Product (category theory) |
In category theory, one defines products to
generalize constructions such as the cartesian product of sets,
the product of groups, the product of rings and the product of
topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to
each of the given objects.
Let C be a category and let {Xi | i ∈ I} be a indexed family of objects in C. The product of the set {Xi} is an object X
together with a collection of morphisms πi : X
→ Xi (called projections) which satisfy a universal property: for any object Y and any collection of morphisms
fi : Y → Xi, there exists a unique morphism f :
Y → X such that fi = πi f. That is, the follow
diagram commutes (for each i):
The product construction given above is actually a special case of a limit in category theory. The product can be defined as the limit of any discrete subcategory in C. Not every family
{Xi} needs to have a product, but if it does, then the product is unique in a strong sense: if
πi : X → Xi and π'i :
X ' → Xi are two products of the family {Xi}, then
there exists a unique isomorphism r : X →
X ' such that π'i r = πi for each i in
I.
An empty product (i.e. I is the empty set) is the same as a terminal object in
C.
If I is a set such that all products for families indexed with I exist, then it is possible to choose the
products in a compatible fashion so that the product turns into a functor
CI → C. The product of the family {Xi} is then often
denoted by ∏i Xi, and the maps πi are known as
the natural projections. We have a natural
isomorphism
-
(where MorC(U,V) denotes the set of all morphisms from U to V in
C, the left product is the one in C and the right is the cartesian product of sets).
If I is a finite set, say I = {1,...,n}, then the product
of objects X1,...,Xn is often denoted by
X1×...×Xn. Suppose all finite products exist in C, product functors
have been chosen as above, and 1 denotes the terminal object of C corresponding to the empty product. We then have
natural isomorphisms
-
-
-
These properties are formally similar to those of a commutative monoid.
See also:
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