Piston motion equations

2014-01-15T04:14:08Z

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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.

Some examples of PROs:
* the discrete category <math>\mathbb{N}</math> of natural numbers,
* the category '''[[FinSet]]''' of natural numbers and functions between them,
* the category '''Bij''' of natural numbers and bijections,
* the category '''BijBraid '''of natural numbers, equipped with the [[braid group]] ''Bn ''as the automorphisms of each ''n ''(and no other morphisms).
* the category '''Inj''' of natural numbers and injections,
* the [[simplex category]] <math>\Delta</math> of natural numbers and [[monotonic function]]s.

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 '''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 <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.

For example:
* an algebra of <math>\mathbb{N}</math> is just an object of <math>C</math>,
* an algebra of '''FinSet''' is a commutative [[monoid object]] of <math>C</math>,
* an algebra of <math>\Delta</math> is a [[monoid object]] in <math>C</math>.
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>.

== See also ==
* [[Lawvere theory]]

== References ==
* {{cite journal
| author = [[Saunders MacLane]]
| year = 1965
| title = Categorical Algebra
| journal = Bulletin of the American Mathematical Society
| volume = 71
| pages = 40–106
| doi = 10.1090/S0002-9904-1965-11234-4
}}

*{{cite book
| author = Tom Leinster
| year = 2004
| title = Higher Operads, Higher Categories
| publisher = Cambridge University Press
| url = http://www.maths.gla.ac.uk/~tl/book.html
}}

{{categorytheory-stub}}
[[Category:Monoidal categories]]

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Piston motion equations