Difference between revisions of "Product and coproduct compared"
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| − | We call the arrow {{M|m}} the | + | We call the arrow {{M|m}} ''the mediating arrow''<ref name="AITCTHS2010"/> ({{AKA}}: ''mediator''<ref name="AITCTHS2010"/>) for the [[wedge (category theory)|wedge]] on {{M|X}} |
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==Notation== | ==Notation== | ||
The [[product (category theory)|product]] is usually denoted {{M|\times}} and the [[coproduct (category theory)|coproduct]] by {{M|+}}, if they agree (are the same) then we use {{M|\oplus}} | The [[product (category theory)|product]] is usually denoted {{M|\times}} and the [[coproduct (category theory)|coproduct]] by {{M|+}}, if they agree (are the same) then we use {{M|\oplus}} | ||
Latest revision as of 16:59, 1 March 2016
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]
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References
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