In mathematics, an equaliser is a set of arguments where two or more functions have equal values.An equaliser is the solution set of an equation.In certain contexts, a difference kernel is the equaliser of exactly two functions.
Let X and Y be sets.Let f and g be functions, both from X to Y.Then the equaliser of f and g is the set of elements x of X such that f(x) equals g(x) in Y.Symbolically:
\operatorname{Eq}(f,g):=\{x\inX\midf(x)=g(x)\}.
The definition above used two functions f and g, but there is no need to restrict to only two functions, or even to only finitely many functions.In general, if F is a set of functions from X to Y, then the equaliser of the members of F is the set of elements x of X such that, given any two members f and g of F, f(x) equals g(x) in Y.Symbolically:
\operatorname{Eq}(l{F}):=\{x\inX\mid\forallf,g\inl{F}, f(x)=g(x)\}.
l{F}
As a degenerate case of the general definition, let F be a singleton .Since f(x) always equals itself, the equaliser must be the entire domain X.As an even more degenerate case, let F be the empty set. Then the equaliser is again the entire domain X, since the universal quantification in the definition is vacuously true.
A binary equaliser (that is, an equaliser of just two functions) is also called a difference kernel. This may also be denoted DiffKer(f, g), Ker(f, g), or Ker(f - g). The last notation shows where this terminology comes from, and why it is most common in the context of abstract algebra: The difference kernel of f and g is simply the kernel of the difference f - g. Furthermore, the kernel of a single function f can be reconstructed as the difference kernel Eq(f, 0), where 0 is the constant function with value zero.
Of course, all of this presumes an algebraic context where the kernel of a function is the preimage of zero under that function; that is not true in all situations.However, the terminology "difference kernel" has no other meaning.
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
f\circeq=g\circeq
f\circm=g\circm
eq\circu=m
A morphism
m:O → X
f
g
f\circm=g\circm
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 equaliser. (And indeed products and equalisers are different concepts: the set-theoretic definition of product doesn't agree with the set-theoretic definition of the equaliser mentioned above, hence they are actually different.) Instead, the appropriate insight is that every equaliser 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 equaliser 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 authors 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 (pullbacks) and products has equalisers.