# nLab equifier

### Context

#### 2-Category theory

2-category theory

## Structures on 2-categories

#### Limits and colimits

limits and colimits

# Contents

## Idea

An equifier is a particular kind of 2-limit in a 2-category, which universally renders a pair of parallel 2-morphism equal.

## Definition

Let $f,g\colon A\rightrightarrows B$ be a pair of parallel 1-morphisms in a 2-category and let $\alpha,\beta\colon f\rightrightarrows g$ be a pair of parallel 2-morphisms. The equifier of $\alpha$ is a universal object $V$ equipped with a morphism $v\colon V\to A$ such that $\alpha v = \beta v$.

More precisely, universality means that for any object $X$, the induced functor

$Hom(X,V) \to Hom(X,A)$

is fully faithful, and its replete image consists precisely of those morphisms $u\colon X\to A$ such that $\alpha u=\beta u$. If the above functor is additionally an isomorphism of categories onto the exact subcategory of such $u$, then we say that $V\xrightarrow{v} A$ is a strict equifier.

Equifiers and strict equifiers can be described as a certain sort of weighted 2-limit, where the diagram shape is the walking parallel pair of 2-morphisms $P$, and the weight $P\to Cat$ is the diagram

$\array{ & \to \\ 1 & \Downarrow\Downarrow & I\\ & \to }$

where $1$ is the terminal category and $I$ is the interval category. Note that this cannot be re-expressed as any sort of conical 2-limit.

An equifier in $K^{op}$ (see opposite 2-category) is called a coequifier in $K$.

## Properties

• The above explicit definition makes it clear that any equifier is a fully faithful morphism.

• Any strict equifier is, in particular, an equifier. (This is not true for all strict 2-limits.)

• Strict equifiers are, by definition, a particular case of PIE-limits.

Revised on December 14, 2010 06:08:11 by Mike Shulman (71.137.3.108)