coequalizer

**limits and colimits**
## 1-Categorical
* limit and colimit
* limits and colimits by example
* commutativity of limits and colimits
* small limit
* filtered colimit
* directed colimit
* sequential colimit
* sifted colimit
* connected limit, wide pullback
* preserved limit, reflected limit, created limit
* product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
* finite limit
* exact functor
* Kan extension
* Yoneda extension
* weighted limit
* end and coend
## 2-Categorical
* 2-limit
* inserter
* isoinserter
* equifier
* inverter
* PIE-limit
* 2-pullback, comma object
## (∞,1)-Categorical
* (∞,1)-limit
* (∞,1)-pullback
* fiber sequence
### Model-categorical
* homotopy Kan extension
* homotopy limit
* homotopy pullback
* mapping cone
* homotopy fiber
* homotopy colimit
* homotopy pushout
* homotopy cofiber
* mapping cocone

In a category $C$ a diagram of morphisms of $C$

$U \underoverset{b}{a}{\rightrightarrows} V \overset{c}{\rightarrow} X$

is called a **coequalizer** diagram if 1. $c a=c b$; and 1. $c$ is universal for this property: i.e. if $f: V \to Y$ is a morphism of $C$ such that $f a=f b$, then there is a unique morphism $f': X \to Y$ such that $f'c=f$.

This concept is a special case of that of colimit; specifically, it’s the colimit of the diagram

$U \underoverset{b}{a}{\rightrightarrows} V .$

A coequalizer in $C$ is an equalizer in the opposite category $C^{op}$.

Revised on June 8, 2011 20:08:27
by Anonymous Coward
(216.239.45.4)