What is a Coproduct in Category Theory?

A coproduct is a mathematical construct that generalizes the notion of a product in a category. It is a way to combine two objects in a category into a new object, similar to how the Cartesian product combines two sets into a new set.

In a category C, a coproduct is a pair of objects A and B, together with a morphism (called a "coprojection") from A to B, such that every morphism from A to C can be factored through this coprojection. In other words, every arrow from A to C can be written as a composite of the coprojection followed by some other arrow.

Here are some key properties of coproducts:

1. Existence: Coproducts exist in any category that has a terminal object (an object that is not the source of any arrows). In particular, every category has a terminal object, which is often denoted by 1 or I.
2. Universal property: The coprojection from A to B is universal in the sense that it is the "best" way to factor out the arrow from A to C. More precisely, if there are two morphisms from A to C, one can be factored through the coprojection, and the other cannot.
3. Associativity: Coproducts are associative, meaning that (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C). This means that we can combine multiple coproducts in any order we like.
4. Distributivity: Coproducts distribute over product, meaning that A ⊕ (B × C) = (A ⊕ B) × (A ⊕ C). This allows us to use coproducts to build more complex structures from simpler ones.

Coproducts are used in many areas of mathematics, including category theory, homological algebra, and sheaf theory. They provide a way to construct new objects by combining existing ones, and they have many interesting properties and applications.