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Let $K$ be a field. The generalized Leibniz rule for higher derivations suggests the definition of a coalgebra ${𝒟}_K^k$ for any positive integer $k$. This is spanned over $K$ by $d_0,...,d_k$, and has comultiplication $\Delta$ and counit $\epsilon$ defined by $\Delta \left(d_i\right)={\sum }_{j=0}^i{d}_j\otimes d_{i-j}$ and $\epsilon \left(d_i\right)={\delta }_{0,i}$ (Kronecker’s delta) for any $i$. This note presents a representation of the coalgebra ${𝒟}_K^k$ 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.

An idea for quantization by means of geometric observables is explained, which is a kind of the sheaf theoretical methods. First the formulation of differential geometry by using the structure sheaf is explained. The point of view to get interesting noncommutative observable algebras of geometric fields is introduced. The idea is to deform the algebra $C^{\infty }(M,ℝ)$ by suitable interaction structures. As an example of such structures the Poisson-structure is mentioned and this leads naturally to deformation...