Yes I have an application from $\mathbb{B}^n$ to $\mathbb{B}^m$, where $\mathbb{B}$ is the $2$-element Boolean algebra and I need a necessary and sufficient condition for this map to be one-to-one.

@BenjaminSteinberg Indeed, it is a system of equations in a semiring.

@AlexDegtyarev : a mistake from me indeed; I was wrongly interpreting "xor" as the disjoint union. With $A=\{a,b\}$ and $B=\{b,c\}$, I had $A\sqcup B=\{a,b_1,b_2,c\}$ which is of course not a subset of $\{a,b,c\}$. I edit the question one more time.

It is not a numerical system of equations (I cannot take a computer and ask it to solve it), but I understand what you mean. A round trip between the 2-element Boolean algebra and $\mathbb{F}_2$ could help. Anyway, any reference about the subject is welcome.

Probably because I am using the terminology in a wrong way: I don't know almost anything in Boolean algebra. I use "or", union and not "xor", for the addition. For me $1+1=1$, and not $0$.

By googling, I have found Is there a standard notation for binary relations in category theory?.

A comment about your motivations: you should read Olschok's PhD which provides more general settings to construct such model structures. The key point is to find an appropriate cylinder (for Cisinski's model categories) or cocylinder functor (for LMW's model categories).

Could you explain with more details what you mean by "analogous to the case of abelian categories" please ? The only thing I can say is that taking a cofibrant (fibrant resp.) replacement in a model category is a little bit like taking a projective (injective resp.) resolution in an abelian category, but in a non-additive setting. But I guess that you already know that.

Is what follows special enough ? The homotopy theory of simplicial sets is the universal homotopy theory generated by one point : http://math.uoregon.edu/~ddugger/univ.html.

You are correct, up to weak homotopy equivalence.

@Charles Rezk I believe that Grothendieck did not like the notion of topological space and wanted a "purely algebraic" representation of homotopy types. Though I cannot remember where I read that.

@AlexDegtyarev Functoriality is not a convenient language. Think for example of Brouwer's fix point theorem, the proof using the fundamental group. It is because of the functoriality of the fundamental group that supposing the nonexistence of a fix point leads to a one-to-one map from $\mathbb{Z}$ to $\{0\}$.