In mathematics, an equaliser is a set of arguments where two or more functions have equal values.
In certain contexts, a difference kernel is the equaliser of exactly two functions.
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 equaliser may be denoted Eq(f, g) or a variation on that theme (such as with lowercase letters "eq").
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: This equaliser may be written as Eq(f, g, h, ...) if
Then the equaliser is again the entire domain X, since the universal quantification in the definition is vacuously true.
In more explicit terms, the equaliser consists of an object E and a morphism eq : E → X satisfying
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.)
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 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.