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Coequalizer

In category theory, a coequalizer of two parallel morphisms f, g: X \to Y in a category \mathcal{C} is an object Q equipped with a morphism q: Y \to Q such that q \circ f = q \circ g, and q is universal with this property: for any morphism h: Y \to Z satisfying h \circ f = h \circ g, there exists a unique morphism u: Q \to Z such that u \circ q = h.^[1]^[2] This construction is a colimit, dual to the equalizer (a limit in the opposite category \mathcal{C}^{op}), and it generalizes quotient constructions across categories.^[1] In the category of sets, the coequalizer Q is the quotient set Y / \sim, where \sim is the smallest equivalence relation on Y such that f(x) \sim g(x) for all x \in X, with q the canonical projection.^[2] In abelian categories, such as those of modules or vector spaces, the coequalizer of f and the zero morphism is the cokernel of f.^[2] Coequalizers play a central role in categorical algebra, enabling the formation of colimits from kernel pairs and characterizing regular epimorphisms as those morphisms that are coequalizers of their kernel pair.^[1] They are preserved by certain functors, such as those reflecting exactness in regular categories, and appear in enriched category theory and topos theory for handling relations and equivalences.^[3]

Definition and Universal Property

General Definition

In category theory, a coequalizer of two parallel morphisms f, g: X \to Y in a category \mathcal{C} is an object Q in \mathcal{C} together with a morphism q: Y \to Q such that q \circ f = q \circ g, where Q is unique up to a unique isomorphism making the relevant triangle commute.^[4] This construction identifies points in Y that are mapped to the same element under the coequalizing morphism q, providing a categorical means to enforce the equality of the images of f and g. Coequalizers generalize quotient constructions across diverse mathematical categories by abstracting the process of collapsing a structure according to an equivalence relation induced by the parallel pair f and g.^[5] In this context, the relation \sim equates f(x) \sim g(x) for all x \in X, and Q serves as the "quotient" object capturing this identification universally. The coequalizer is commonly denoted as \mathrm{coeq}(f, g) or Y / \sim, emphasizing its role as the quotient of Y by the equivalence \sim generated by f and g.^[4]^[5]

Universal Property

In category theory, the coequalizer of a pair of parallel morphisms f, g: X \rightrightarrows Y in a category \mathcal{C} is an object Q together with a morphism q: Y \to Q such that q \circ f = q \circ g, and this pair (Q, q) satisfies the following universal property: for any object Z in \mathcal{C} and any morphism h: Y \to Z such that h \circ f = h \circ g, there exists a unique morphism u: Q \to Z making the diagram

\begin{tikzcd} X \arrow[r, shift left, "f"] \arrow[r, shift right, "g"'] & Y \arrow[d, "q"] \arrow[dr, "h", dashed] \\ & Q \arrow[d, "u", dashed] & Z \end{tikzcd}

commute, i.e., u \circ q = h.^[1] This universal property characterizes the coequalizer uniquely up to unique isomorphism: if (Q', q') is another such pair, then there is a unique isomorphism \phi: Q \to Q' such that \phi \circ q = q'.^[1] Moreover, the morphism q: Y \to Q is a regular epimorphism and an epimorphism.^[1] Categorically, this property encodes the process of identifying elements in Y that are made equal via the images of f and g from X, ensuring that Q is the "freest" such quotient object mediating the equality imposed by the parallel pair.^[1]

Constructions in Specific Categories

In the Category of Sets

In the category of sets, denoted \mathbf{Set}, the coequalizer of two parallel morphisms f, g: X \to Y is constructed as the quotient set Y / \sim, where \sim is the smallest equivalence relation on Y such that f(x) \sim g(x) for all x \in X.^[1]^[2] This equivalence relation is generated by identifying elements via the pairs (f(x), g(x)), ensuring reflexivity, symmetry, and transitivity.^[1] The coequalizer morphism is the canonical quotient map q: Y \to Y / \sim, which sends each element to its equivalence class and satisfies q \circ f = q \circ g, as elements identified by f and g lie in the same class.^[2] This construction verifies the universal property in \mathbf{Set}: for any morphism h: Y \to Z such that h \circ f = h \circ g, the relation \sim ensures that h is constant on equivalence classes, inducing a unique morphism \overline{h}: Y / \sim \to Z with \overline{h} \circ q = h, since h is constant on each equivalence class.^[1]^[6]

