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In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
, the coproduct, or categorical sum, is a construction which includes as examples the
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injective function, injection of each A_i into A, such that the image (mathematics), images of th ...
of sets and of topological spaces, the
free product In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
of groups, and the
direct sum The direct sum is an Operation (mathematics), operation between Mathematical structure, structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct ...
of modules and
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a
morphism In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
. It is the category-theoretic dual notion to the
categorical product In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic top ...
, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.

# Definition

Let $C$ be a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...
and let $X_1$ and $X_2$ be objects of $C.$ An object is called the coproduct of $X_1$ and $X_2,$ written $X_1 \sqcup X_2,$ or $X_1 \oplus X_2,$ or sometimes simply $X_1 + X_2,$ if there exist morphisms $i_1 : X_1 \to X_1 \sqcup X_2$ and $i_2 : X_2 \to X_1 \sqcup X_2$ satisfying the following
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
: for any object $Y$ and any morphisms $f_1 : X_1 \to Y$ and $f_2 : X_2 \to Y,$ there exists a unique morphism $f : X_1 \sqcup X_2 \to Y$ such that $f_1 = f \circ i_1$ and $f_2 = f \circ i_2.$ That is, the following diagram commutes: The unique arrow $f$ making this diagram commute may be denoted $f_1 \sqcup f_2,$ $f_1 \oplus f_2,$ $f_1 + f_2,$ or The morphisms $i_1$ and $i_2$ are called , although they need not be injections or even monic. The definition of a coproduct can be extended to an arbitrary
family Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...
of objects indexed by a set $J.$ The coproduct of the family $\left\$ is an object $X$ together with a collection of
morphism In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
s $i_j : X_j \to X$ such that, for any object $Y$ and any collection of morphisms $f_j : X_j \to Y$ there exists a unique morphism $f : X \to Y$ such that $f_j = f \circ i_j.$ That is, the following diagram commutes for each $j \in J$: The coproduct $X$ of the family $\left\$ is often denoted $\coprod_X_j$ or $\bigoplus_ X_j.$ Sometimes the morphism $f : X \to Y$ may be denoted $\coprod_ f_j$ to indicate its dependence on the individual $f_j$s.

# Examples

The coproduct in the
category of sets In the mathematical field of category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundatio ...
is simply the
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injective function, injection of each A_i into A, such that the image (mathematics), images of th ...
with the maps ''ij'' being the
inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function (mathematics), function ι, \iota that sends each element x of A to x, treated as an element of B: \iota : A\ ...
s. Unlike
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying Set (mathematics), sets, together with a suitably defined structure on the product set. More ...
s, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be dramatically different from each other. For example, the coproduct in the
category of groups In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
, called the
free product In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
, is quite complicated. On the other hand, in the
category of abelian groups In mathematics, the category theory, category Ab has the abelian groups as object (category theory), objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every Small category, small abelian category can ...
(and equally for
vector spaces In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
), the coproduct, called the
direct sum The direct sum is an Operation (mathematics), operation between Mathematical structure, structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct ...
, consists of the elements of the direct product which have only finitely many nonzero terms. (It therefore coincides exactly with the direct product in the case of finitely many factors.) Given a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring proper ...
''R'', the coproduct in the category of commutative ''R''-algebras is the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
. In the category of (noncommutative) ''R''-algebras, the coproduct is a quotient of the tensor algebra (see free product of associative algebras). In the case of
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s, coproducts are disjoint unions with their disjoint union topologies. That is, it is a disjoint union of the underlying sets, and the
open set In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s are sets ''open in each of the spaces'', in a rather evident sense. In the category of pointed spaces, fundamental in
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent disciplin ...
, the coproduct is the
wedge sum In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the Quo ...
(which amounts to joining a collection of spaces with base points at a common base point). The concept of disjoint union secretly underlies the above examples: the direct sum of abelian groups is the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero), similarly for vector spaces: the space spanned by the "almost" disjoint union; the free product for groups is generated by the set of all letters from a similar "almost disjoint" union where no two elements from different sets are allowed to commute. This pattern holds for any variety in the sense of universal algebra. The coproduct in the category of
Banach spaces In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a Complete metric space, complete normed vector space. Thus, a Banach space is a vector space with a Metric (mathematics), metric that allows the computation ...
with
short map In the mathematics, mathematical theory of metric spaces, a metric map is a Function (mathematics), function between metric spaces that does not increase any distance (such functions are always continuous function, continuous). These maps are the m ...
s is the sum, which cannot be so easily conceptualized as an "almost disjoint" sum, but does have a
unit ball Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
almost-disjointly generated by the unit ball is the cofactors. The coproduct of a poset category is the join operation.

