Quantum Functor $Mor$

Let $Top_c$ be the category of compact spaces and continuous maps and $Top_f\subset Top_c$ be the full subcategory of finite spaces. Consider the covariant functor $Mor:Top_f^{op}\times Top_c\to Top_c$ that associates any pair $(X,Y)$ with the space of all morphisms from $X$ to $Y$. In this paper, we describe a non commutative version of $Mor$. More pricelessly, we define a functor $\mathfrak{M}\mathfrak{o}\mathfrak{r}$, that takes any pair $(B,C)$ of a finitely generated unital C*-algebra $B$ and a finite dimensional C*-algebra $C$ to the quantum family of all morphism from $B$ to $C$.

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