PROP (category theory)

In category theory, a branch of mathematics, a PROP is a symmetric strict monoidal category whose objects are the natural numbers "n" identified with the finite sets formula_1 and whose tensor product is given on objects by the addition on numbers. Because of “symmetric”, for each "n", the symmetric group on "n" letters is given as a subgroup of the automorphism group of "n". The name PROP is an abbreviation of "PROduct and Permutation category".
The notion was introduced by Adams and MacLane; the topological version of it was later given by Boardman and Vogt. Following them, J. P. May then introduced the notion of “operad”, a particular kind of PROP.
There are the following inclusions of full subcategories:
where the first category is the category of (symmetric) operads.
Examples and variants.
An important "elementary" class of PROPs are the sets formula_3 of "all" matrices (regardless of number of rows and columns) over some fixed ring formula_4. More concretely, these matrices are the "morphisms" of the PROP; the objects can be taken as either formula_5 (sets of vectors) or just as the plain natural numbers (since objects "do not have to" be sets with some structure). In this example:
There are also PROPs of matrices where the product formula_10 is the Kronecker product, but in that class of PROPs the matrices must all be of the form formula_20 (sides are all powers of some common base formula_21); these are the coordinate counterparts of appropriate symmetric monoidal categories of vector spaces under tensor product.
Further examples of PROPs:
If the requirement “symmetric” is dropped, then one gets the notion of PRO category. If “symmetric” is replaced by braided, then one gets the notion of 
PROB category.
is a PROB but not a PROP.
is an example of PRO that is not even a PROB.
Algebras of a PRO.
An algebra of a PRO formula_24 in a monoidal category formula_25 is a strict monoidal functor from formula_24 to formula_25. Every PRO formula_24 and category formula_25 give rise to a category formula_30 of algebras whose objects are the algebras of formula_24 in formula_25 and whose morphisms are the natural transformations between them.
For example:
More precisely, what we mean here by "the algebras of formula_36 in formula_25 are the monoid objects in formula_25" for example is that the category of algebras of formula_24 in formula_25 is equivalent to the category of monoids in formula_25.