In category theory, a branch of mathematics, a PROP is a symmetricstrict monoidal category whose objects are the natural numbersn identified with the finite sets 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 Broadman and Vogt. Following them, J. P. May then introduced the notion of “operad”. There are the following inclusions of full subcategories: where the first category is the category of operads.
Examples and variants
An important elementary class of PROPs are the sets of all matrices over some fixed ring. More concretely, these matrices are the morphisms of the PROP; the objects can be taken as either or just as the plain natural numbers. In this example:
The identity morphism of an object is the identity matrix with side.
The product acts on objects like addition and on morphisms like an operation of constructing block diagonal matrices:.
* The compatibility of composition and product thus boils down to
*:.
* As an edge case, matrices with no rows or no columns are allowed, and with respect to multiplication count as being zero matrices. The identity is the matrix.
The permutations in the PROP are the permutation matrices. Thus the left action of a permutation on a matrix is to permute the rows, whereas the right action is to permute the columns.
There are also PROPs of matrices where the product is the Kronecker product, but in that class of PROPs the matrices must all be of the form ; these are the coordinate counterparts of appropriate symmetric monoidal categories of vector spaces under tensor product. Further examples of PROPs:
the category FinSet of natural numbers and functions between them,
the category Bij of natural numbers and bijections,
the category Inj of natural numbers and injections.
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.
the category BijBraidof natural numbers, equipped with the braid groupBnas the automorphisms of each n .
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.
