Monads and Graphs ∗

The simplicial category ∆ plays many roles all over mathematics. In one of these roles it is a monad freely generated by a single object (see [9], Section 3, [2], Section 4, and references therein). That ∆ is isomorphic to this free monad may be understood as a coherence result connecting the syntax brought by the equational presentation of monads and the semantics given by the order preserving functions on finite ordinals, which can be graphically presented as pictures, or diagrams, or graphs, made of threads connecting the points at the top of the picture with some points at the bottom. However, this isomorphism is more than just coherence, for which it would be enough to have the faithfulness of a functor going from the free monad to the graphical category ∆. An analogous result, which may be understood as a coherence result in the sense of category theory, is the isomorphism of the commutative Frobenius monad freely generated by a single object with the skeleton of the category 2Cob, whose arrows are cobordisms in dimension 2 (see [1] and [13]). Still another example of such a result is the relation between commutative separable Frobenius monads and the category Cospan(Setsfin), whose arrows are cospans in the base category Set (see [14] and [16]). In logical terms, this is like proving completeness with respect to a manageable model, which helps us to solve the decision problem for commuting of diagrams of arrows. The categories we envisage that serve as manageable models are either categories whose arrows are of a geometric kind (tangles, oriented tangles, Temperley-Lieb diagrams, Kelly-Mac Lane graphs, Brauer diagrams),