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
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
of
all matrices (regardless of number of rows and columns) over some fixed ring
. More concretely, these matrices are the
morphisms of the PROP; the objects can be taken as either
(sets of vectors) or just as the plain natural numbers (since
objects do not have to be sets with some structure). In this example:
of morphisms is ordinary
matrix multiplication.
- The identity morphism of an object
(or
) is the
identity matrix with side
.
acts on objects like addition (
or
) and on morphisms like an operation of constructing block diagonal matrices:
\alpha ⊗ \beta=\begin{bmatrix}\alpha&0\ 0&\beta\end{bmatrix}
.
- The compatibility of composition and product thus boils down to
(A ⊗ B)\circ(C ⊗ D)=\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)
.
- As an edge case, matrices with no rows (
matrices) or no columns (
matrices) are allowed, and with respect to multiplication count as being zero matrices. The
identity is the
matrix.
- The permutations in the PROP are the permutation matrices. Thus the left action of a permutation on a matrix (morphism of this PROP) is to permute the rows, whereas the right action is to permute the columns.
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
(sides are all powers of some common
base
); these are the coordinate counterparts of appropriate symmetric monoidal categories of vector spaces under tensor product.
Further examples of PROPs:
of natural numbers,
- the category FinSet of natural numbers and functions between them,
- the category Bij of natural numbers and bijections,
- the category Inj of natural numbers and injections.
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.
- the category BijBraid of natural numbers, equipped with the braid group Bn as the automorphisms of each n (and no other morphisms).
is a PROB but not a PROP.
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
in a
monoidal category
is a strict
monoidal functor from
to
. Every PRO
and category
give rise to a category
of algebras whose objects are the algebras of
in
and whose morphisms are the natural transformations between them.
For example:
is just an object of
,
,
is a
monoid object in
.More precisely, what we mean here by "the algebras of
in
are the monoid objects in
" for example is that the category of algebras of
in
is
equivalent to the category of monoids in
.
See also
References
- Saunders . Mac Lane . Saunders Mac Lane . 1965 . Categorical Algebra . Bulletin of the American Mathematical Society . 71 . 40–106 . 10.1090/S0002-9904-1965-11234-4 . free .
- Book: Martin . Markl . Steve Shnider . Steve . Shnider . Jim Stasheff . Jim . Stasheff . 2002 . Operads in Algebra, Topology and Physics . American Mathematical Society . 978-0-8218-4362-8 .
- Book: Leinster
, Tom . 2004 . Higher Operads, Higher Categories . Cambridge University Press . math/0305049. 2004hohc.book.....L . 978-0-521-53215-0 .
Notes and References
- 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.
- 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