In Algebraic Categories

In algebraic categories, such as those of groups, abelian groups, and rings, the coequalizer of parallel morphisms f, g: A \to B is constructed as a quotient object that respects the underlying algebraic operations, ensuring that the resulting structure forms a valid algebra in the category. This involves factoring out a suitable congruence relation generated by the "differences" imposed by f and g, which preserves the category's homomorphisms and operations like addition or multiplication.^[4] In the category of groups \mathbf{Grp}, the coequalizer of group homomorphisms f, g: G \to H is the quotient group H / N, where N is the normal subgroup of H generated by the set \{f(x) g(x)^{-1} \mid x \in G\}. The canonical projection \pi: H \to H/N equalizes f and g via \pi \circ f = \pi \circ g, and it satisfies the universal property: for any group homomorphism h: H \to K with h \circ f = h \circ g, there exists a unique \overline{h}: H/N \to K such that h = \overline{h} \circ \pi. This construction ensures that the quotient inherits the group structure, with N acting as the kernel of the induced homomorphism.^[4]^[1] In the category of abelian groups \mathbf{Ab}, the construction simplifies due to commutativity. The coequalizer of f, g: A \to B is the factor group B / M, where M is the subgroup generated by \{f(a) - g(a) \mid a \in A\} for a \in A, using the additive notation. The projection p: B \to B/M equalizes f and g, and the universal property holds similarly, with any equalizing morphism factoring uniquely through p. This reflects the additive structure, where M = \operatorname{im}(f - g).^[4] For the category of rings \mathbf{Rng}, the coequalizer of ring homomorphisms f, g: R \to S is the quotient ring S / I, where I is the two-sided ideal of S generated by \{f(r) - g(r) \mid r \in R\}. The natural projection \pi: S \to S/I satisfies \pi \circ f = \pi \circ g and the universal property among ring homomorphisms. This ideal ensures that the quotient preserves both addition and multiplication, distinguishing it from mere set-theoretic quotients by requiring compatibility with the ring operations.^[4]^[1]

In Topological Spaces

In the category of topological spaces, denoted \mathbf{Top}, the coequalizer of two parallel continuous morphisms f, g: X \to Y is obtained by first forming the coequalizer in the underlying category of sets, which yields the quotient set Y / \sim. Here, the equivalence relation \sim on Y is the smallest equivalence relation such that f(x) \sim g(x) for every x \in X; thus, two points y_1, y_2 \in Y satisfy y_1 \sim y_2 if they can be connected by a finite chain of such identifications, meaning there exist x_1, \dots, x_n \in X with y_1 = f(x_1) or g(x_1), and alternating applications of f and g link to y_2.^[7]^[8] This quotient set Y / \sim is then endowed with the quotient topology, defined such that a subset U \subseteq Y / \sim is open if and only if its preimage q^{-1}(U) under the canonical projection q: Y \to Y / \sim is open in Y. This topology is the finest one on Y / \sim that renders q continuous, ensuring that the coequalizer morphism q: Y \to Y / \sim in \mathbf{Top} satisfies q \circ f = q \circ g and is universal with respect to continuous maps to other topological spaces.^[7] Unlike the coequalizer in the category of sets, which is purely set-theoretic, the topological version imposes the quotient topology to preserve continuity; this construction guarantees the existence of coequalizers in \mathbf{[Top](/page/Top)}, as the category is cocomplete.^[7] The map q is always continuous and surjective by definition, but the resulting space Y / \sim inherits Hausdorff separation properties from Y only if the equivalence relation \sim, viewed as a subset of Y \times Y, is closed.^[8] In general, additional conditions on f and g or on Y may be needed to ensure desirable topological features like openness of q or regularity of the quotient.^[9]

