# Category theory

- Category (mathematics)
A description of how a collection of mathematical objects are related to one another.

- Equaliser (category theory)
- Product (Category Theory)
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.

- Product is unique up to isomorphism

- Universal property of the product
- Object identity via interactions
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!

- Universal property
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.

- The empty set is the only set which satisfies the universal property of the empty set
- 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.

- Product is unique 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.

- Universal property of the empty set
- Category of finite 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.