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Cokernel

In mathematics, the cokernel of a morphism f: A \to B between abelian groups, modules over a ring, or more generally objects in an abelian category, is defined as the quotient B / \operatorname{im}(f), where \operatorname{im}(f) is the image of f, providing a measure of how f fails to be surjective.^[1]^[2] This construction is the categorical dual to the kernel, obtained by reversing the arrows in the kernel's universal property diagram.^[3] The cokernel satisfies a universal property: given any morphism p: B \to C such that p \circ f = 0, there exists a unique morphism \phi: \operatorname{coker}(f) \to C making the diagram commute, i.e., p = \phi \circ i where i: B \to \operatorname{coker}(f) is the canonical projection.^[3] In additive categories with zero morphisms, this ensures the cokernel is unique up to unique isomorphism.^[3] Cokernels are fundamental in homological algebra, appearing in exact sequences where a short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 implies that g is the cokernel of f and f is the kernel of g.^[3] They also feature in the definition of abelian categories, where every morphism has a cokernel and every epimorphism is the cokernel of its kernel.^[3] In non-abelian settings, such as groups, the cokernel is the quotient by the normal closure of the image.^[4]

Definition and Properties

Formal Definition

In category theory, the cokernel of a morphism f: A \to B in a category \mathcal{C} with zero morphisms is an object Q in \mathcal{C} together with a morphism i: B \to Q such that i \circ f = 0, where $0 denotes the zero morphism from A to Q. This pair (Q, i) satisfies the universal property: for every object Q' in \mathcal{C} and every morphism g: B \to Q' with g \circ f = 0, there exists a unique morphism h: Q \to Q' such that h \circ i = g.^[5] The cokernel is denoted \operatorname{coker}(f) = Q, and the canonical morphism is typically written as \pi: B \to \operatorname{coker}(f). Cokernels do not exist in every category; their existence depends on the presence of zero morphisms and the category supporting the relevant colimits.^[5] This construction is standard in preadditive categories, where the hom-sets are abelian groups, thereby providing addition of morphisms and a well-defined zero morphism for each pair of objects. The cokernel is the categorical dual of the kernel.^[5]

Universal Property

The universal property of the cokernel provides a categorical characterization that defines it up to unique isomorphism, independent of any concrete construction.^[6] For a morphism f: A \to B in a category with zero morphisms, the cokernel is a morphism i: B \to Q such that i \circ f = 0, and it is universal with respect to this property: for any morphism g: B \to Q' satisfying g \circ f = 0, there exists a unique morphism u: Q \to Q' such that g = u \circ i.^[6] This universality ensures that the cokernel captures the essential "quotient" structure induced by f, making it the initial object among all objects receiving a zero-composing morphism from B.^[6] In categories with zero morphisms, such as abelian categories, the cokernel can be explicitly realized as the coequalizer of the pair consisting of f: A \to B and the zero morphism $0: A \to B.^[6] The coequalizer property states that i: B \to Q equalizes f and $0 (i.e., i \circ f = i \circ 0 = 0), and for any other morphism j: B \to Q'' that equalizes them, there is a unique morphism v: Q \to Q'' such that j = v \circ i.^[6] This formulation aligns directly with the universal property above, as the zero condition is the defining equalizer relation in this context.^[6] To illustrate, consider the commutative diagram depicting the universal property:

