In category theory
, a PRO
is a strict monoidal category
whose objects are the natural integers and whose tensor product is given on objects by the addition on integers. By an integer
, we mean here the set
Some examples of PROs:
- the discrete category of integers,
- the category FinSet of integers and functions between them,
- the category Bij of integers and bijections,
- the category Inj of integers and injections,
- the simplicial category of integers and monotonic functions.
The name PRO is an abbreviation of "PROduct category". PROBs (resp. PROPs) are defined similarly with the additional requirement for the category to be braided (resp. to have a symmetry, or a permutation).
Algebras of a PRO
An algebra of a PRO
in a monoidal category
is a strict monoidal functor
. Every PRO
give rise to a category
of algebras whose objects are the algebras of
and whose morphisms are the natural transformations between them.
- an algebra of is just an object of ,
- an algebra of FinSet is a commutative monoid object of ,
- an algebra of is a monoid object in .
More precisely, what we mean here by "the algebras of in are the monoid objects in " for example is that the category of algebras of in is equivalent to the category of monoids in .