A graph is reflexive if for each vertex$v$ there is a (specified) edge$v \to v$.

A reflexive quiver has a specified identity edge $i_X: X \to X$ on each object (vertex) $X$. The free category on a reflexive quiver has the same objects, identity morphisms corresponding to the identity edges, and non-identity morphisms consisting of paths of non-identity edges.

Properties

Proposition

The category of reflexive directed graphs $RefGph$, i.e., reflexive quivers, equipped with the functor $U: RefGph \to Set$ which sends a graph to its set of edges, is monadic over $Set$.

Proof

$RefGph$ is the category of functors $R \to Set$ where $R$ is the walkingreflexive fork, consisting of two objects $0, 1$ and generated by arrows $i: 0 \to 1$ and $s, t: 1 \to 0$ subject to $s i = 1_0 = t i$ and no other relations. This $R$ is in turn the Cauchy completion of a monoid $M$ consisting of two elements $e_0 = i s, e_1 = i t$ and an identity, with multiplication $e_0 e_0 = e_0 = e_1 e_0$ and $e_1 e_1 = e_1 = e_0 e_1$, and therefore $RefGph$ is equivalent to the category of functors $M \to Set$, i.e., the category of $M$-sets $Set^M$. This is the category of algebras of the monad $M \times -$ whose monad structure is induced from the monoid structure of $M$.

$Cat$ is monadic over $RefGph$ and $RefGph$ is monadic over $Set$ but $Cat$ is not monadic over $Set$; this is a nice example of how the relation ‘being monadic over’ is not transitive.