Product . category theory

Collection

 context $F:\mathrm{Ob}_{{\bf C}^{\bf 2}}$ range $a,b:\mathrm{Ob}_{\bf 2},\ a\neq b$ definition $\langle Fa\times Fb, \pi\rangle := \mathrm{lim}\,F$

Discussion

Elaboration

${\bf C}^{\bf 2}$ is the functor category with objects being functors in ${\bf 2}\longrightarrow{\bf C}$ and the morphisms are natural transformations, i.e. families of ${\bf 2}$-indexed arrows in ${\bf C}$.

So

$\pi:\prod_{x:\mathrm{Ob}_{\bf 2}}\left(Fa\times Fb\right)\to Fx$

I.e. $\pi_a:(Fa\times Fb)\to Fa$ and $\pi_b:(Fa\times Fb)\to Fb$.

Idea

We first discuss the concept in the category of sets.

Say we want to specify the binary product of $A$ and $B$ in ${\bf C}$. We can do this in the language of cones, by considering the discrete category with only two objects $a,b$ and no non-identity arrows, and define a functor $Fa:=A$ and $F:=B$. A cone is any object $N$ with two arrows $\psi_A:N\to A$ and $\psi_B:N\to B$. If there is a limit cone, let's call its tip $A\times B$, then you can put the two arrows together to define a map $u(n):=\langle\psi_A(n),\psi_B(n)\rangle$ from $N$ to $A\times B$, and then $\psi_A(n)=\pi_A(u(n))$ and $\psi_B(n)=\pi_B(u(n))$.

If the objects of a category are propositions, then the product is $\land$ i.e. 'and': from $A\land B$ you can derive $A$ as well as $B$. The coproduct turns out to be $\lor$, i.e. 'or': From both $A$ or $B$, you can derive $A\lor B$.

While the category ${\bf C}$ might have a billion ways to “look at” $A$ and $B$, category theory works out that these will always just be some arrow concatenated with the limit cones binoculars - that's sort of the “why” answer to why projection operators are an ubiquitous concept.

Alternative definitions

If ${\bf C}$ has a terminal object $T$, the product $A\times B$ is the pullback $A\times_T B$.