The paper studies the representation of small categories as monoids in the categories \(\mathcal{S}pan\), \(\mathcal{R}el\) and \(\mathcal{S}up\). For a small category \(\mathcal C\) let the underlying set of a monoid be the set of all morphisms of \(\mathcal C\), the unit object of monoid be the set of all identity morphisms of \(\mathcal C\) (with the inclusion mapping) and a composition be a partial mapping such that it is defined on a pair \((f,g)\) just when the domain of \(f\) is equal to a codomain of \(g\), and it maps \((f,g)\) to \(f\circ g\). Clearly, it is a monoid in the category \(\mathcal{S}pan\).
Functors between small categories correspond to monoid homomorphisms. There is a natural functor from \(\mathcal{S}pan\) into \(\mathcal{R}el\) preserving monoids (the converse is not true). The paper characterizes monoids in \(\mathcal{S}pan\) and monoids in \(\mathcal{R}el\) corresponding small categories. The category \(\mathcal{R}el\) is isomorphic to a full subcategory of \(\mathcal{S}up\) and monoids of \(\mathcal{S}up\) are quantales – this motivates these constructions. The appropriate ways to express other categorical constructions (natural transformations, profunctors) are studied in these categories.
Some illustrating examples are presented.