Properties

Epimorphisms and Kernel Pairs

A fundamental property of coequalizers is that the coequalizer morphism is always an epimorphism. Specifically, if q: Y \to Z is the coequalizer of parallel morphisms f, g: X \rightrightarrows Y, then q is right-cancellative: for any morphisms h, k: Z \to W, if h \circ q = k \circ q, then h = k.^[1]^[10] To see this, suppose h \circ q = k \circ q. Then h and k both coequalize the pair (f, g), since (h \circ q) \circ f = (k \circ q) \circ f and similarly for g. By the universal property of the coequalizer, there exists a unique morphism m: Z \to W such that m \circ q = h \circ q = k \circ q, which implies h = m = k. This establishes that q is an epimorphism.^[1] The kernel pair provides a canonical pair of parallel morphisms whose coequalizer recovers a given epimorphism in suitable categories. The kernel pair of a morphism q: Y \to Z is the pullback of q along itself, yielding an object R = Y \times_Z Y equipped with projections p_1, p_2: R \rightrightarrows Y such that q \circ p_1 = q \circ p_2. This R represents the equivalence relation on Y induced by q, where elements of Y are identified if they map to the same element in Z; formally, R is a subobject of Y \times Y consisting of pairs (y_1, y_2) with q(y_1) = q(y_2).^[11] In a regular category (or more generally, a topos), every epimorphism arises in this way: if q: Y \to Z is an epimorphism, then q is the coequalizer of its kernel pair p_1, p_2: R \rightrightarrows Y. This follows because regular epimorphisms—those that are coequalizers of some parallel pair—are precisely the coequalizers of their own kernel pairs in categories with pullbacks, and in regular categories, all epimorphisms are regular. Thus, the kernel pair captures the "relations" that the epimorphism enforces, and the coequalizer construction quotients Y by this relation to obtain Z.^[12]^[13]

Preservation under Functors

Absolute coequalizers are those coequalizers preserved by every functor out of the category.^[14] They arise purely from the diagrammatic structure of the defining fork, independent of the ambient category. A characterizing feature is that the parallel pair f, g: X \to Y admits splittings making the coequalizer diagram absolute.^[4] Split coequalizers provide the canonical example of absolute coequalizers. Consider parallel morphisms f, g: X \to Y with a coequalizer q: Y \to Q. The diagram is split if there exist morphisms s: Q \to Y and t: Y \to X such that q \circ s = \mathrm{id}_Q, f \circ t = \mathrm{id}_Y, and g \circ t = s \circ q.^[4] These conditions ensure the coequalizer is preserved under any functor, as the splittings are preserved diagrammatically. In abelian categories, such a split coequalizer admits an explicit description: Q \cong Y / \mathrm{im}(f - g), where f - g: X \to Y is the difference morphism.^[14] More generally, left adjoint functors preserve all colimits, including coequalizers. If F \dashv G: \mathcal{C} \to \mathcal{D} with F left adjoint, then for any parallel pair in \mathcal{C}, the image under F of their coequalizer in \mathcal{C} is the coequalizer in \mathcal{D}.^[14] This holds because left adjoints preserve the universal property of colimits. In abelian categories, left adjoints are right exact, preserving finite colimits such as coequalizers of finite presentations.^[14] In varieties of universal algebras, the forgetful functor to \mathbf{Set} creates reflexive coequalizers, meaning that coequalizers of reflexive pairs (where the parallel morphisms admit a common section) in the variety are obtained by computing the coequalizer in \mathbf{Set} and equipping it with the induced algebra structure.^[4] Since varieties are monadic over \mathbf{Set}, and the free functor (left adjoint to the forgetful) preserves colimits, these reflexive coequalizers align with the absolute ones when split. Split coequalizers, being absolute, are uniformly preserved across such algebraic categories.^[14]

