Equalizer

Co - equalizer is a categorical generalization of the concept of a factor with respect to the equivalence relation . This concept is dual to the concept of equalizer , hence the name.

Definition

A co - equalizer is a colimit of a diagram consisting of two objects - X and Y , and two parallel morphisms f , g : X → Y.

More explicitly, a co-equalizer is an object Q together with a morphism q : Y → Q such that q ∘ f = q ∘ g . Moreover, the pair ( Q , q ) has the universal property : for any other pair ( Q ′, q ′) with the same property there exists a unique morphism u : Q → Q ′ that closes the following diagram to a commutative one :

Like any universal constructions, the co-equalizer, if one exists, is defined up to isomorphism. It can be shown that the co-equalizer q is an epimorphism in any category.

Examples

In the category of sets, the co-equalizer of two functions f , g : X → Y is the factor Y in the weakest equivalence relation $\sim$ ${\ displaystyle \ sim}$ such that for any $x\in X$ ${\ displaystyle x \ in X}$ , right $f(x)\sim g(x)$ ${\ displaystyle f (x) \ sim g (x)}$ .

In the category of groups, the situation is very similar: if f , g : X → Y are group homomorphisms, their co-equalizer is a factor Y in the normal closure of the set:
$S=\{f(x)g(x)^{-1}\ |\ x\in X\}$ ${\ displaystyle S = \ {f (x) g (x) ^ {- 1} \ | \ x \ in X \}}$ .

For abelian groups, the co-equalizer is especially simple. This is simply the quotient group Y / im ( f - g ) (the cokernel of the morphism f - g ).

In the category of topological spaces, the circle $S^{1}$ ${\ displaystyle S ^ {1}}$ can be considered as a co-equalizer of two embeddings of the standard 0-dimensional simplex into the standard 1-dimensional simplex.

Co-equalizers can be quite large: there are exactly two functors from category 1 with one object and one morphism, into category 2 with two objects and exactly one non-identical morphism. The co-equalizer of these functors is a monoid of natural numbers by addition, considered as a category of one element. This shows that although each co-equalizer is epimorphic, it is not necessarily surjective .

Literature

MacLane S. Chapter 3. Universal constructions and limits // Categories for the working mathematician = Categories for the working mathematician / Per. from English under the editorship of V.A. Artamonova. - M .: Fizmatlit, 2004. - S. 68-94. - 352 p. - ISBN 5-9221-0400-4 .

Equalizer

Definition

Examples

Literature

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