Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
An equifier is a particular kind of 2-limit in a 2-category, which universally renders a pair of parallel 2-morphisms equal.
Let be a pair of parallel 1-morphisms in a 2-category and let be a pair of parallel 2-morphisms. The equifier of is a universal object equipped with a morphism such that .
More precisely, universality means that for any object , the induced functor
is fully faithful, and its replete image consists precisely of those morphisms such that . If the above functor is additionally an isomorphism of categories onto the exact subcategory of such , then we say that 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 , and the weight is the diagram
where is the terminal category and is the interval category. Note that this cannot be re-expressed as any sort of conical 2-limit.
An equifier in (see opposite 2-category) is called a coequifier in .
In the strict 2-category of categories, equifiers can be computed as follows. The input data is two categories and , two functors , and two natural transformations . The equifier of and is the full subcategory of consisting of those objects for which .
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.
Last revised on June 3, 2020 at 08:43:43. See the history of this page for a list of all contributions to it.