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 ${\displaystyle \{0,1,\ldots ,n-1\}}$ 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".

There are the following inclusions of full subcategories:^[3]

${\displaystyle {\mathsf {Operads}}\subset {\tfrac {1}{2}}{\mathsf {PROP}}\subset {\mathsf {PROP}}}$

where the first category is the category of (symmetric) operads.