1 Definition
A coend is a categorical construction that associates to a functor a universal object obtained by “integrating out” an indexing category. It is widely used to package algebraic operations and to express generalized tensor-like combinations.
1.1 Functors and bifunctors
Let \(\mathcal{A}\) be a small category and \(\mathcal{C}\) a category with suitable colimits. A typical input for a coend is a bifunctor \[ F : \mathcal{A}^{op}\times \mathcal{A}\to \mathcal{C}, \] which assigns an object \(F(a',a)\) for each pair \((a',a)\) and is functorial in both variables. In many applications \(F\) is built from a pair of ordinary functors, such as \(F(a',a)=G(a')\otimes H(a)\) in settings with a monoidal product.
1.2 Coends as a universal construction
The coend of \(F\) is an object \(\int^{a\in \mathcal{A}} F(a,a)\) equipped with a family of morphisms \[ \iota_a : F(a,a)\to \int^{a\in \mathcal{A}} F(a,a) \] subject to a dinaturality condition. Dinaturality requires that for every morphism \(f:a\to b\) in \(\mathcal{A}\), the induced map along \(F\) in the two variables agrees after passing through the structure morphisms. Concretely, the relation identifies the images of \(F(f,1_a)\) and \(F(1_b,f)\) through the corresponding \(\iota\)’s.
The defining feature is universality: any dinatural family of morphisms from \(F(a,a)\) factors uniquely through the coend.
1.3 Coequalizer formulation
A common way to construct the coend is as a coequalizer. Under mild conditions ensuring the required colimits exist, \(\int^{a}F(a,a)\) can be realized as the coequalizer of two parallel morphisms between coproducts: \[ \coprod_{f:a\to b} F(b,a)\rightrightarrows \coprod_{a\in \mathcal{A}} F(a,a)\to \int^{a\in \mathcal{A}} F(a,a). \] The two arrows are induced by the functoriality of \(F\) in the left and right variables, respectively: one uses the action of \(F\) on \(f\) in \(\mathcal{A}^{op}\), and the other uses the action on \(f\) in \(\mathcal{A}\). The coend is then the quotient that enforces the dinaturality relations.
1.4 Notation and terminology
The phrase coend of \(F\) over \(\mathcal{A}\) refers to the universal object \(\int^{a\in\mathcal{A}}F(a,a)\). The superscript on the integral sign emphasizes “co”-variance in both directions through dinaturality rather than ordinary evaluation. The dual construction, end, uses similar data but reversed universal property (limits rather than colimits).
2 Basic properties
Coends inherit many structural features from their universal characterization and from the general behavior of colimits.
2.1 Universal property
If \(X\) is an object of \(\mathcal{C}\), providing a morphism \[ \int^{a\in\mathcal{A}}F(a,a)\to X \] is equivalent to giving a dinatural transformation from \(F(a,a)\) to the constant functor at \(X\). Equivalently, it is the same as giving a family of maps \(F(a,a)\to X\) satisfying the dinaturality constraints for each morphism in \(\mathcal{A}\). This characterization is the basis for most calculations and proofs involving coends.
2.2 Functoriality
Coends behave well under functorial changes of the input:
- Change of the indexing category (via functors): A functor between indexing categories that transports \(F\) appropriately can induce canonical morphisms between coends.
- Change of the target category: If \(\mathcal{C}\to\mathcal{D}\) preserves the relevant colimits, it takes coends in \(\mathcal{C}\) to coends in \(\mathcal{D}\).
These properties let one compute coends after applying colimit-preserving functors such as left adjoints.
2.3 Duality with ends
Coends are formally dual to ends. If ends can be described as universal objects characterized by dinatural transformations and constructed via equalizers, then coends are the corresponding colimit version characterized by dinaturality and constructed via coequalizers. Many identities about coends can be obtained from the corresponding end identities by reversing arrows and replacing limit-style conditions with colimit ones.
2.4 Existence conditions
A coend \(\int^{a\in\mathcal{A}}F(a,a)\) exists when \(\mathcal{C}\) has the colimits required by its coequalizer presentation, in particular:
- coproducts of the objects \(F(a,a)\) over \(\mathrm{Ob}(\mathcal{A})\),
- coproducts of \(F(b,a)\) over \(\mathrm{Mor}(\mathcal{A})\),
- the corresponding coequalizer.
