Product

So, in the equation

C + O₂ -> CO₂

CO₂ is the product.

In category theory, a product of two objects A and B is an object AxB and a pair of morphisms p₁:AxB->A and p₂:AxB->B, called projections. The object AxB, if it exists, has the property that for any object C and projections f:C->A and g:C->B, there is exactly one morphism <f,g> such that that p₁<f,g> = f and p₂<f,g> = g. Or, to make it a little more intuitive, <f,g> makes the following diagram commute:

          --------C--------
        f/        |        \g
        /       <f,g>       \
       |          |          |
       V          V          V
       A <------ AxB ------> B
            p1         p2

(That is, any path from one object to another which follows the arrows is equal.) The uniqueness of <f,g> implies that AxB is unique up to isomorphism.

Although this definition is not particularly straightforward, a little work shows that it's identical to Cartesian product when the objects are sets and the morphisms are functions.

If the category is a partially ordered set, then a product is a greatest lower bound. (There is a morphism from A to B if and only if A <= B. The definition then says that AxB is the object which is less than or equal to both A and B, yet is greater than or equal to any other object which is <= A and B.)

Some categories don't have all products. For example, the finite category 2 (consisting of two objects 0 and 1 with just their identity morphisms) has the products 1x1 == 1 and 0x0 == 0, but no object 0x1. Products are just one example of a broader category theory concept called limits.

Prod"uct (?), n. [L. productus, p. pr. of producere. See Produce.]

1.

Anything that is produced, whether as the result of generation, growth, labor, or thought, or by the operation of involuntary causes; as, the products of the season, or of the farm; the products of manufactures; the products of the brain.

There are the product Of those ill-mated marriages. Milton.

These institutions are the products of enthusiasm. Burke.

2. Math.

The number or sum obtained by adding one number or quantity to itself as many times as there are units in another number; the number resulting from the multiplication of two or more numbers; as, the product of the multiplication of 7 by 5 is 35. In general, the result of any kind of multiplication. See the Note under Multiplication.

Syn. -- Produce; production; fruit; result; effect; consequence; outcome; work; performance.

© Webster 1913.

Pro*duct" (?), v. t.

1.

To produce; to bring forward.

Foxe.

2.

To lengthen out; to extend.

He that doth much . . . products his mortality. Hackett.

3.

To produce; to make.

Holinshed.

© Webster 1913.