Product and coproduct compared
From Maths
Contents
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[1] and do a two-column layout showing the differences and similarities.
Definition
Given a pair [ilmath]A[/ilmath], [ilmath]B[/ilmath] of objects in a category [ilmath]\mathscr{C} [/ilmath] a:
Product | Coproduct |
---|---|
is a wedge | |
[ilmath]\xymatrix{ A \\ S \ar[u]_{p_A} \ar[d]^{p_B} \\ B}[/ilmath] | [ilmath]\xymatrix{ A \ar[d]_{i_A} \\ S \\ B \ar[u]^{i_B} }[/ilmath] |
with the following universal property; for each wedge: | |
[ilmath]\xymatrix{ & A\\ X \ar[ur]^{f_A} \ar[dr]_{f_B} & \\ & B }[/ilmath] | [ilmath]\xymatrix{ A \ar[dr]^{f_A} & \\ & X \\ B \ar[ur]_{f_B} &}[/ilmath] |
there exists a unique arrow | |
[ilmath]X\mathop{\longrightarrow}^m S[/ilmath] | [ilmath]S\mathop{\longrightarrow}^m X[/ilmath] |
such that the following diagram commutes | |
[ilmath]\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}[/ilmath] | [ilmath]\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}[/ilmath] |
We call the arrow [ilmath]m[/ilmath] the mediating arrow[1] (AKA: mediator[1]) for the wedge on [ilmath]X[/ilmath]
Notation
The product is usually denoted [ilmath]\times[/ilmath] and the coproduct by [ilmath]+[/ilmath], if they agree (are the same) then we use [ilmath]\oplus[/ilmath]
(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
References
|