Quantum functor \({\mathcal M}or\). (English) Zbl 1240.46079

Summary: Let \({\mathcal T}op_c\) be the category of compact spaces and continuous maps and \({\mathcal T}op_f\subset {\mathcal T}op_c\) be the full subcategory of finite spaces. Consider the covariant functor \({\mathcal M}or:{\mathcal T}op^{op}_f\times {\mathcal T}op_c\to {\mathcal T}op_c\) 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 \({\mathcal M}or\). More precisely, 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 \(C\). As an application, we introduce a non-commutative version of the path functor.


46L05 General theory of \(C^*\)-algebras
46L85 Noncommutative topology
46M15 Categories, functors in functional analysis
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