equifier in nLab

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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:A⇉Bf,g\colon A\rightrightarrows B be a pair of parallel 1-morphisms in a 2-category and let α,β:f⇉g\alpha,\beta\colon f\rightrightarrows g be a pair of parallel 2-morphisms. The equifier of α\alpha is a universal object VV equipped with a morphism v:V→Av\colon V\to A such that αv=βv\alpha v = \beta v.

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

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

is fully faithful, and its replete image consists precisely of those morphisms u:X→Au\colon X\to A such that αu=βu\alpha u=\beta u. If the above functor is additionally an isomorphism of categories onto the exact subcategory of such uu, then we say that V→vAV\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 PP, and the weight P→CatP\to Cat is the diagram

→ 1 ⇓⇓ I →\array{ & \to \\ 1 & \Downarrow\Downarrow & I\\ & \to }

where 11 is the terminal category and II is the interval category. Note that this cannot be re-expressed as any sort of conical 2-limit.

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

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 AA and BB, two functors F,G:A→BF,G\colon A\to B, and two natural transformations p,q:F→Gp,q\colon F\to G. The equifier of pp and qq is the full subcategory of AA consisting of those objects X∈AX\in A for which p X=q Xp_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.