1 Definition and basic idea

The coimage of a morphism is a canonical quotient of the domain that records the part of the object not annihilated by the morphism. It is designed to separate the “kernel part” from the essential data carried forward by the map. In many familiar algebraic settings, the coimage provides the natural bridge between the source object and the image.

1.1 Morphisms in algebra

A morphism is a structure-preserving map between algebraic objects such as groups, modules, or vector spaces. Such maps often admit decompositions that isolate what is lost, what is preserved, and what is produced by the mapping process. The coimage is one of the standard tools used to express this decomposition.

1.2 Coimage as a quotient by the kernel

Given a morphism, its kernel consists of all elements sent to the zero element, or the identity element in multiplicative notation. The coimage is formed by quotienting the domain by this kernel. This construction identifies elements that differ only by something invisible to the morphism.

1.3 Canonical factorization through the image

Once the domain has been divided by the kernel, the original morphism factors through the resulting quotient. The induced map from the coimage to the codomain lands in the image of the morphism, and the original map can be recovered as a composite of a quotient map followed by this induced map. This factorization is canonical, meaning it is determined naturally by the morphism itself.

1.4 Relationship with image

The image is the subobject of the codomain consisting of all values attained by the morphism. The coimage and the image are distinct constructions in general, since one comes from the domain and the other from the codomain. In many standard algebraic categories, however, they are canonically isomorphic, so the quotient description and the subobject description match in a precise way.

2 Coimage in category theory

Category theory treats the coimage abstractly through universal properties rather than through elementwise formulas. This viewpoint makes the construction applicable across a wide range of algebraic contexts. It also clarifies how the coimage fits into the general theory of limits, colimits, kernels, and cokernels.

2.1 Kernel and cokernel framework

In a category with suitable structure, the kernel of a morphism is defined by a universal property that characterizes maps annihilated by the given morphism. Dually, the cokernel captures the largest quotient through which the morphism vanishes. The coimage is often described in relation to the cokernel of the kernel, giving it a categorical origin.

2.2 Construction via universal properties

The coimage can be constructed as the universal quotient of the domain through which the morphism factors and on which the kernel acts trivially. This means that any other factorization through a quotient must uniquely pass through the coimage. Universal properties make the construction independent of elementwise notation and suitable for abstract categories.

2.3 The comparison morphism from coimage to image

The coimage and image are connected by a natural comparison morphism. This map measures the extent to which the quotient-by-kernel description agrees with the subobject-of-codomain description. In well-behaved categories, it is often an isomorphism.

2.3.1 Definition of the canonical map

The canonical map from the coimage to the image is induced by the original morphism after passing to the quotient by the kernel. It sends each equivalence class in the coimage to the corresponding element in the image. Because the original map already lands in the image, this induced map is defined without ambiguity.

2.3.2 When the map is an isomorphism

The comparison morphism is an isomorphism in many common algebraic categories, including abelian categories. In such cases, the coimage and image are not merely related but effectively the same object up to canonical identification. This property is central to exactness arguments and many standard proofs in algebra.

3 Coimage in abelian categories

Abelian categories provide the setting in which coimages behave especially well. They are designed so that kernels and cokernels interact harmoniously, and every morphism admits a robust factorization. In this environment, the coimage is a standard component of the structural theory.

3.1 Exactness properties

Exactness describes how sequences of morphisms fit together when the image of one map matches the kernel of the next. The coimage appears naturally in these sequences because it reflects the portion of the domain that survives after quotienting out the kernel. Its behavior helps organize exact arguments in a clean categorical form.

3.2 Coimage-image coincidence

In an abelian category, the canonical map from coimage to image is an isomorphism. This coincidence is one of the defining features that makes abelian categories so useful in homological algebra. It ensures that factoring through the quotient by the kernel gives precisely the same information as taking the image as a subobject.

3.3 Role in exact sequences

Coimages help express morphisms in terms of exact sequences, where kernels and images are linked step by step. They are especially useful when analyzing how a map decomposes into its essential components. In this way, the coimage serves as an intermediate object connecting domain data to codomain data.

