based on *Gelfand duality*.

\[thm:gelfandduality\] The category of commutative $C^*$-algebras and -morphisms is equivalent to the opposite of the category of compact Hausdorff spaces and continuous functions.

The equivalence takes a space to the algebra of continuous complex-valued functions on that space; in the other direction, a $C^*$-algebra is taken to its spectrum. In some sense, therefore, one may consider the theory of compact Hausdorff spaces — classical topology — to be the theory of commutative $C^*$-algebras.

The idea of noncommutative geometry is to consider *noncommutative* in light of Gelfand duality, thus enabling us to study the ‘noncommutative’ or ‘quantum’ spaces to which they would be dual. In this paper, we consider only finite discrete spaces and finite-dimensional $C^*$-algebras. In this case, Theorem \[thm:gelfandduality\] reduces to the following statement.

\[cor:finiteGelfandduality\] The category of finite-dimensional commutative $C^*$-algebras and $*$-homomorphisms is equivalent to the opposite of the category of finite sets and functions.

The string diagram calculus
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In order to investigate finite Gelfand duality and its noncommutative generalisation, we make use of the string diagram calculus for monoidal categories. This calculus is well established and has been treated in detail elsewhere [@Joyal1991a; @Joyal1991b; @Selinger2010; @Coecke2010]; here we only provide a brief 