Category theory

A description of how a collection of mathematical objects are related to one another.

How a product is characterized rather than how it’s constructed

• Universal property of the product

  The product can be defined in a very general way, applicable to the natural numbers, to sets, to algebraic structures, and so on.

  • Product is unique up to isomorphism

    If something satisfies the universal property of the product, then it is uniquely specified by that property, up to isomorphism.

If we think of objects as opaque “black boxes”, how can we tell whether two objects are different? By looking at how they interact with other objects!

A universal property is a way of defining an object based purely on how it interacts with other objects, rather than by any internal property of the object itself.

• Universal property of the empty set

  The empty set can be characterised by how it interacts with other sets, rather than by any explicit property of the empty set itself.

  • The empty set is the only set which satisfies the universal property of the empty set

    This theorem tells us that the universal property provides a sensible way to define the empty set uniquely.

• Universal property of the product

  The product can be defined in a very general way, applicable to the natural numbers, to sets, to algebraic structures, and so on.

  • Product is unique up to isomorphism

    If something satisfies the universal property of the product, then it is uniquely specified by that property, up to isomorphism.

• Universal property of the disjoint union

  Just as the empty set may be described by a universal property, so too may the disjoint union of sets.

The category of finite sets is exactly what it claims to be. It’s a useful training ground for some of the ideas of category theory.