# Can every monoid action be turned into a group action?

Let $\mathbf{Mon}$ be the category of monoids. Let $\mathbf{Grp}$ be the category of groups. There is the inclusion functor $i : \mathbf{Grp} \to \mathbf{Mon}$. It has both a left and a right adjoint; see here.

Now given a group $G \in \mathbf{Grp}$ and a right group action $X \times G \to X$ on a set $X$, we have an obvious monoid action $X \times i(G) \to X$ on $X$. On the other hand, given a monoid $M \in \mathbf{Mon}$ and a right monoid action $X \times M \to X$, is it possible to turn this canonically into a group action, using either the left adjoint or the right adjoint of $i$? Preferably both?

• I don't think I have seen this version of the free group functor before. Do you have a reference for it being right adjoint to the inclusion? (inverting element in arbitrary monoids is very poorly behaved usually). – Tobias Kildetoft Mar 30 '16 at 17:55
• I just came up with it myself... Is it not the right one? – rwols Mar 30 '16 at 17:56
• It is just not clear to me that it is at all well-behaved. Note that for example a cancellative monoid need not embed into a group (so adding inverses of all elements will cause at least some elements to be killed). – Tobias Kildetoft Mar 30 '16 at 17:57
• Ah. Then maybe I need to study these functors better first :-) – rwols Mar 30 '16 at 17:59
• I will edit the question so that the functors make more sense – rwols Mar 30 '16 at 18:03

The right adjoint takes monoid elements to units. (This behavior is analogous to constructing the quotient of a ring with one of its maximal ideals.) For the multiplicative monoid $\Bbb{N}$, this takes everyone to 1. Reversing that is hopeless.
For the left adjoint, I refer to one of my favorite monoid actions: differentiation applied to a vector space of polynomials (of degree less than or equal to $N$ and coefficients in something nice, like $\Bbb{Z}$). The action is not injective. Given that, the inverse action of the single generator is not a map from a polynomial to a polynomial, but to an equivalence class of polynomials. I.e., just because I know the action of a monoid element does not mean the inverse action is well-definable. So you can construct a formal object containing those inverses, but there is no guarantee that these formal inverses are the right kinds of maps, or can be made so.