A cotriple construction of a simplicial algebra used in the definition of higher Chow groups (Q2808149)

Let \(\Delta \) be the category of finite ordinals and order preserving functions. There is an obvious monoidal structure in \(\Delta ^{\mathrm{op}}\) that makes it a free (strict) monoidal category with a comonoid, meaning that it is the \(2\)-initial object in the \(2\)-category of monoidal categories with (a chosen) comonoid object. If \(\mathfrak X \) is a category and \([\mathfrak X,\mathfrak X] \) is its usual (strict monoidal) category of endofunctors, a comonad (also called cotriple) on \(\mathfrak X\) is a comonoid in \([\mathfrak X,\mathfrak X] \). Hence a comonad determines and is determined by a strict monoidal functor \(\mathcal T:\Delta^{\mathrm{op}}\to[\mathfrak X,\mathfrak X] \) (see, for instance, [\textit{S. Schanuel} and \textit{R. Street}, Cah. Topologie Géom. Différ. Catégoriques 27, No. 1, 81--83 (1986; Zbl 0592.18002)]). Therefore, given such a comonad, it has a mate \(\mathcal T^\ast:\mathfrak X\to[\Delta^{\mathrm{op}},\mathfrak X]\) which associates each object of \(\mathfrak X \) to a simplicial object of \(\mathfrak X\), which can be called the free simplicial \(\mathcal T\)-resolution of \(X\) (see [\textit{J.M. Beck}, Repr. Theory Appl. Categ. 2003, No. 2, 1--59 (2003; Zbl 1022.18004)]).NEWLINENEWLINEThere is a simplicial \(R\)-algebra \(\Delta^{\mathrm{op}}\to\mathrm{R-}\mathsf{Alg}\) which arises in the definition of higher Chow groups of [\textit{S. Bloch}, Adv. Math. 61, 267--304 (1986; Zbl 0608.14004)].NEWLINENEWLINEThe present paper shows that this (not so simple) simplicial \(R\)-algebra, denoted by \(D\) and defined in the first page, comes in a natural way from the free simplicial \(\bot\)-resolution of \((\mathrm R,-1) \) composed with \(Q\) in which \(\bot (A,a) = (A[t], t+a)\) is a simple enough comonad on the category of pointed \(\mathrm R\)-algebras and \(Q (A,a) = A/(a)\).