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

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

is fully faithful, and its replete image consists precisely of those morphisms $u:X\to A$ 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 $V\stackrel{v}{\to }A$ 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 $P\to \mathrm{Cat}$ is the diagram

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 $K^{\mathrm{op}}$ (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.