Projective resolutions for commutative monoids

What is the right notion of a projective resolution of a commutative monoid?

The category Mon of commutative monoids has plenty of projective (and even free) objects. Indeed, for every set $X$ one can consider the commutative monoid $\mathbb{N}[X]$, with generators $\delta_x$ for $x\in X$. Then, for every commutative monoid $M$ one can consider the free monoid $P_0:=\mathbb{N}[M]$ together with the canonical surjective homomorphism $\varphi_0\colon P_0\to M$, which sends the (formal) sum $\sum_m c_m \delta_m$ in $P_0$ (with $c_m\in\mathbb{N}$, and all but finitely many $c_m$ zero) to the sum $\sum_m c_m m$ in $M$.

One can consider this as the first step in building a projective resolution for $M$. In the category of groups, one would next consider the kernel of $\varphi_0$ and find a free group surjecting onto that kernel.

However, the notion of kernel in Mon is more subtle. If one simply considers $L:=\{x\in P_0 : \varphi_0(x)=0\}$, then it is in general not true that $M$ is isomorphic to $P_0 / L$. One needs to consider kernel pairs, which categorically are the pullbacks of the two maps $(\varphi_0,\varphi_0)$. Thus, a "better" definition of kernel for the map $\varphi_0$ is $$K_0 := \{(x,y)\in P_0\oplus P_0 : \varphi_0(x)=\varphi_1(y) \},$$ together with the two natural maps $p_0,p_1\colon K_0\to P_0$ satisfying $p_0(x,y)=x$ and $p_1(x,y)=y$. Then, it is true that $M$ is isomorphic to the quotient of $P_0$ by the congruence relation generated by $p_0$ and $p_1$. Categorically, $\varphi_0$ is the co-equalizer of $p_0$ and $p_1$.

Next, I can set $P_1:=\mathbb{N}[K_1]$ and consider the natural surjection $\varphi_1\colon P_1\to K_1$. Doing this, I obtain two maps from $P_1$ to $P_0$, namely $p_0\varphi_1$ and $p_1\varphi_1$. Thus, we have obtained the following situation: $$P_1 \rightrightarrows P_0 \to M.$$

What is the next step? I could consider the kernels of the two maps from $P_1$ to $P_0$. This would mean to construct two free monoids with two morphisms each. Thus, at each next level it seems that this (naive) approach doubles the number of arrows. Is this really how it should be done? Has this been considered? Are there better (simpler) notions of projective resolutions for commutative monoids? In either case, the following seems like an interesting question to aks as well:

Do commutative monoids have projective resolutions of finite length?

The motivation for this question is to define derived functors in the category Mon.

Just to avoid some misunderstanding: I am not interested in homological algebra over the field with one element. I am aware of some other questions that have been asked about homological algebra of commuative monoids, such as:

Homological Algebra for Commutative Monoids?

Structure Theorem for finitely generated commutative cancellative monoids?

• While the links given by Vladimir Dotsenko are probably more useful, it is worth noting that you can certainly get a "bar resolution" from general machinery (going back to categorists in the 1960s I think). The general machinery applies whenever you have a pair of adjoint functors, and allows you to build a "simplicial object" which is some kind of "free resolution". Look in Chapter 8 of Weibel's book, for instance. Here we'd look at the forgetful functor U from Mon to Set and its left adjoint F (the free commutative monoid functor). Mar 2, 2016 at 11:49
• I suspect that Mon is also what Borceux, Bourn and others call a "semi-abelian category", which is roughly speaking a category that has many of the good properties of abelian categories (mono and epi implies iso) but is not additive. So one may also be able to use their machinery to define and study derived functors, however I don't remember right now what exactly they do Mar 2, 2016 at 11:53
• @YemonChoi I think Mon is not semi-abelian. I think it is regular and exact (in the sense of Barr), but I think it is not protomodular. A problem (for me) is that such concrete statements are difficult to find in the literature. Mar 2, 2016 at 12:56