부산대학교 도서관

The elementary functions $f(x) = \exp(x)-1$, $x/(1-x)$ and ${-\log(1-x)}$ are characterized by classical functional equations. In the article under review, the author considers functors $F\colon {\scr C}\rightarrow {\rm Ch}({\bf K})$, from a category with coproducts ${\scr C}$ to the category of chain complexes over a ring ${\bf K}$, characterized by analogous equations. \par More explicitly, for commutative algebras without unity, the free algebra functor satisfies the identity $F(X\oplus Y) = F(X)\otimes F(Y)\oplus F(X)\oplus F(Y)$ derived from the functional equation of the exponential function $f(x) = \exp(x)-1$. The author gives a similar connection between the free associative algebra functor (respectively, the free Lie algebra functor) and the function $f(x) = {x/(1-x)}$ (respectively, $f(x) = -\log(1-x)$). The author also considers free $n$-Poisson algebra functors $F_n$ which are associated to the homology operads of little $n$-cubes. In this case, he proves that $F_n$ satisfies the same functional equation as the function $f_n(x) = {1/(1-x/2^{n-1})^{2^{n-1}}-1}$. Then, the author uses the functional equations to obtain explicit formulas for Goodwillie's derivatives $D_n F$ of the given functors. \par In the exponential case, one can observe furthermore that the forgetful functor $U$ from the category of commutative algebras without unity, denoted by ${\scr N}_{\bf K}$ in the article under review, to the category of chain complexes ${\rm Ch}({\bf K})$ defines a universal functor $F\colon {\scr C}\rightarrow {\rm Ch}({\bf K})$ that satisfies the exponential identity. Precisely, any such functor arises from a composite of $U$ with a reduced functor $\tilde{F}\colon {\scr C}\rightarrow{\scr N}_{\bf K}$ that preserves coproducts. The author re-proves this result and obtains a similar theorem for the function $f(x) = {x/(1-x)}$ and the forgetful functor on the category of associative algebras.