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The Lumenary
Early undergraduateCategory TheoryPlanned

Universal properties

Describing objects by their relationships rather than their elements: universal properties of products, quotients, free objects and fields of fractions, and the fact that an object with a universal property is unique up to unique isomorphism.

math.cat.universal-properties

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Builds on

CategoriesProducts by maps: a first universal property

Required prerequisites, all the way down: 16

You will learn to

  • State the universal properties of products in Set, Grp and Top, of quotient sets and quotient groups, and of free monoids and free groups.
  • Prove that any two objects with the same universal property are isomorphic by a unique isomorphism compatible with the structure maps.
  • Recognise when a familiar construction, such as the tensor product or the field of fractions, is best defined by a universal property.

Definition

  • A universal arrow from an object X of C to a functor U : D → C is an object F of D with a morphism η : X → U(F) such that every morphism X → U(E) equals U(g) ∘ η for a unique g : F → E.
  • A free monoid on a set S is a monoid F(S) with a map S → F(S) through which every map from S to a monoid factors uniquely via a monoid homomorphism.

Theorem

  • Objects defined by a universal property are unique up to a unique isomorphism compatible with the structure maps.
  • The free monoid on S is the monoid of finite words in the elements of S under concatenation.
  • The quotient map X → X/~ is universal among maps out of X that are constant on equivalence classes.

Common misconceptions

  • Thinking a universal property constructs an object; it characterises one, and existence must be proved separately. For example, the fields 𝔽₂ and 𝔽₃ have no product in the category of fields.
  • Forgetting the uniqueness clause, which is what makes the object unique up to isomorphism.

Leads to

Products, coproducts, initial and terminal objectsAdjunctions

Optional discoveries

Further reading

  • Tom Leinster, Basic Category Theory (2014)
  • Emily Riehl, Category Theory in Context (2016)