Product and coproduct compared

Overview

The pages product and coproduct pages make it hard to see just how similar the two definitions are. As a result I shall steal the format from and do a two-column layout showing the differences and similarities.

Definition

Given a pair $A$, $B$ of objects in a category $\mathscr{C}$ a:

Product	Coproduct
is a wedge
$\xymatrix{ A \\ S \ar[u]_{p_A} \ar[d]^{p_B} \\ B}$	$\xymatrix{ A \ar[d]_{i_A} \\ S \\ B \ar[u]^{i_B} }$
with the following universal property; for each wedge:
$\xymatrix{ & A\\ X \ar[ur]^{f_A} \ar[dr]_{f_B} & \\ & B }$	$\xymatrix{ A \ar[dr]^{f_A} & \\ & X \\ B \ar[ur]_{f_B} &}$
there exists a unique arrow
$X\mathop{\longrightarrow}^m S$	$S\mathop{\longrightarrow}^m X$
such that the following diagram commutes
$\begin{xy}\xymatrix{ & A \\ X \ar[ur]^{f_A} \ar[dr]_{f_B} \ar[r]^m & S \ar[u]_{p_A} \ar[d]^{p_B} \\ & B}\end{xy}$	$\begin{xy}\xymatrix{A \ar[d]_{i_A} \ar[dr]^{f_A} & \\ S \ar[r]^m & X \\ B \ar[u]^{i_B} \ar[ur]_{f_B} & }\end{xy}$

We call the arrow $m$ the mediating arrow (AKA: mediator) for the wedge on $X$

Notation

The product is usually denoted $\times$ and the coproduct by $+$, if they agree (are the same) then we use $\oplus$

(Unknown grade)

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

This notation thing needs expanding