universal algebra - Defining 'free monoid' without Nat?

As I pointed out in the comments, the theory of free monoids is somewhat ill defined. It is still unclear what your logic is and what you really want, but you have two basic options which were proposed by Pete Clark and sigfpe. Here are a few additional remarks that may help you sort things out.

Universal Property à la Pete Clark. The free monoid functor $F:Setto Mon$ is left-adjoint to the forgetful functor $G:Mon to Set$. This adjunction completely determines the isomorphism classes of free monoids. Moreover, many of the essential properties of free monoids follow directly from properties of sets and adjoint functors.

Alternately, the bijection between $Set(A,GB)$ and $Mon(FA,B)$ can be described as Pete Clark suggested, without any reference to adjoint functors. This latter option preferable if your logic can handle the categories $Set$ and $Mon$, but not functors between them.

The obvious advantage of this approach is that this is the true freeness property of free monoids (the fact that word monoids are free is nothing but an interesting accident from this point of view). Other than the fact that $F1$ is a natural number object, there is no mention of natural numbers here. However, I always thought that there was a lot of unnecessary tedium to show that free monoids can be viewed as word monoids in this context.

Induction Axiom à la sigfpe. The idea here is to attempt to capture freeness via induction. Here is a variant that fixes the minor problem I pointed out in a comment to sigfpe's answer -- free monoids are structures $(F,A,e,{cdot})$ where $(F,e,{cdot})$ is a monoid and $A subseteq F$ (understood as a unary predicate) is a distinguished set of generators. The axioms to describe this setup are the usual monoid axioms together with the obvious modifications of sigfpe's two axioms

• $A(a) land A(a') land xcdot a = x' cdot a' to x = x' land a = a'$,


• $forall P,(P(e) land forall x,a,(P(x) land A(a) to P(xcdot a)) to forall x,P(x))$,

where $P$ varies over all unary predicates $P subseteq F$. In first-order logic, quantification over such $P$ is not possible and the induction axiom must be replaced by the corresponding induction scheme where $P$ varies over all formulas of the language instead. (In higher-order logics without predicate types, you can usually replace $P(x)$ by $f(x) = g(x)$ for appropriate $f$ and $g$.)

In the standard semantics for second-order logic, these axioms completely describe free monoids. However you must be careful since not all formulations of higher-logic are equal. Much like first-order logic, some of these formulations will implicitly allow for models of the above theory which are not free in the categorical sense. In truth, this is unlikely to be a problem for you but I am compelled to warn you of this possibility.