Morgan Rogers said:

The literature I've been able to find on this stuff, cf for example this nLab page, seems to treat biproducts as colimits rather than limits. There are probably results about limits that you can deduce thanks to the monadicity results presented there.
As for irreducibility, this is also a concept more typically applied to coproducts, drawing intuition from situations where these behave as disjoint or orthogonal sums. The dual concept is typically called "prime", for reasons that you can probably guess.

Ah yes, I didn't think of the issue that in many categories of algebraic structures, you actually have biproducts.
I guess it makes sense to refer to the 'irreducibility' in the coproduct as being irreducible while irreducibility in the product means it is 'prime'.

John van de Wetering (Aug 09 2020 at 09:32):

In my situation however (category of effect algebras), the product acts as Cartesian product, while the coproduct is really weird, so it seems weird to call the algebras 'prime' if they are irreducible with respect to the product.

John van de Wetering (Aug 09 2020 at 09:33):

More informal question: if a paper where to say "monoid that is irreducible", what would you intuitively think it meant?

Morgan Rogers (he/him) (Aug 09 2020 at 14:42):

Because of what I said before, I would assume you meant with respect to coproducts (so that the free monoid on two elements is reducible but $\mathbb{N}$ is irreducible). I would definitely look for it to be defined somewhere in there, though.