Editing Equaliser (mathematics)

{{short description|Set of arguments where two or more functions have the same value}}
{{More citations needed|date=August 2024}}
In [[mathematics]], an '''equaliser''' is a set of arguments where two or more [[function (mathematics)|function]]s have [[equality (math)|equal]] values.
An equaliser is the [[solution set]] of an [[equation]].
In certain contexts, a '''difference kernel''' is the equaliser of exactly two functions.

== Definitions ==

Let ''X'' and ''Y'' be [[Set (mathematics)|sets]].
Let ''f'' and ''g'' be [[function (mathematics)|function]]s, 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:
: <math> \operatorname{Eq}(f, g) := \{x \in X \mid f(x) = g(x)\}. </math>
The equaliser may be denoted Eq(''f'', ''g'') or a variation on that theme (such as with lowercase letters "eq").
In informal contexts, the notation {''f'' = ''g''} is common.

The definition above used two functions ''f'' and ''g'', but there is no need to restrict to only two functions, or even to only [[finite set|finite]]ly many functions.
In general, if '''F''' is a [[Set (mathematics)|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:
: <math> \operatorname{Eq}(\mathcal{F}) := \{x \in X \mid \forall f,g \in \mathcal{F}, \; f(x) = g(x)\}. </math>
This equaliser may be written as Eq(''f'', ''g'', ''h'', ...) if <math> \mathcal{F}</math> is the set {''f'', ''g'', ''h'', ...}.
In the latter case, one may also find {''f'' = ''g'' = ''h'' = ···} in informal contexts.

As a [[degenerate (math)|degenerate]] case of the general definition, let '''F''' be a [[singleton (set theory)|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]].

== Difference kernels ==

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'' &minus; ''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 (algebra)|kernel]] of the difference ''f'' &minus; ''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 [[0 (number)|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.

== 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 [[category theory|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 [[commutative diagram|diagram]] in the category in question, and the equaliser is simply the [[limit (category theory)|limit]] of that diagram.

In more explicit terms, the equaliser consists of an object ''E'' and a morphism ''eq'' : ''E'' → ''X'' satisfying <math>f \circ eq = g \circ eq</math>,
and such that, given any object ''O'' and morphism ''m'' : ''O'' → ''X'', if <math>f \circ m = g \circ m</math>, then there exists a [[unique (mathematics)|unique]] morphism ''u'' : ''O'' → ''E'' such that <math>eq \circ u = m</math>.

<div style="text-align: center;">[[Image:Equalizer-01.png|class=skin-invert]]</div>

A morphism <math>m:O \rightarrow X</math> is said to '''equalise''' <math>f</math> and <math>g</math> if <math>f \circ m = g \circ m</math>.<ref>{{cite book|last1=Barr|first1=Michael|url=http://www.tac.mta.ca/tac/reprints/articles/22/tr22.pdf|title=Category theory for computing science|last2=Wells|first2=Charles|publisher=[[Prentice Hall International Series in Computer Science]]|year=1998|page=266|format=PDF|author-link1=Michael Barr (mathematician)|author-link2=Charles Wells (mathematician)}}</ref>

In any [[universal algebra]]ic 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 (category theory)|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 (logic)|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 category|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 category|enriched]] over the category of [[Abelian group]]s), 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 [[kernel (category theory)|category-theoretic kernel]].

Any category with [[Pullback (category theory)|fibre products]] (pullbacks) and products has equalisers.

=== Category of topological spaces (''Top'') ===
In [[Category of topological spaces|'''Top''']] where objects are [[Topological space|topological spaces]] and [[Morphism|morphisms]] are [[continuous maps]], the equaliser of two continuous maps <math>f, g : X \rightarrow Y</math> retains the [[underlying set]] <math>E = \{x \in X | f(x) = g(x)\}</math>, but endows it with the subspace topology inherited from <math>X</math>. The inclusion <math>e : E \rightarrow X</math> is continuous, and the universal property is preserved: any continuous map <math>h : Z \rightarrow X</math> coalescing with <math>f</math> and <math>g</math> factors uniquely through <math>e</math>. This exemplifies the principle that topological structure, when subordinate to the equaliser's defining condition, is inherited without contradiction.<ref>{{Cite book |last=Riehl |first=Emily |title=Category Theory in Context |date=November 2016 |publisher=[[Dover Publications]] |isbn=978-0486809038 |language=en |chapter=Chapter 3 (Basic Category Theory)}}</ref>

== See also ==

*[[Coequalizer|Coequaliser]], the [[dual (category theory)|dual]] notion, obtained by reversing the arrows in the equaliser definition.
*[[Coincidence theory]], a topological approach to equaliser sets in [[topological space]]s.
*[[Pullback (category theory)|Pullback]], a special [[Limit (category theory)|limit]] that can be constructed from equalisers and products.

==Notes==
{{reflist}}

==References==
* {{nlab|id=equalizer|title=Equalizer}}

== External links ==

*[https://web.archive.org/web/20080916162345/http://www.j-paine.org/cgi-bin/webcats/webcats.php Interactive Web page ] which generates examples of equalisers in the category of finite sets. Written by [https://web.archive.org/web/20081223001815/http://www.j-paine.org/ Jocelyn Paine].

{{Category theory}}

[[Category:Set theory]]
[[Category:Limits (category theory)]]