In mathematics, an **equaliser**, or **equalizer**, 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.

Table of contents |

2 Difference kernels 3 In category theory 4 Coequalisers |

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:

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:

As a degenerate case of the general definition, let **F** be a singleton {*f*}.
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*.
Conversely, 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 its preimage under zero; 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 from *E* to *X* satisfying *f*·eq = *g*·eq (where "·" denotes composition of morphisms);
and such that, given any *other* object *O* and morphism *m* from *O* to *X*, if *f*·*m* = *g*·*m*, then there exists a unique morphism *u* from *O* to *E* such that eq·*u* = *m*.

*There should be a picture here.*

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 degenerate case of no morphisms at all might be confusing at first; it may seem that the diagram in question consists of only the objects *X* and *Y*, with no morphsims.
The limit of that diagram is the product of *X* and *Y*, which doesn't agree with the set-theoretic definition above.
However, the correct interpretation is that the diagram is based on *X* and includes *Y* only because *Y* is the codomain of a morphism in the diagram.
Then if there are no morphisms involved, then the diagram consists of *X* alone, so the limit is again 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.

A *coequaliser*, or *coequalizer*, is the dual notion, obtained by reversing the arrows in the equaliser definition.