# 5.Products and Coproducts

## 05.06.2020

### Contents/Index

1.Definition
2.Dual Category
3.Mono-, Epi- and Isomorphisms
4.Initial and Terminal Objects
@5.Products and Coproducts
6.Exponentation and Cartesian Closed Categories

The usual definition on sets of a Cartesian product is $$A \times B = \{ (a,b) | a \in A \land b \in B \}$$ Now we want to reason about products in a function like manner. To do this we introduce projections $$\pi_1 : A \times B \rightarrow A$$ and $$\pi_2 : A \times B \rightarrow B$$ If we consider the set of all tuples of the form $(X,f_1, f_2)$, we find that $(A \times B,\pi_1, \pi_2)$ is optimal in the following sense. For some set $C$ let $f : C \rightarrow A$ and $g : C \rightarrow B$. Now we can form the product function $$\langle f,g \rangle : C \rightarrow A \times B$$ given as $$\langle f,g \rangle (x) = (f(x),g(x))$$ We can recover the functions of the product as thus $$f = \pi_1 \circ \langle f , g \rangle$$ and $$g = \pi_2 \circ \langle f , g \rangle$$ A quick proof of the first $$(\pi_1 \circ \langle f, g \rangle)(x) = \pi_1(f(x),g(x)) = f(x)$$

We have other optimal choices. That is $(B \times A,\pi_2,\pi_2)$ is just as good. But these alternatives are isomorphic to each other.

### Definition - Product of Objects

A product of two objects is an object $A \times B$ together with the two projection arrows $$\pi_1 : A \times B \rightarrow A$$ and $$\pi_2 : A \times B \rightarrow B$$ such that for any object $C$ and pair of arrows $f : C \rightarrow A$ and $g : C \rightarrow B$, there is exactly one mediating arrow $$\langle f,g \rangle : C \rightarrow A \times B$$ such that $$\pi_1 \circ \langle f,g \rangle = f$$ and $$\pi_2 \circ \langle f,g \rangle = g$$

If a category $\mathcal{C}$ has a product $A \times B$ for every pair of objects, $A$ and $B$, we say that $\mathcal{C}$ has all binary products.

### Definition - Product Map

If $A \times C$ and $B \times D$ are product objects, then for every pair of arrows, $f : A \rightarrow B$ and $g : C \rightarrow D$, the product map $$f \times g : A \times C \rightarrow B \times D$$ is the arrow $$\langle f \circ \pi_1 , g \circ \pi_2 \rangle$$

### Definition - Coproduct

A coproduct of two objects, $A$ and $B$, is an object $A + B$ and two injection arrows $$\iota_1 : A \rightarrow A + B$$ and $$\iota_2 : B \rightarrow A + B$$ such that for any object $C$ and pair of arrows, $f : A \rightarrow C$ and $g : B \rightarrow C$, there is exactly one mediating arrow $$[f,g] : A + B \rightarrow C$$ such that $$[f,g] \circ \iota_1 = f$$ and $$[f,g] \circ \iota_2 = g$$