# Discussion

The coproduct construction given above is actually a special case of a colimit in category theory. The coproduct in a category $C$ can be defined as the colimit of any
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
from a
discrete category In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms: :hom''C''(''X'', ''X'') = {id''X''} for all objects ''X'' :hom''C''(''X'', ''Y'') = ∅ for all objects ''X'' ≠ ''Y ...
$J$ into $C$. Not every family $\lbrace X_j\rbrace$ will have a coproduct in general, but if it does, then the coproduct is unique in a strong sense: if $i_j:X_j\rightarrow X$ and $k_j:X_j\rightarrow Y$ are two coproducts of the family $\lbrace X_j\rbrace$, then (by the definition of coproducts) there exists a unique
isomorphism In mathematics, an isomorphism is a structure-preserving Map (mathematics), mapping between two Mathematical structure, structures of the same type that can be reversed by an inverse function, inverse mapping. Two mathematical structures are iso ...
$f:X\rightarrow Y$ such that $f \circ i_j = k_j$ for each $j \in J$. As with any
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
, the coproduct can be understood as a universal morphism. Let $\Delta : C\rightarrow C\times C$ be the
diagonal functor In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a) = \langle a,a \rangle, which maps Object (category theory), objects as well as morphisms. This functor can be empl ...
which assigns to each object $X$ the
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
$\left\left(X, X\right\right)$ and to each morphism $f : X\rightarrow Y$ the pair $\left\left(f, f\right\right)$. Then the coproduct $X + Y$ in $C$ is given by a universal morphism to the functor $\Delta$ from the object $\left\left(X, Y\right\right)$ in $C\times C$. The coproduct indexed by the
empty set In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
(that is, an ''empty coproduct'') is the same as an
initial object In category theory, a branch of mathematics, an initial object of a category (mathematics), category is an object in such that for every object in , there exists precisely one morphism . The dual (category theory), dual notion is that of a t ...
in $C$. If $J$ is a set such that all coproducts for families indexed with $J$ exist, then it is possible to choose the products in a compatible fashion so that the coproduct turns into a functor $C^J\rightarrow C$. The coproduct of the family $\lbrace X_j\rbrace$ is then often denoted by :$\coprod_ X_j$ and the maps $i_j$ are known as the natural injections. Letting $\operatorname_C\left\left(U, V\right\right)$ denote the set of all morphisms from $U$ to $V$ in $C$ (that is, a
hom-set In mathematics, particularly in category theory, a morphism is a structure-preserving Map (mathematics), map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set ...
in $C$), we have a natural isomorphism :$\operatorname_C\left\left(\coprod_X_j,Y\right\right) \cong \prod_\operatorname_C\left(X_j,Y\right)$ given by the
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...
which maps every
tuple In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
of morphisms :$\left(f_j\right)_ \in \prod_\operatorname\left(X_j,Y\right)$ (a product in Set, the
category of sets In the mathematical field of category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundatio ...
, which is the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two Set (mathematics), sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notatio ...
, so it is a tuple of morphisms) to the morphism :$\coprod_ f_j \in \operatorname\left\left(\coprod_X_j,Y\right\right).$ That this map is a
surjection In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
follows from the commutativity of the diagram: any morphism $f$ is the coproduct of the tuple :$\left(f\circ i_j\right)_.$ That it is an injection follows from the universal construction which stipulates the uniqueness of such maps. The naturality of the isomorphism is also a consequence of the diagram. Thus the contravariant hom-functor changes coproducts into products. Stated another way, the hom-functor, viewed as a functor from the
opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given Category (mathematics), category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the ...
$C^\operatorname$ to Set is continuous; it preserves limits (a coproduct in $C$ is a product in $C^\operatorname$). If $J$ is a
finite set In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
, say $J = \lbrace 1,\ldots,n\rbrace$, then the coproduct of objects $X_1,\ldots,X_n$ is often denoted by $X_1\oplus\ldots\oplus X_n$. Suppose all finite coproducts exist in ''C'', coproduct functors have been chosen as above, and 0 denotes the
initial object In category theory, a branch of mathematics, an initial object of a category (mathematics), category is an object in such that for every object in , there exists precisely one morphism . The dual (category theory), dual notion is that of a t ...
of ''C'' corresponding to the empty coproduct. We then have natural isomorphisms :$X\oplus \left(Y \oplus Z\right)\cong \left(X\oplus Y\right)\oplus Z\cong X\oplus Y\oplus Z$ :$X\oplus 0 \cong 0\oplus X \cong X$ :$X\oplus Y \cong Y\oplus X.$ These properties are formally similar to those of a commutative
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ar ...
; a category with finite coproducts is an example of a symmetric
monoidal category In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an Object (categor ...
. If the category has a
zero object In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic top ...
$Z$, then we have a unique morphism $X\rightarrow Z$ (since $Z$ is terminal) and thus a morphism $X\oplus Y\rightarrow Z\oplus Y$. Since $Z$ is also initial, we have a canonical isomorphism $Z\oplus Y\cong Y$ as in the preceding paragraph. We thus have morphisms $X\oplus Y\rightarrow X$ and $X\oplus Y\rightarrow Y$, by which we infer a canonical morphism $X\oplus Y\rightarrow X\times Y$. This may be extended by induction to a canonical morphism from any finite coproduct to the corresponding product. This morphism need not in general be an isomorphism; in Grp it is a proper
epimorphism In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic top ...
while in Set* (the category of pointed sets) it is a proper
monomorphism In the context of abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, fiel ...
. In any
preadditive category In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category (mathematics), category that is enriched category, enriched over the category of abelian groups, Ab. That is, an Ab-cate ...
, this morphism is an isomorphism and the corresponding object is known as the
biproduct In category theory and its applications to mathematics, a biproduct of a finite collection of Object (category theory), objects, in a category (mathematics), category with zero objects, is both a product (category theory), product and a coproduct. ...
. A category with all finite biproducts is known as a semiadditive category. If all families of objects indexed by $J$ have coproducts in $C$, then the coproduct comprises a functor $C^J\rightarrow C$. Note that, like the product, this functor is ''covariant''.

*
Product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Pr ...
* Limits and colimits * Coequalizer *
Direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be Group (mathematics), groups, Ring (mathematics), rings, Vector spac ...

*