PROP (category theory) explained

In category theory, a branch of mathematics, a PROP is a symmetric strict monoidal category whose objects are the natural numbers n identified with the finite sets

\{0,1,\ldots,n-1\}

and whose tensor product is given on objects by the addition on numbers. Because of “symmetric”, for each n, the symmetric group on n letters is given as a subgroup of the automorphism group of n. The name PROP is an abbreviation of "PROduct and Permutation category".

The notion was introduced by Adams and Mac Lane; the topological version of it was later given by Boardman and Vogt.[1] Following them, J. P. May then introduced the term “operad”, which is a particular kind of PROP, for the object which Boardman and Vogt called the "category of operators in standard form".

There are the following inclusions of full subcategories:[2]

Operads\subset\tfrac{1}{2}PROP\subsetPROP

where the first category is the category of (symmetric) operads.

Examples and variants

An important elementary class of PROPs are the sets

l{R}\bullet x \bullet

of all matrices (regardless of number of rows and columns) over some fixed ring

l{R}

. More concretely, these matrices are the morphisms of the PROP; the objects can be taken as either
infty
\{l{R}
n=0
(sets of vectors) or just as the plain natural numbers (since objects do not have to be sets with some structure). In this example:

\circ

of morphisms is ordinary matrix multiplication.

n

(or

l{R}n

) is the identity matrix with side

n

.

acts on objects like addition (

mn=m+n

or

l{R}ml{R}n=l{R}m+n

) and on morphisms like an operation of constructing block diagonal matrices:

\alpha\beta=\begin{bmatrix}\alpha&0\ 0&\beta\end{bmatrix}

.

(AB)\circ(CD)=\begin{bmatrix}A&0\ 0&B\end{bmatrix}\circ\begin{bmatrix}C&0\ 0&D\end{bmatrix}=\begin{bmatrix}AC&0\ 0&BD\end{bmatrix}=(A\circC)(B\circD)

.

0 x n

matrices) or no columns (

m x 0

matrices) are allowed, and with respect to multiplication count as being zero matrices. The

identity is the

0 x 0

matrix.

There are also PROPs of matrices where the product

is the Kronecker product, but in that class of PROPs the matrices must all be of the form

km x kn

(sides are all powers of some common base

k

); these are the coordinate counterparts of appropriate symmetric monoidal categories of vector spaces under tensor product.

Further examples of PROPs:

N

of natural numbers,

If the requirement “symmetric” is dropped, then one gets the notion of PRO category. If “symmetric” is replaced by braided, then one gets the notion of PROB category.

is a PROB but not a PROP.

\Delta+

of natural numbers and order-preserving functions.

is an example of PRO that 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
C
Alg
P
of algebras whose objects are the algebras of

P

in

C

and whose morphisms are the natural transformations between them.

For example:

N

is just an object of

C

,

C

,

\Delta

is a monoid object in

C

.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

Notes and References

  1. Boardman . J.M. . Vogt . R.M. . Homotopy-everything H -spaces . Bull. Amer. Math. Soc. . 74 . 1968 . 6 . 1117–22 . 10.1090/S0002-9904-1968-12070-1 . 0236922.
  2. Markl . Martin. 2006 . Operads and PROPs . Handbook of Algebra . 5 . 1. 87–140. 10.1016/S1570-7954(07)05002-4. 978-0-444-53101-8 . 3239126. pg 45