Relations to Other Concepts

As a Type of Colimit

In category theory, the coequalizer of two parallel morphisms f, g: A \to B is defined as the colimit of the diagram consisting of the objects A and B together with the two morphisms f and g.^[4] This diagram, often denoted as A \rightrightarrows B, captures the structure where the coequalizer object Q comes equipped with a morphism q: B \to Q such that q \circ f = q \circ g, and this cocone is universal in the sense that any other morphism h: B \to C satisfying h \circ f = h \circ g factors uniquely through q.^[15] More generally, a colimit of a diagram D: \mathcal{J} \to \mathcal{C} is an object \varinjlim D in the category \mathcal{C} together with a universal cocone from D to the constant diagram at \varinjlim D; that is, for any other cocone from D to an object X, there exists a unique morphism \varinjlim D \to X making the diagram commute.^[4] In the case of the coequalizer, this universal cocone specializes to the coequalizing property over the parallel pair diagram, aligning directly with the colimit construction for finite-indexed shapes.^[15] In categories with finite colimits, such as the category of sets or groups, coequalizers exist as basic finite colimits. More generally, finite colimits can be constructed using coproducts and coequalizers.^[4] Since the parallel pair diagram is finite and small, the resulting coequalizer is a small colimit, which is computable in many concrete categories where finite colimits exist.^[4]

Connection to Cokernels and Pushouts

In abelian categories, the cokernel of a morphism h: A \to B is defined as the coequalizer of the parallel pair consisting of h and the zero morphism $0: A \to B.^[4] This identification leverages the zero object to treat cokernels as a special case of coequalizers, where the universal property ensures that any morphism factoring through the image of h uniquely extends to the cokernel quotient B / \operatorname{im}(h).^[4] More generally, in preadditive categories—where hom-sets form abelian groups, allowing subtraction of morphisms—the coequalizer of two parallel morphisms f, g: A \to B coincides with the cokernel of their difference f - g: A \to B.^[4] This equivalence simplifies computations in settings like modules over a ring, where the coequalizer is the quotient B / \operatorname{im}(f - g), bridging equalizer-like constructions to colimit structures via additive inverses.^[16] These relations extend the general notion of coequalizers from arbitrary categories to additive ones, facilitating homological algebra; for instance, in the category of chain complexes of abelian groups (an abelian category), coequalizers compute cokernels that appear in short exact sequences and homology groups.^[4] A pushout is a special case of a coequalizer: given morphisms f: A \to B and g: A \to C, the pushout of f and g is the coequalizer of the two morphisms B \sqcup_A C \to B \sqcup C induced by the inclusions into the coproduct B \sqcup C composed with f and g.^[1] This relation highlights how coequalizers generalize quotient constructions to amalgamated free products in categories with coproducts.

Examples and Applications

Concrete Examples

In the category of sets, a concrete example of a coequalizer is the singleton set obtained from the two parallel functions f, g: \{*\} \to \{a, b\}, where f(*) = a and g(*) = b. The equivalence relation \sim on \{a, b\} is the smallest equivalence containing a \sim b, so the coequalizer is the quotient set \{a, b\}/{\sim} with a single equivalence class. In the category of groups, the free abelian group \mathbb{Z} \oplus \mathbb{Z} on two generators x and y arises as the coequalizer of the free group F_2 on x, y modulo the relation xy = yx. This is the quotient F_2 / \ll [x, y] \gg, where [x, y] = xyx^{-1}y^{-1} and \ll [x, y] \gg denotes the normal closure of the commutator subgroup generated by [x, y]. Equivalently, it is the coequalizer of the two group homomorphisms \iota, \kappa: K \to F_2, with K the free group on a single generator z, \iota(z) = [x, y], and \kappa(z) = e the identity element.^[17]^[18] In the category of topological spaces, the circle S^1 is the coequalizer of the two continuous maps f, g: \{*\} \to [0, 1], where f(*) = 0 and g(*) = 1. The coequalizing map is the quotient map q: [0, 1] \to S^1 identifying the endpoints $0 \sim 1, endowed with the quotient topology, which yields the standard circle homeomorphic to the unit circle in \mathbb{R}^2.^[19] In the category of small categories, the one-object category whose endomorphism monoid is the natural numbers \mathbb{N} under addition is a concrete coequalizer. Consider the walking arrow category J with objects s (source) and t (target), and a single non-identity morphism \alpha: s \to t. The two parallel functors F, G: J \to \mathbf{Cat} are defined such that F and G both send s and t to the terminal category (one object with only identity morphism), but differ on \alpha by identifying its source and target in complementary ways, forcing compositions \alpha \circ \alpha = \alpha^2, \alpha^2 \circ \alpha = \alpha^3, and so on in the coequalizer. This identifies iterations of \alpha, yielding the monoid \mathbb{N} of finite iterations.^[20]

