S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 8: Categories: An Interlude

8.2 Universal Properties

Many constructions in mathematics are characterized not by explicit formulas but by universal properties—conditions that specify the construction uniquely up to isomorphism. We’ve already seen this with quotient spaces: the quotient is characterized by factoring any linear map that kills . Products and coproducts provide two more fundamental examples.

8.2.1 Products and Coproducts

Definition 8.2.1 (Product). Let be a category and . A product of and is an object together with morphisms and (called projections) satisfying the following universal property:

For any object and morphisms , , there exists a unique morphism such that and .

Universal property of the product: α from T to A and β from T to B factor through a unique dashed map φ from T to P followed by the projections π1 to A and π2 to B.

We write and for the unique induced map.

Note. The universal property says: to give a map into the product is the same as giving a map into each factor. This is the categorical abstraction of the familiar fact that a function is determined by its components and .

The dual notion reverses all arrows.

Definition 8.2.2 (Coproduct). Let be a category and . A coproduct (or sum) of and is an object together with morphisms and (called inclusions or coprojections) satisfying:

For any object and morphisms , , there exists a unique morphism such that and .

Universal property of the coproduct: α from A to T and β from B to T factor through the inclusions ι1 and ι2 into S followed by a unique dashed map φ from S to T.

We write or and for the unique induced map.

Note. The universal property of the coproduct says: to give a map out of the coproduct is the same as giving a map out of each summand. This duality between products and coproducts—reversing all arrows—is a general phenomenon in category theory.

Theorem 8.2.1 (Uniqueness of products and coproducts). If a product (resp. coproduct) of and exists, it is unique up to unique isomorphism. That is, if and are both products of and , then there is a unique isomorphism with and .

Proof. By the universal property of , applied to the maps and , there exists a unique with and .

By the universal property of , applied to and , there exists a unique with and .

Now satisfies and . But also satisfies these equations. By uniqueness in the universal property of , we have . Similarly, .

The proof for coproducts is dual (reverse all arrows). ∎

Note. This proof pattern—using universal properties to construct inverse maps—is standard in category theory. The key is that universal properties give uniqueness, which forces the composites to be identities.

Products and coproducts need not exist in every category. When working in a new category, one of the first questions is whether these constructions exist and what form they take.

8.2.2 Examples in

Example 8.2.1 (Products and coproducts in ). In the category of sets:

  • The product is the Cartesian product with projections and .
  • The coproduct is the disjoint union with inclusions and .

The universal properties are immediate from the definitions of functions on Cartesian products and disjoint unions.

Example 8.2.2 (Products and coproducts in ). In the category of groups:

  • The product is the direct product with componentwise operations and projections.
  • The coproduct is the free product—a more complicated construction involving alternating words from and .

In the subcategory of abelian groups, both product and coproduct are the direct sum (with the same underlying group but different universal properties).

Example 8.2.3 (Products and coproducts in ). In the category of finite-dimensional vector spaces over , both the product and coproduct of and are realized by the direct sum .

As a product: Define by and by . Given linear maps and , define by . This is the unique map with and .

As a coproduct: Define by and by . Given linear maps and , define by . This is the unique map with and .

Note. The coincidence of products and coproducts in (and ) reflects the additive structure of these categories. In categories without such structure (like or ), products and coproducts are genuinely different constructions.

Exercise 8.2.1 (Products and coproducts exercises).

  1. Verify explicitly that the Cartesian product satisfies the universal property of products in .
  2. Show that in , the coproduct of and is , while in , the coproduct is the free group on two generators.
  3. Let and be finite-dimensional vector spaces over . Show that using the universal property (not by direct computation).
Show solution

(1) Let be sets and let be any set with functions and . We must show there exists a unique function with and .

Existence: Define . Then and .

Uniqueness: If satisfies and , then for any , .

(2) In : The coproduct of and has inclusions such that any pair of homomorphisms (for abelian) factors uniquely through . Taking with and , the map defined by is the unique such factorization.

In : The coproduct must handle non-commuting images. Given any group and homomorphisms , the elements and generate a subgroup that need not be abelian. The free group with and satisfies the universal property: any choice of and in extends uniquely to a homomorphism .

(3) By the universal property of coproducts, for any . Taking : Thus . Since for finite-dimensional spaces, we get .

Note. The argument in (3) illustrates a general principle: universal properties translate into isomorphisms of Hom-sets. The functor sends coproducts to products, which is why . This is part of a larger story involving adjoint functors.