Operads in the category of convexors. I. (Q1398527)

A convexor is a set \(A\) together with a family of operations \(\alpha: A\times A\to A\), \(\alpha\in [0,1]\), such that for all \(\alpha,\beta\in [0,1]\), \(x,y,z\in A\), \hskip17mm (1) \(\alpha(\beta(x, y),z)= (\alpha\cdot\beta)(x, \alpha-(1\cdot\beta)/(1- \alpha\cdot\beta)(y, z))\), \hskip17mm (2) \(\alpha(x,y)= (1-\alpha)(y,x)\), \hskip17mm (3) \(\alpha(x,x)= x,\quad 1(x,y)= x\). Convexors play a role in the theory of probabilistic automata and stochastic matrices. It is clear from the definition that convexors can be described as algebras over a suitable algebraic theory, which cannot come from an operad in the classical sense because condition (3) requires diagonals. For this reason the author extends the notion of an operad: instead of actions of the symmetric group \(\Sigma_n\) on the \(n\)th set of an operad he allows actions of suitable subcategories of the category \({\mathcal F}{\mathcal S}et\) of finite sets \([n]= \{1,2,\dots, n\}\). The permutations are an example of such a subcategory. The author then defines an operad \(\Delta\) whose \(n\)th set is the \((n- 1)\)-simplex \(\Delta^{n-1}\) (it can be viewed as the topological realization of \textit{C. Berger}'s complete graph operad [Contemp. Math. 202, 37--52 (1997; Zbl 0860.18001)]) and extends its structure to an \({\mathcal F}{\mathcal S}et\)-operad. He then shows that the category of its algebras is the category \({\mathcal C}onv\) of convexors. Finally, he introduces the tensor product of two convexors and proves that \(({\mathcal C}onv,\otimes)\) is a symmetric monoidal closed category.