product

Formally, a product of two objects X and Y in a category C is an object Z of C together with two morphisms, called the projections, p:Z→X and q:Z→Y such that—and this is the universal property—for all objects W with morphisms f:W→X and g:W→Y, there is a unique morphism h:W→Z such that p∘h=f and q∘h=g.

Note that we have defined a product for X and Y and not the product for X and Y. Indeed, products and other objects with a universal property are defined only up to a (unique) isomorphism.

实例

• 范畴Set中对象的product就是集合的笛卡尔积
• 一个偏序集构成的范畴中对象product就是上确界(least upper bound)