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?