\begin{CD} A @>f>> B @>i>> Q \\ @V0VV @. @. \\ A @>0>> B @>g>> Q' \end{CD}

Here, the solid arrows form the cokernel, and the existence of the dashed unique morphism u: Q \to Q' (such that g = u \circ i) follows from the universality whenever the left square commutes (i.e., g \circ f = 0).^[6] A proof sketch of the uniqueness up to unique isomorphism proceeds as follows: suppose i': B \to Q' is another cokernel of f. By the universal property of i, there exists a unique morphism v: Q \to Q' such that i' = v \circ i. Similarly, by the universal property of i', there exists a unique morphism w: Q' \to Q such that i = w \circ i'. Composing yields i = w \circ v \circ i = i and i' = v \circ w \circ i' = i', confirming that v and w are inverses, hence Q \cong Q' via the unique isomorphism v.^[6] This argument relies on the category having zero morphisms to ensure the compositions align with the zero conditions.^[6] In categories lacking zero morphisms, cokernels may still be defined as coequalizers of f and some other designated morphism, though this approach is less standard and typically restricted to specific contexts where such a pair is canonically chosen.^[6] The universal property nonetheless guarantees that any such cokernels are unique up to unique isomorphism, preserving the abstract essence of the construction across different categorical settings.^[6]

Relations to Other Concepts

Comparison with Kernel

In category theory, the cokernel of a morphism f: A \to B in a category \mathcal{C} is the dual concept to the kernel, precisely defined as the kernel of the opposite morphism f^{\mathrm{op}} in the opposite category \mathcal{C}^{\mathrm{op}}.^[4] This duality underscores their symmetric yet opposing roles: the kernel operates "backwards" from the domain A, identifying a subobject that nullifies under f, whereas the cokernel proceeds "forwards" from the codomain B, constructing a quotient object that annihilates the image of f.^[4] In abelian categories, this opposition manifests concretely, where the cokernel of f is isomorphic to the quotient of the codomain by the image of f, \coker(f) \cong B / \operatorname{im}(f), providing a mirror to the kernel's structure as a subobject of the domain, though the kernel itself arises as the equalizer rather than a direct quotient.^[7] In categories with a duality functor, such as finite-dimensional vector spaces, the dual map f^\vee relates \ker(f) \cong A / \operatorname{im}(f^\vee), emphasizing contravariant symmetry. Kernels and cokernels were formalized as categorical constructs in the 1960s, with their duality prominently featured in Saunders Mac Lane's Categories for the Working Mathematician (1971), which axiomatized their behavior in abelian categories.^[8]

Aspect	Kernel of f: A \to B	Cokernel of f: A \to B
Domain/Codomain Role	Subobject of domain A (incoming morphisms to A that compose to zero with f)	Quotient of codomain B (outgoing morphisms from B that compose to zero with f)
Universal Property	Universal among morphisms g: K \to A such that f \circ g = 0, with unique factorization through the kernel inclusion	Universal among morphisms h: B \to C such that h \circ f = 0, with unique factorization through the cokernel projection

Coimage and Exactness

In abelian categories, the coimage of a morphism f: A \to B is defined as the cokernel of its kernel, denoted \operatorname{coim}(f) = \coker(\ker(f)).^[9]^[10] Equivalently, the coimage is the quotient of the domain by its kernel, \operatorname{coim}(f) = A / \ker(f).^[9] The first isomorphism theorem in abelian categories establishes a natural isomorphism \operatorname{coim}(f) \cong \operatorname{im}(f), where \operatorname{im}(f) is the kernel of the cokernel, \operatorname{im}(f) = \ker(\coker(f)).^[9]^[10] This isomorphism arises explicitly from the canonical maps in the (epi, mono) factorization of f, with the coimage serving as the codomain of the epimorphic part and the image as the domain of the monomorphic part.^[11] In such categories, the cokernel of f is given by the formula \coker(f) = B / \operatorname{im}(f), where the quotient is taken with respect to the subobject \operatorname{im}(f) \subseteq B.^[10]^[11] This reflects the coimage's role as the "effective image" of f, isomorphic to the kernel of the canonical map i: B \to \coker(f), since \operatorname{coim}(f) \cong \operatorname{im}(f) = \ker(i).^[9] A sequence A \xrightarrow{f} B \xrightarrow{g} C is exact at B if \operatorname{im}(f) = \ker(g).^[10]^[11] The canonical map B \to \coker(f) is always exact at B in this sense, since \ker(i) = \operatorname{im}(f) by construction, linking the cokernel directly to local exactness conditions.^[9]^[10]

