Equaliser (mathematics) - in Category Theory

In Category Theory

Equalisers can be defined by a universal property, which allows the notion to be generalised from the category of sets to arbitrary categories.

In the general context, X and Y are objects, while f and g are morphisms from X to Y. These objects and morphisms form a diagram in the category in question, and the equaliser is simply the limit of that diagram.

In more explicit terms, the equaliser consists of an object E and a morphism eq : E → X satisfying, and such that, given any object O and morphism m : O → X, if, then there exists a unique morphism u : O → E such that .

In any universal algebraic category, including the categories where difference kernels are used, as well as the category of sets itself, the object E can always be taken to be the ordinary notion of equaliser, and the morphism eq can in that case be taken to be the inclusion function of E as a subset of X.

The generalisation of this to more than two morphisms is straightforward; simply use a larger diagram with more morphisms in it. The degenerate case of only one morphism is also straightforward; then eq can be any isomorphism from an object E to X.

The correct diagram for the degenerate case with no morphisms is slightly subtle: one might initially draw the diagram as consisting of the objects X and Y and no morphisms. This is incorrect, however, since the limit of such a diagram is the product of X and Y, rather than the equalizer. (And indeed products and equalizers are different concepts: the set-theoretic definition of product doesn't agree with the set-theoretic definition of the equalizer mentioned above, hence they are actually different.) Instead, the appropriate insight is that every equalizer diagram is fundamentally concerned with X, including Y only because Y is the codomain of morphisms which appear in the diagram. With this view, we see that if there are no morphisms involved, Y does not make an appearance and the equalizer diagram consists of X alone. The limit of this diagram is then any isomorphism between E and X.

It can be proved that any equaliser in any category is a monomorphism. If the converse holds in a given category, then that category is said to be regular (in the sense of monomorphisms). More generally, a regular monomorphism in any category is any morphism m that is an equaliser of some set of morphisms. Some authorities require (more strictly) that m be a binary equaliser, that is an equaliser of exactly two morphisms. However, if the category in question is complete, then both definitions agree.

The notion of difference kernel also makes sense in a category-theoretic context. The terminology "difference kernel" is common throughout category theory for any binary equaliser. In the case of a preadditive category (a category enriched over the category of Abelian groups), the term "difference kernel" may be interpreted literally, since subtraction of morphisms makes sense. That is, Eq(f,g) = Ker(f - g), where Ker denotes the category-theoretic kernel.

Any category with fibre products (pull backs) and products has equalisers.