===== Equalizer . category theory =====
==== Tuple ====
| @#55CCEE: context    | @#55CCEE: $F:(a{\overset{\rightarrow}{\rightarrow}}b)\longrightarrow{\bf C}$ |
| @#FF9944: definition | @#FF9944: $\langle E,\langle Fa\mapsto e,Fb\mapsto e'\rangle\rangle := \mathrm{lim}\,F$ |

Here $a{\overset{\rightarrow}{\rightarrow}}b$ denotes the two object category with two parallel arrows. 

Let $A:=Fa$, $B:=Fb$ and $f$ be one of the fmap images. Then $e:{\bf C}[E,A]$ and the other arrow must be $e\circ f$ and is hence usually ignored.

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{{ category_theory_equalizer_diagram.png?X200}}

=== Examples ===
In ${\bf{Set}}$, if $f,g:A\to B$, then their equalizer is the set 

$E=\{a\in A\ |\ f(a)=g(a)\}$ 

and $e:E\to A$ is the obvious injection.

Remark: In the image, $e=E$ and $i=e$.
=== Reference ===
nLab: [[http://ncatlab.org/nlab/show/equalizer|Equalizer]]

Wikipedia: [[http://en.wikipedia.org/wiki/Equaliser_%28mathematics%29|Equalizer]]

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=== Context ===
[[Functor]]
=== Subset of ===
[[A triangle limit]]