In practice, existence often follows when \(\mathcal{A}\) is small and \(\mathcal{C}\) is cocomplete (has all small colimits), or when \(F\) has restricted support and only finite colimits are needed.
3 Examples
Examples illustrate how coends express standard constructions.
3.1 Coend of a representable functor
Let \(\mathcal{C}\) be cocomplete and consider a representable-type bifunctor built from a functor \(H:\mathcal{A}\to \mathcal{C}\). A typical pattern is that coends involving representables collapse to evaluating a functor at an object, reflecting the Yoneda-style behavior. For instance, when \(F(a',a)=\mathcal{A}(a,a_0)\cdot X\) for a fixed \(a_0\) (with suitable scalar/copower conventions), the dinaturality relations identify all contributions except those corresponding to the object \(a_0\), yielding an expression equivalent to \(X\).
The precise statement depends on how representability is inserted and on the enrichment or tensoring structure available in \(\mathcal{C}\).
3.2 Coend of a constant functor
If \(F(a',a)\) is constant, say \(F(a',a)=K\) for all objects \(a',a\), then the coend forms the colimit quotient that identifies the two legs induced by every morphism of \(\mathcal{A}\). In many categories this yields a “coinvariants” object: the result records \(K\) modulo the relations generated by the morphisms of \(\mathcal{A}\). In simple cases where \(\mathcal{A}\) has only identity morphisms, the coend reduces to a coproduct of copies of \(K\).
3.3 Coends in Set
In \(\mathbf{Set}\), coends can be understood concretely using quotient sets from the coequalizer description. If \(F:\mathcal{A}^{op}\times\mathcal{A}\to\mathbf{Set}\), then \(\int^{a}F(a,a)\) is constructed from the disjoint union of \(F(a,a)\) by imposing the identifications dictated by morphisms \(f:a\to b\). This makes many coend calculations combinatorial: elements in different fibers are equated precisely when they are related by the action of \(F\) along some arrow of \(\mathcal{A}\).
3.4 Coends in vector spaces
In the category of vector spaces over a field, \(\mathbf{Vect}\), coends typically become tensor products modulo relations. When \(F(a',a)\) is bilinear in a suitable sense, the coend quotient corresponds to imposing dinaturality relations linearly. This is the categorical mechanism behind many familiar linear-algebraic constructions such as tensoring and balancing.
4 Calculational forms
Coends admit several equivalent descriptions that facilitate explicit computation.
4.1 Coend as a quotient
Via the coequalizer presentation, a coend can be described as a quotient of a coproduct.
4.1.1 Generators and relations
One may view the coproduct \(\coprod_a F(a,a)\) as generated by symbols representing elements (or vectors, morphisms, etc.) in each \(F(a,a)\). The parallel maps from \(\coprod_{f:a\to b}F(b,a)\) specify the relations: an element coming from \(F(b,a)\) maps to two generators in \(F(a,a)\) and \(F(b,b)\). The coend is obtained by enforcing that these two images coincide.
This “generators-and-relations” viewpoint is especially transparent in \(\mathbf{Set}\) and \(\mathbf{Vect}\).
4.1.2 Equivalence relations induced by morphisms
In categories like \(\mathbf{Set}\), the coequalizer corresponds to forming an equivalence relation generated by the dinaturality identifications. Specifically, whenever a morphism \(f:a\to b\) induces corresponding maps from the bifunctor, elements connected through those maps become identified. The resulting quotient set is the coend.
In additive settings, the same idea yields a quotient by the submodule generated by the differences of the two induced maps.
4.2 Fubini theorem for coends
When coends are iterated over product indexing categories, there is a Fubini-type interchange law. Under suitable existence conditions, coends over independent variables can be computed in either order: \[ \int^{a\in\mathcal{A}}\left(\int^{b\in\mathcal{B}} F(a,b,a,b)\right)\cong \int^{(a,b)\in\mathcal{A}\times\mathcal{B}} F(a,b,a,b). \] This theorem is often proved by comparing the corresponding coequalizer presentations and using the fact that colimits commute in appropriate ways.