Applications in Mathematics

In homotopy theory, coequalizers play a fundamental role in computing homotopy colimits within model categories. For a diagram D in a model category, the simplicial replacement \operatorname{srep}(D) allows the ordinary colimit to be expressed as the coequalizer \operatorname{colim} D = \operatorname{coeq}[\operatorname{srep}(D)_1 \rightrightarrows \operatorname{srep}(D)_0], and under suitable cofibrancy conditions—such as when the diagram is Reedy cofibrant—the homotopy colimit \operatorname{hocolim} D is weakly equivalent to this coequalizer.^[21] This construction extends to localizations in model categories, where Bousfield localizations can be realized via homotopy colimits involving coequalizers of simplicial objects, enabling the computation of derived functors and resolutions in stable homotopy categories.^[21] In universal algebra, coequalizers are instrumental in classifying varieties of algebras through Mal'cev conditions that ensure structural properties like the commutation of products with coequalizers. Specifically, in pointed varieties, the property that binary products commute with arbitrary coequalizers is characterized by the existence of certain Mal'cev terms: binary terms b_i(x, y) and unary terms c_i(x) (for i = 1 to m), along with (m+2)-ary terms p_1, \dots, p_n satisfying identities such as p_1(x, y, b_1(x, y), \dots, b_m(x, y)) = x and p_i(0, 0, c_1(z), \dots, c_m(z)) = z.^[22] These conditions delineate varieties where coequalizers preserve algebraic structure, facilitating the study of congruence permutability and related equational classes. In computer science, particularly in type theory, coequalizers manifest as higher inductive types that model quotients and support the construction of inductive types with built-in equivalences. A coequalizer type \operatorname{coeq}_{A,B}(f,g) for functions f, g: A \to B is defined by the constructor \operatorname{in}: B \to \operatorname{coeq}_{A,B}(f,g) and the path constructor \operatorname{glue}(x): \operatorname{in}(f(x)) = \operatorname{in}(g(x)) for x: A, enabling the formalization of quotient types in homotopy type theory (HoTT) and their integration with inductive definitions like sums, circles, and suspensions.^[23] This framework underpins domain-theoretic constructions of quotient domains by enforcing equivalence relations categorically, as seen in generalizations of partial orders to categories for denotational semantics.^[24] Historically, coequalizers were pivotal in Grothendieck's development of toposes for sheafification, where epimorphisms in the category of sheaves \operatorname{Sh}(\mathcal{C}) on a site \mathcal{C} are precisely coequalizers of their kernel pairs; for a surjective sheaf morphism F \to G, G is the coequalizer of F \times_G F \rightrightarrows F.^[25] In descent theory, coequalizers enforce gluing conditions for objects over sites by characterizing connected groupoids—where the coequalizer of source and target maps d_0, d_1: B_1 \to B_0 is terminal—and facilitating descent data via coequalizers in slice categories, as in the tensor product construction for modules over a descent morphism.^[26]

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