===== A triangle limit =====
==== Collection ====
{{ category_theory_a_triangle_pullback.jpg?X400}}

Consider the circle graph with 3 vertices $a,b,c$ and 3 edges. 

There are, up to relabeling, two directed versions of it: 
  * The graph where all arrows go in one direction, say

$h:a\to c$

$g:c\to b$

$f:b\to a$

  * The graph where one of the arrows point in another direction, say

$h:a\to c$

$g:c\to b$

$f:a\to b$

so that both $f$ and $g$ point at $b$.

{{ category_theory_equalizer_diagram.png?X200}}
If $h$ is an isomorphism $h:c\simeq a$, then we can replace $c$ by $a$ and we're left with a pullback. This can also be viewed as a two-parallel-arrows situation, and it is called the [[equalizer]].

In ${\bf{Set}}$, if $h$ is an iso, the object is $e=\{x\in a\ |\ f(x)=g(x)\}$.
For general $h$, the object is a subset of the pullback $a\times_b c$, and thus in particular a subset of the Cartesian product $a\times c$.

$e=\{\langle x,y\rangle\in a\times_b c\,\mid\,h(y)=x\}$

or

$e=\{\langle x,y\rangle\in a\times c\,\mid\,f(x)=g(y)\land h(y)=x\}$

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=== Reference ===

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=== Related ===
[[Pullback . category theory]], 
[[Equalizer . category theory]]
=== Context ===
[[Functor]]
=== Refinement of ===
[[Limit . category theory]]