4.3 Change of variables
Coend expressions are invariant under appropriate reindexing. If \(u:\mathcal{A}\to\mathcal{B}\) is a functor and \(F\) is expressed in terms of objects of \(\mathcal{B}\), then coend calculations can be transferred along \(u\) when the bifunctor is pulled back or pushed forward in a way compatible with dinaturality. The resulting change-of-variables is typically a canonical isomorphism between coends with different index categories.
5 Applications
Coends serve as a unifying language for constructions resembling integration, tensoring, and balancing in categorical contexts.
5.1 Tensor products of functors
In many enriched or monoidal settings, coends encode tensoring of functors. Given compatible left and right actions, one forms a coend that produces the balanced tensor product by quotienting out relations induced by morphisms in the indexing category. This captures the idea that tensoring should “forget” how intermediate objects are chosen, while respecting the action structures.
5.2 Profunctors and distributors
A profunctor (also called a distributor) from \(\mathcal{A}\) to \(\mathcal{B}\) can be viewed as a bifunctor \(\mathcal{B}^{op}\times\mathcal{A}\to\mathbf{Set}\) (or into another suitable base). Composition of profunctors admits a coend formula: the composite is obtained by integrating over the intermediate category, typically written as a coend that sums over all possible intermediate objects while imposing identifications along morphisms. This makes coends central to the categorical study of relations and generalized composition.
5.3 Weighted colimits
Weighted colimits are defined using ends or coends depending on variance. In the standard enriched setup, a weighted colimit can be expressed as a coend that combines the diagram with a weight functor. The coend packages the universal property of weighted colimits into a compact formula, allowing one to deduce properties from known facts about coends and colimits.
5.4 Enriched category theory
In enriched categories, coends frequently appear as the mechanism for defining composition, identity, and Kan extension-like operations when expressed via tensoring of hom-objects. The enrichment replaces sets of morphisms by objects of a monoidal base, and the coend ensures the correct balancing and dinaturality across the enriched structure.
6 Related constructions
Coends are closely connected to several other categorical notions.
6.1 Ends
Ends are the dual notion to coends. While coends are built from coequalizers and represent dinatural transformations into a colimit object, ends are built from equalizers and represent dinatural transformations from a limit object. Many formulas occur in both “end form” and “coend form,” with corresponding direction changes.
6.2 Kan extensions
Kan extensions, which generalize extending a functor along a map of categories, can often be expressed using ends or coends. In common formulations, the left Kan extension uses a colimit-like (coend) expression, while the right Kan extension uses a limit-like (end) expression. This provides a powerful way to compute or reason about Kan extensions in concrete cases.
6.3 Coequalizers
Since coends can be realized as coequalizers, they are built from the same basic colimit machinery. Understanding the coequalizer representation clarifies how dinaturality relations become explicit identifications. In computational settings, one can often reduce coend calculations to straightforward quotienting.
6.4 Yoneda lemma
The Yoneda lemma provides the representability underpinning that makes many coends collapse to simpler expressions. When a bifunctor in a coend is representable in one variable, dinaturality relations frequently force the coend to evaluate the other variable at a specific object. Thus, Yoneda-style reasoning is frequently used to simplify or interpret coend formulas.
7 Notation and conventions
Precise notation varies across textbooks and research areas, but common patterns recur.
7.1 Common symbols
The standard symbol for a coend of \(F:\mathcal{A}^{op}\times\mathcal{A}\to\mathcal{C}\) is \[ \int^{a\in\mathcal{A}} F(a,a). \] The integration variable indicates which objects are being identified and quotiented out via dinaturality. Sometimes authors suppress the “\(a\in\mathcal{A}\)” part when the indexing category is clear from context.
7.2 Variants in enriched settings
In enriched category theory, one typically replaces ordinary hom-sets by enriched hom-objects, and the target of the coend may live in the enriching monoidal category. Notation may include an explicit enrichment base, and the coend may be written with enriched tensoring or copowers present in the integrand.
7.3 Historical and textbook usage
Coends emerged from efforts to formalize constructions in topology and algebra using universal properties, later becoming standard in categorical and enriched frameworks. Textbooks typically present them through either the universal dinaturality definition or the coequalizer/quotient construction, depending on the intended audience and the applications emphasized.