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| 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.
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| Some examples of PROs:
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| * the discrete category <math>\mathbb{N}</math> of natural numbers,
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| * the category '''[[FinSet]]''' of natural numbers and functions between them,
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| * the category '''Bij''' of natural numbers and bijections,
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| * the category '''Bij<sub>Braid</sub> '''of natural numbers, equipped with the [[braid group]] ''B<sub>n</sub> ''as the automorphisms of each ''n ''(and no other morphisms).
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| * the category '''Inj''' of natural numbers and injections,
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| * the [[simplex category]] <math>\Delta</math> of natural numbers and [[monotonic function]]s.
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| The name PRO is an abbreviation of "PROduct category". '''PROB'''s and '''PROP'''s are defined similarly with the additional requirement for the category to be [[braided monoidal category|'''b'''raided]], and to have a [[symmetric monoidal category|symmetry]] (that is, a '''p'''ermutation), respectively. All of the examples above are '''PROP'''s, except for the simplex category and '''Bij<sub>Braid</sub>'''; the latter is a '''PROB '''but not a '''PROP''', and the former is not even a '''PROB'''.
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| == Algebras of a PRO ==
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| An algebra of a PRO <math>P</math> in a [[monoidal category]] <math>C</math> is a strict [[monoidal functor]] from <math>P</math> to <math>C</math>. Every PRO <math>P</math> and category <math>C</math> give rise to a category <math>\mathrm{Alg}_P^C</math> of algebras whose objects are the algebras of <math>P</math> in <math>C</math> and whose morphisms are the natural transformations between them.
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| For example:
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| * an algebra of <math>\mathbb{N}</math> is just an object of <math>C</math>,
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| * an algebra of '''FinSet''' is a commutative [[monoid object]] of <math>C</math>,
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| * an algebra of <math>\Delta</math> is a [[monoid object]] in <math>C</math>.
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| More precisely, what we mean here by "the algebras of <math>\Delta</math> in <math>C</math> are the monoid objects in <math>C</math>" for example is that the category of algebras of <math>P</math> in <math>C</math> is [[equivalence of categories|equivalent]] to the category of monoids in <math>C</math>.
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| == See also ==
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| * [[Lawvere theory]]
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| == References ==
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| * {{cite journal
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| | author = [[Saunders MacLane]]
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| | year = 1965
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| | title = Categorical Algebra
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| | journal = Bulletin of the American Mathematical Society
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| | volume = 71
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| | pages = 40–106
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| | doi = 10.1090/S0002-9904-1965-11234-4
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| }}
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| *{{cite book
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| | author = Tom Leinster
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| | year = 2004
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| | title = Higher Operads, Higher Categories
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| | publisher = Cambridge University Press
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| | url = http://www.maths.gla.ac.uk/~tl/book.html
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| }}
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| {{categorytheory-stub}}
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| [[Category:Monoidal categories]]
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I am Oscar and I completely dig that name. She is a librarian but she's usually needed her personal company. Puerto Rico is where he's usually been living but she requirements to transfer because of her family. One of the very very best things in the world for me is to do aerobics and now I'm attempting to make cash with it.
Feel free to visit my site ... www.breda.nl