nLab equifier

Idea

An equifier is a particular kind of 2-limit in a 2-category, which universally renders a pair of parallel 2-morphisms 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 $K$.

Example: equifiers in the strict 2-category of categories

In the strict 2-category of categories, equifiers can be computed as follows. The input data is two categories $A$ and $B$, two functors $F,G\colon A\to B$, and two natural transformations $p,q\colon F\to G$. The equifier of $p$ and $q$ is the full subcategory of $A$ consisting of those objects $X\in A$ for which $p_X=q_X$.

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.