PRO (category Theory)

PRO (category Theory)

In category theory, a PRO is a strict monoidal category whose objects are the natural numbers (incl. zero), and whose tensor product is given on objects by the addition on numbers.

Some examples of PROs:

• the discrete category of natural numbers,
• 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,
• the simplex category of natural numbers and monotonic functions.

The name PRO is an abbreviation of "PROduct category". PROBs and PROPs are defined similarly with the additional requirement for the category to be braided, and to have a symmetry (that is, a permutation), respectively.