Categories as monoids in \({\mathcal S}pan,{\mathcal R}el\) and \({\mathcal S}up\). (English) Zbl 1242.18003
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.
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.
Reviewer: Václav Koubek (Praha)
MSC:
18B10 | Categories of spans/cospans, relations, or partial maps |
06F07 | Quantales |
18D35 | Structured objects in a category (MSC2010) |
18B40 | Groupoids, semigroupoids, semigroups, groups (viewed as categories) |