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
and whose tensor product is given on objects by the addition on numbers.
[1] 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 Mac Lane; the topological version of it was later given by Boardman and Vogt.
[2] Following them, J. P. May then introduced the term “operad”, which is a particular kind of PROP, for the object which Boardman and Vogt called the "category of operators in standard form".
An important elementary class of PROPs are the sets
of all matrices (regardless of number of rows and columns) over some fixed ring
More concretely, these matrices are the morphisms of the PROP; the objects can be taken as either
In this example: 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
(sides are all powers of some common base
); 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.
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