Difference between revisions of "Types of category arrows"
From Maths
(Created page with "==Monic & Epic== Monic captures the idea of an injection {| class="wikitable" border="1" |- | align=center | <span style="font-size:1.7em;">{{M|\xymatrix{X \ar@<-.5ex>[r]_...") |
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| + | ==Overview== | ||
| + | There are many kinds of [[arrow|arrows]] in a [[category]], here are some common terms compared, and a diagram showing how they relate. | ||
| + | {{:Types of category arrows/Diagram}} | ||
==Monic & Epic== | ==Monic & Epic== | ||
Monic captures the idea of an [[injection]] | Monic captures the idea of an [[injection]] | ||
Latest revision as of 14:43, 6 February 2016
Overview
There are many kinds of arrows in a category, here are some common terms compared, and a diagram showing how they relate. [ilmath]\xymatrix{ & \text{Arrow} \\ \text{Monic} \ar@{^{(}->}[ur] & & \text{Epic} \ar@{^{(}->}[ul] \\ & \text{Bimorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \\ {\begin{array}{c}\text{Section}\\ \text{(Split monic)} \end{array} } \ar@{^{(}->}[uu] & & {\begin{array}{c}\text{Retraction}\\ \text{(Split epic)} \end{array} } \ar@<-0.75ex>@{^{(}->}[uu] \\ & \text{Isomorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \ar@{^{(}->}[uu] }[/ilmath]
Monic & Epic
Monic captures the idea of an injection
| [ilmath]\xymatrix{X \ar@<-.5ex>[r]_g \ar@<.5ex>[r]^f & B \ar[r]^m & A} [/ilmath] |
| (warning: non-commutative diagram) |
|---|
[ilmath]m[/ilmath] is monic if:
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And epic, which captures the idea of a surjection
| [ilmath]\xymatrix{A \ar[r]^e & B \ar@<-.5ex>[r]_g \ar@<.5ex>[r]^f & X} [/ilmath] |
| (warning: non-commutative diagram) |
|---|
[ilmath]m[/ilmath] is epic if:
|
