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 BijBraid; the latter is a PROB but not a PROP, and the former is not even a PROB.

Algebras of a PRO

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

For example:

More precisely, what we mean here by "the algebras of \Delta in C are the monoid objects in C" for example is that the category of algebras of P in C is equivalent to the category of monoids in C.

See also

References

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