Products and Coproducts

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 $$ [f,g] \circ \iota_2 = g $$