Products of Objects

Products

Product of Objects

In a category , a product for the objects and consists of an object and arrows and : satisfying the following universal property: for all with and , there exists unique such that the following diagram commutes: We write such product as , and write for .

e.g.

• Products in is Cartesian products.
• Products in is the meet (i.e., greatest lower bound) as we require that , and .
• Products in is the product topology.

Proposition

Products are unique up to isomorphism.

Proof Suppose we have and are products of : Then there is unique such that and . Similarly, exists unique such that and . Therefore,It follows that and , thus .

Proposition

The binary product of objects is associative up to isomorphism:

Def All Finite Products A category is said to have all finite products if it has a terminal object and all binary products. The category has all (small) products if every set of objects in has a product.

Duality Principle

Formal Duality

For any statement in the language of category theory, if follows from the axioms for categories, then so does :

Conceptual Duality

For any statement about categories, if holds for all categories, then so does the dual statement .

Proof If holds for all categories , then it also holds in all categories , but then holds in all categories , thus in all categories .

Coproducts

Coproduct

In any category , is a coproduct (dual product) of objects and if for any and and , there is a unique with for , all as indicated in: In other words, a coproduct of two objects is exactly their product in the opposite category. We usually write for the coproduct, and for the uniquely determined morphism . And are usually called coproduct injections.

e.g.

• In , the coproduct of two sets is their disjoint union with evident coproduct injections and . And we we have So every finite set is a coproduct: This is because a function is uniquely determined by its values for all .
• If and are free monoids on sets and , then in we can construct their coproduct as .
• In a fixed poset , a coproduct of two elements is the join (i.e., supremum) of and .
• In , given groups and , the free group of words is usually called the free product of and , written or . The free product is actually a coproduct.
• In the category of abelian groups, the coproduct of and is the Cartesian product of the underlying sets and .
• The tensor product of algebras is the coproduct in the category of commutative algebras .

Proposition

In the category of abelian groups, there is a canonical isomorphism between the binary coproduct and product,

Proof Let . Then for all , $$\begin{aligned}\pi(a,b)&=\langle1_A,0_B\rangle,\langle0_A,1_B\rangle\&=\langle1_A,0_B\rangle(a)+\langle0_A,1_B\rangle(b)\&=(1_A(a),0_B(a))+(0_A(b),1_B(b))\&=(a,0_B)+(0_A,b)\&=(a+0_A,0_B+b)\&=(a,b)\end{aligned}$$$\square$

Corollary

The following properties hold for coproducts:

• Coproducts are unique up to isomorphism.
• Coproducts are associative. That is .

Proof By duality, and the fact that the dual of an isomorphism is an isomorphism.

Coproduct Functor

Since we define the coproduct of two morphisms as which leads to a coproduct functor on categories with binary coproducts.