Examples in Specific Categories

In Abelian Groups

In the category of abelian groups, denoted Ab, the cokernel of a group homomorphism f: A \to B is the quotient group B / \operatorname{im}(f), where \operatorname{im}(f) is the image subgroup generated by the elements f(a) for a \in A.^[12] This construction inherits the abelian structure from B, ensuring that the cokernel is always an abelian group.^[13] A concrete example illustrates this: consider the homomorphism f: \mathbb{Z} \to \mathbb{Z} defined by multiplication by 2, so f(n) = 2n. The image is $2\mathbb{Z}, the even integers, and the cokernel is \mathbb{Z} / 2\mathbb{Z}, which is the cyclic group of order 2.^[14] Here, the non-surjectivity of f introduces torsion in the cokernel, as the coset of 1 has order 2.^[15] For finitely generated abelian groups, the structure of the cokernel can be analyzed using the Smith normal form of the matrix representing f, which decomposes the presentation matrix into diagonal form and reveals the invariant factors or elementary divisors of the quotient.^[16] Every abelian group arises as the cokernel of a homomorphism between free abelian groups, corresponding to a presentation where the domain and codomain are free on chosen generators and relations.^[17]

In Vector Spaces

In the category of vector spaces over a field K, denoted \mathbf{Vect}_K, the cokernel of a linear map f: V \to W is defined as the quotient vector space W / \operatorname{im}(f).^[1] This construction measures the failure of f to be surjective, as elements of the cokernel correspond to cosets of the image subspace in W.^[18] By the rank-nullity theorem, the dimension of the cokernel satisfies \dim(\operatorname{coker}(f)) = \dim(W) - \operatorname{rank}(f), where \operatorname{rank}(f) = \dim(\operatorname{im}(f)).^[19] This formula highlights the direct relationship between the codomain's dimension and the map's image size, providing a quantitative assessment of surjectivity deficiency.^[18] For a concrete example, consider the linear map f: \mathbb{R}^2 \to \mathbb{R}^2 defined by rotation through 90 degrees, with matrix representation

\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.

Since f is invertible, \operatorname{im}(f) = \mathbb{R}^2, and thus \operatorname{coker}(f) = \{0\}, the trivial vector space.^[1] The canonical projection \pi: W \to W / \operatorname{im}(f) is a surjective linear map whose kernel is precisely \operatorname{im}(f).^[20] In \mathbf{Vect}_K, cokernels always exist and form vector spaces, as the category is abelian and supports quotient constructions for subspaces.^[3] This setup specializes the more general notion of cokernels in abelian groups to the field-structured context of vector spaces.^[21]

Applications and Extensions

In Exact Sequences

In an abelian category, for any morphism f: A \to B, there exists a short exact sequence $0 \to \operatorname{im}(f) \to B \to \operatorname{coker}(f) \to 0, where the map \operatorname{im}(f) \to B is the inclusion morphism and the map B \to \operatorname{coker}(f) is the canonical quotient morphism induced by f. This construction arises because, by definition, \operatorname{im}(f) is the kernel of the quotient map B \to \operatorname{coker}(f), ensuring exactness at B, while the inclusion is a monomorphism and the quotient is an epimorphism. A short exact sequence in an abelian category is a sequence of the form $0 \to X \xrightarrow{i} Y \xrightarrow{p} Z \to 0 such that i is a monomorphism, p is an epimorphism, and \operatorname{im}(i) = \ker(p). In the sequence constructed from f, the cokernel \operatorname{coker}(f) is the zero object if and only if f is an epimorphism (surjective), meaning the sequence is exact at B precisely when f covers all of B. Such a short exact sequence $0 \to K \to G \to Q \to 0 is said to split if there exists a morphism s: Q \to G (a section on the cokernel side) such that the composition p \circ s = \operatorname{id}_Q, where p: G \to Q is the quotient map. This splitting condition is equivalent to G \cong K \oplus Q as objects in the category, and on the cokernel side, it holds whenever Q is a projective object, since projectivity ensures that the identity morphism on Q lifts through the epimorphism p. A classic example of a short exact sequence that does not split is $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0, where the first map sends n \mapsto 2n and the second is the canonical projection \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}. Here, \mathbb{Z}/2\mathbb{Z} is not projective as a \mathbb{Z}-module, and there is no section s: \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z} satisfying the splitting condition, as \mathbb{Z} \not\cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}. In abelian categories, cokernels play a key role in completing partial sequences to exact ones; for instance, given any morphism f: A \to B, adjoining the cokernel yields the exact sequence A \to B \to \operatorname{coker}(f) \to 0, and combining with the kernel produces the full short exact sequence $0 \to \ker(f) \to A \to B \to \operatorname{coker}(f) \to 0.

