Multiplicative group of a ring as a morphism of theories

Question

On nlab I read

For instance, there is a morphism of theories from the theory of commutative rings to the theory of abelian groups which sends a ring to its multiplicative group of units, but this is not induced by any morphism of monads because it does not preserve the underlying set.

Can anyone tell me: what theories and what morphisms are we talking about? From the point of view of Lawvere single-sorted theories, there is no such morphism either, because the functors induced by them also preserve the underlying set (actually, the category of Lawvere theories is equivalent to the category of finitary monads on $\mathrm{Set}$).

Probably, we are talking about theories with some more powerful logic? (like geometric?)

Answer 1

The functor sending a (not necessarily commutative) ring to its group of units is induced by a morphism of cartesian (= finite limit) theories.

More generally, suppose given (small!) cartesian theories $\mathcal{T}$ and $\mathcal{T}'$. Let $\mathcal{M}$ and $\mathcal{M}'$ be the respective categories of models. These are locally finitely presentable categories, so given a functor $G : \mathcal{M} \to \mathcal{M}'$; the following are equivalent:

• $G$ preserves (small) limits and filtered colimits.

• $G$ has a left adjoint that sends finitely presentable objects in $\mathcal{M}'$ to finitely presentable objects in $\mathcal{M}$.

But $\mathcal{T}^\textrm{op}$ (resp. $\mathcal{T}'{}^\textrm{op}$) is equivalent to the full subcategory of finitely presentable objects in $\mathcal{M}$ (resp. $\mathcal{M}'$), so we deduce the following is also equivalent:

• $G$ is induced by a morphism $\Phi : \mathcal{T}' \to \mathcal{T}$ of cartesian theories, in the sense that $G \cong \Phi^*$, where $\Phi^* (M) = M \Phi$.

(Note the contravariance!)