3.3.1 Short exact sequences

A short exact sequence contains three objects and two morphisms arranged so that the image of the first equals the kernel of the second. In this setting, the coimage of the first morphism identifies with its image, making the quotient description transparent. Short exact sequences are among the clearest contexts in which the coimage appears implicitly.

3.3.2 First isomorphism theorem analogy

The coimage formalizes the idea behind the first isomorphism theorem: a map factors through a quotient by its kernel, and the resulting quotient corresponds to the image. This analogy is especially familiar in group theory and linear algebra. The categorical formulation generalizes that familiar result to broader algebraic settings.

4 Examples

Concrete examples show how the coimage behaves in familiar algebraic systems. Although the language may vary from one category to another, the underlying idea remains the same: remove the kernel, then compare the resulting quotient with the image. These examples illustrate why the construction is so widely used.

4.1 Groups

For a group homomorphism, the coimage is the quotient of the domain by the kernel. The induced map sends each coset to its image in the codomain, landing in the subgroup that is the image of the homomorphism. This reproduces the usual quotient form of the first isomorphism theorem.

4.2 Modules

For a module homomorphism, the coimage is obtained by dividing the source module by the kernel submodule. The induced map to the image is linear and respects the module structure. In many module categories, this map is an isomorphism, so the quotient module and the image module are canonically identified.

4.3 Vector spaces

For a linear transformation between vector spaces, the coimage is the quotient of the domain by the null space. The induced map sends each equivalence class to the corresponding vector in the image subspace. This is one of the most familiar examples, often introduced as a standard linear algebra theorem.

4.4 Abelian groups

For a homomorphism of abelian groups, the coimage is the quotient by the kernel subgroup. The resulting quotient group maps naturally onto the image subgroup. Because abelian groups form an abelian category, the comparison map is an isomorphism.

Several closely related notions are essential for understanding the coimage. Each highlights a different aspect of the same morphism: what is killed, what is retained, and what quotient or subobject represents the structure. Together they form a basic toolkit of abstract algebra and category theory.

5.1 Image

The image is the set or subobject consisting of values attained by a morphism. It describes the actual output of the map inside the codomain. The coimage and image are linked by a canonical comparison morphism.

5.2 Kernel

The kernel is the collection of elements mapped to zero or the identity. It measures the failure of injectivity. The coimage is formed by quotienting out this kernel.

5.3 Cokernel

The cokernel is a quotient of the codomain that measures what remains after the image has been accounted for. It is the dual notion to the kernel in many categorical settings. Coimage constructions often appear alongside cokernels in exactness arguments.

5.4 Factor object

A factor object is an object obtained by dividing another object by a specified equivalence relation or subobject. The coimage is a particular kind of factor object, namely the quotient of the domain by the kernel. This terminology emphasizes the role of the coimage as an intermediate quotient.

5.5 First isomorphism theorem

The first isomorphism theorem states that a homomorphism factors through the quotient by its kernel and that the resulting quotient is isomorphic to the image. The coimage is the categorical expression of this principle. It provides a unified formulation across many algebraic categories.

6 Uses and significance

The coimage is important because it clarifies how morphisms decompose and how algebraic information passes from one object to another. Its utility extends beyond elementary algebra into the deeper structure of categories and exact sequences. As a result, it is a standard device in modern algebraic reasoning.

6.1 Structural decomposition of morphisms

Coimages help break a morphism into simpler pieces: a quotient by the kernel, followed by a map onto the image. This decomposition isolates the essential content of the morphism while removing redundancy. It is useful for comparing maps and analyzing their behavior.

6.2 Applications in homological algebra

In homological algebra, coimages appear in the study of complexes, exact sequences, and derived constructions. They help formalize the passage from kernels to images in settings where exactness is central. Their compatibility with abelian categories makes them particularly valuable in this field.

6.3 Role in diagram chasing

Diagram chasing is a method for proving statements about morphisms in commutative diagrams. Coimages can simplify such arguments by replacing a morphism with its quotient-by-kernel factorization. This often makes it easier to track elements or subobjects through a diagram and verify exactness conditions.