References/literature for pushouts in category of commutative monoids? [ed. - amalgams]

Question

This is more of a request for pointers to relevant literature than a question per se. I am, erm, looking at a paper which uses a kind of iterated pushout construction to obtain a commutative monoid with certain desired properties. The particular "gluing" construction the authors want to do is handled by quite direct means, and while that's probably fine for this particular problem, it would be nice to put it in the proper context.

My copy of Howie's "Introduction to Semigroup Theory" discusses the general notion of "semigroup amalgams" but doesn't seem to say anything about the pushouts in the smaller category of commutative monoids. The closest I can find via MathSciNet is a 1968 paper of Howie, which seems to be working in the category of commutative semigroups, but I can't get hold of a copy at the moment.

Anyway: I'm hoping that someone reading might know of a sensible place to look for a summary of some general results. (The issue is whether the constituent pieces "embed" into the pushout, not the mere existence of the pushout.)

EDIT: On further reflection, while pushouts are undoubtedly relevant, for the purposes of the paper I'm looking at, faithfulness of the embedding is more important than the universal property of pushouts. So I'm changing the title of the question to reflect this. I also think that while the suggestions below have been useful in a broad sense they don't quite address the case I need - which could just be an illustration that said case falls between the two extremes most commonly looked at by semigroup theorists.

Answer 1

Arthur Ogus wrote a book on logarithmic geometry, apparently soon to be published, and there is a preprint version on his webpage. The first chapter is about commutative monoids, and in particular, it has a bit about pushouts (starting on page 12, and you can tell your computer to search for other instances of the word).

General pushouts can be quite pathological, but you can say interesting things if the monoids satisfy some properties such as integrality. For example, if all of the monoids are integral and one of the monoids is a group, then the pushout is integral, and its group completion is the pushout of the group completions. This reduces Reid's example to a calculation with groups.

Answer 2

I'm not sure exactly what you're looking for, but here is a somewhat weird example of a pushout in commutative monoids I recently came across:

$\begin{matrix} \mathbb{N}&\to&\mathbb{Z}\\\\ \downarrow&&\downarrow\\\\ 0&\to&0 \end{matrix}$

This can't happen in abelian groups (more precisely a pushout square with 0 in the lower left whose top arrow is a monomorphism is also a pullback square).

Answer 3

I don't know a reference, but in case you never find one, here's a possible strategy for figuring it out yourself.

Let $F$ be the functor from commutative monoids to abelian groups that adjoins formal inverses to every element. Then $F$ is left adjoint to the forgetful functor $U$, so preserves colimits and in particular pushouts. I believe it's the case that for a commutative monoid $M$, the canonical map $M \to UF(M)$ is injective iff $M$ is cancellable, i.e. $m + n = m' + n$ implies $m = m'$ (for $m, m', n \in M$). This might allow you to transfer known results about pushouts of abelian groups over to the category of commutative monoids.

Answer 4

Pushouts are expressible in terms of coproducts and coequalizers (just as all other colimits).

Since commutative monoids are the same as $\mathbb{N}$-semimodules, the coproduct $A+B$ is the direct sum with pairs $(a,b)$ as elements along with the injections $i : A \to A+B$ sending $a \mapsto (a,0_B)$ and $j : B \to A+B$ sending $b \mapsto (0_A,b)$.

The pushout of $A \stackrel{f}{\leftarrow} X \stackrel{g}{\rightarrow} B$ is then the coequalizer of the parallel maps $X \stackrel{f}{\rightarrow} A \stackrel{i}{\rightarrow} A+B$ and $Y \stackrel{g}{\rightarrow} B \stackrel{j}{\rightarrow} A+B$. Following the recipe for Set given in MacLane, CWM, Sec. III.3, the coequalizer is $(A+B)/E$ where $E \subseteq (A+B) \times (A+B)$ is the least monoid congruence relation containing pairs $\langle i(f(x)), j(g(y)) \rangle$ for all $x \in X$.

Except for replacing "equivalence relation" by "monoid congruence relation", there is nothing mysterious here.

[Note added: If one were working with non-commutative monoids, the coproduct would be the free product instead of the direct sum, but everything else would be the same.]

Answer 5

Have you looked at the book Monoids, Acts and Categories by Mati Kilp, Ulrich Knauer and Alexander V. Mikhalev? Pushouts are discussed in chapter II.