In Homological Algebra

In a chain complex C_\bullet with differentials d_n: C_n \to C_{n-1}, the homology group is defined as H_n(C) = \ker(d_n) / \operatorname{im}(d_{n+1}). The cokernel of the map d_{n+1}: C_{n+1} \to C_n is C_n / \operatorname{im}(d_{n+1}), which fits into a short exact sequence $0 \to \ker(d_n) / \operatorname{im}(d_{n+1}) \to C_n / \operatorname{im}(d_{n+1}) \to \operatorname{im}(d_n) \to 0 with the homology H_n(C) as the kernel of the induced map \coker(d_{n+1}) \to \operatorname{im}(d_n). This structure highlights how cokernels encode the failure of exactness at each degree, relating directly to homology via subquotients. Furthermore, when considering short exact sequences of chain complexes $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0, the cokernels in each degree contribute to the long exact sequence in homology through the connecting homomorphisms, which arise from diagram chases involving kernels and cokernels across degrees.^[22]^[23] Cokernels play a central role in the computation of derived functors like Ext and Tor via resolutions. For Tor, a projective resolution \cdots \to P_1 \to P_0 \to M \to 0 of a module M is tensored with another module N, yielding a chain complex whose homology groups are the Tor functors; here, the cokernels of the resolution maps ensure the complex is exact except at the end, with boundaries defined as images that implicitly rely on cokernel properties for the tensor product functor's right exactness. Similarly, for Ext, applying the Hom functor to an injective resolution of N produces a cochain complex whose cohomology is Ext; the cokernels in the resolution guarantee that the cohomology vanishes in positive degrees before localization, allowing the derived functor to measure deviations from exactness. These constructions underscore cokernels' necessity in building acyclic resolutions that compute the functors.^[24]^[25] In derived categories, the cofiber—functioning as a cokernel in the triangulated structure—of a quasi-isomorphism is acyclic, meaning its homology vanishes in all degrees. This follows from the mapping cone construction: for a quasi-isomorphism f: X^\bullet \to Y^\bullet, the cone complex \operatorname{cone}(f) fits into a distinguished triangle

X^\bullet \to Y^\bullet \to \operatorname{cone}(f) \to X^\bullet{{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}}

, and since f induces isomorphisms on homology, the long exact sequence from the triangle implies \operatorname{cone}(f) is acyclic. This acyclicity is pivotal for inverting quasi-isomorphisms to form the derived category, as it identifies morphisms up to homotopy equivalence. The snake lemma extends this to short exact sequences of complexes, propagating cokernels degreewise to yield long exact sequences in homology, where the connecting maps are constructed by lifting elements through cokernels and kernels.^[26]^[27] Cokernels were instrumental in Alexander Grothendieck's foundational development of abelian categories and derived functors, as detailed in his 1957 Tôhoku paper, where they axiomatize the existence of kernels and cokernels essential for exactness and functorial derivations. Beyond abelian settings, cokernels extend to non-abelian categories such as pointed sets or groups, where the cokernel of a homomorphism f: G \to H is the quotient H / \langle \operatorname{im}(f) \rangle^N by the normal closure of the image, enabling homological constructions like non-abelian cohomology despite the absence of full exactness. This generalization supports higher categorical structures without relying on commutativity.^[28]^[4]

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