Free coalgebras in a category of rings

Abstract:

Let $\mathcal {R}$ be the category of commutative rings with unity and unity-preserving homomorphisms, and let $\Pi$ be a small algebraic theory, i.e., an algebraic theory with a rank in the sense of Linton. The category $\mathcal {A}$ of $\Pi$-coalgebras in $\mathcal {R}$ is the category of coproduct-preserving functors ${\Pi ^{\ast }} \to \mathcal {R}$. We prove that the standard forgetful functor $U:\mathcal {A} \to \mathcal {R}$ has a right adjoint $V$.