Summary:

Let $K$ be a field. The generalized Leibniz rule for higher derivations suggests the definition of a coalgebra ${\cal D}^k_K$ for any positive integer $k$. This is spanned over $K$ by $d_0,\ldots,d_k$, and has comultiplication $\Delta$ and counit $\varepsilon$ defined by $\Delta(d_i)=\sum_{j=0}^id_j\otimes d_{ij}$ and $\varepsilon(d_i)=\delta_{0,i}$ (Kronecker's delta) for any $i$. This note presents a representation of the coalgebra ${\cal D}^k_K$ by using smooth spaces and a procedure of microlocalization. The author gives an interpretation of this result following the principles of the quantum theory of geometric spaces. (English) 