Representability of the functor \(K_ 0\) \((\tilde K_ 0)\) of the ring of 0-forms (Q1068186)

Let R be a regular ring, \(\Omega^ 0_ R: Set^{\Delta^{op}}\to_ R{\mathcal A}\) the cofunctor, which associates with a simplicial set the R- algebra of polynomial 0-forms. \(K_ 0\), \(\tilde K_ 0\) denote the Grothendieck functors. Let Ho Set\(^{\Delta^{op}}\) denote the homotopy category of the category of simplicial sets \(Set^{\Delta^{op}}\). The main theorem states that the functors \(K_ 0 \Omega^ 0_ R\) and \(\tilde K_ 0 \Omega^ 0_ R: Ho Set^{\Delta^{op}}\to {\mathcal A}b\) are representable on finite simplicial sets. The proof essentially uses the representation theorem of \textit{A. Heller}, J. Lond. Math. Soc., II. Ser. 23, 551-567 (1981; Zbl 0477.55013).