Similarly, we show that the monoidal category of quantum automorphisms of a graph is the category of finite-dimensional representations of the Hopf $C^*$-algebra associated to Banica’s quantum automorphism group (Proposition \[prop:Banica\], [@Banica2005]).

The $2$-categories $\QSet$ and $\QGraph$ therefore extend these quantum groups analogously to the way in which categories extend groups or monoids.

The results in this work serve as a foundation for finite quantum set and quantum graph theory. In a subsequent paper [@paper1b], we use this framework to classify quantum isomorphic graphs in terms of algebraical and group theoretical data.

#### A new approach to quantum functions.

While noncommutative topology is formulated in the language of operator algebras, we present a compositional, linear algebraic framework based on the graphical calculus of string diagrams. In our diagrams, wires represent finite-dimensional Hilbert spaces, and vertices represent linear maps between them. Wiring diagrams, read from bottom to top, represent composite linear maps in the obvious way. We start from the following simple graphical observation:[^3]

\[eq:basicidea\]

(1.5,0) to node\[zxnode=, pos=0.25,scale=1.2\](a) [$a$]{} node\[zxnode=, pos=0.75,scale=1.2\] (b) [$b$]{}(0,2); (\[xshift=+, yshift=-\]b.center) to \[out=20, in=70,looseness=1.1\] (\[xshift=-,yshift=+\]a.center); (\[xshift=--, yshift=-+\] a.center) to \[out=-160, in=-110, looseness=1.1\] (\[xshift=-+, yshift=--\] b.center) ;

  =  

(1.5,0) to node\[zxnode=, pos=0.25,scale=1.2\][$b$]{} node\[zxnode=, pos=0.75,scale=1.2\] [$a$]{}(0,2);

&

(1.5,0) 