PRO (category theory)

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

Some examples of PROs:

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. All of the examples above are PROPs, except for the simplex category and Bij_Braid; the latter is a PROB but not a PROP, and the former is not even a PROB.

An algebra of a PRO ${\displaystyle P}$ in a monoidal category ${\displaystyle C}$ is a strict monoidal functor from ${\displaystyle P}$ to ${\displaystyle C}$. Every PRO ${\displaystyle P}$ and category ${\displaystyle C}$ give rise to a category ${\displaystyle \mathrm {Alg} _{P}^{C}}$ of algebras whose objects are the algebras of ${\displaystyle P}$ in ${\displaystyle C}$ and whose morphisms are the natural transformations between them.

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