proof of Gelfand duality in this setting [@Coecke2009] are important starting points and guide posts for our work (see Section \[sec:background\]). Many of our proofs and constructions can be understood as simple quantisations, in the sense of equation , of constructions from categorical quantum mechanics.

A related 2-categorical framework for quantum theory based on the 2-category $\mathrm{2Hilb}$ of categorified Hilbert spaces was studied by Vicary and the second author [@Vicary2012hq; @Vicary2012_2; @Reutter2016]; we remark that there is a locally faithful 2-functor $\QSet \to \mathrm{2Hilb}$ allowing us to translate most of our results into this setting.

#### Monads.

The 2-category of quantum sets $\QSet$ is an instance of Street’s 2-category of monads [@Street1972] in the $2$-category $\BHilb$, the delooping of the monoidal category $\Hilb$ (see Remark \[rem:street\]). Hinze and Marsden [@Marsden2014; @Hinze2016] give an analogous graphical treatment of the 2-category $\mathrm{Mnd}(\mathrm{Cat})$ of monads in the 2-category $\mathrm{Cat}$ of categories, functors and natural transformations.

Recent work by Abramsky and others [@Abramsky2017] uses monads to study binary constraint systems and quantum graph homomorphisms. There are many similarities between their work and ours; for example, composition in their Kleisli category corresponds to the composition of $1$-morphisms in $\QSet$ (see Remark \[rem:ppmcomposition\]).

Definitions and conventions
