# Support, with “up”

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Last seen on: LA Times Crossword 20 Apr 2018, Friday

### Random information on the term “PROP”:

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. The name PRO is an abbreviation of “PROduct category”.

Some examples of PROs:

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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.

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