In

_{j}'' being the

_{*} (the category of pointed sets) it is a proper

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which generates examples of coproducts in the category of finite sets. Written b

Jocelyn Paine

{{Authority control Limits (category theory)

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 acategory
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 $X\_1\; \backslash sqcup\; X\_2,$ or $X\_1\; \backslash oplus\; X\_2,$ or sometimes simply $X\_1\; +\; X\_2,$ if there exist morphisms $i\_1\; :\; X\_1\; \backslash to\; X\_1\; \backslash sqcup\; X\_2$ and $i\_2\; :\; X\_2\; \backslash to\; X\_1\; \backslash sqcup\; X\_2$ 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 $f\_1\; :\; X\_1\; \backslash to\; Y$ and $f\_2\; :\; X\_2\; \backslash to\; Y,$ there exists a unique morphism $f\; :\; X\_1\; \backslash sqcup\; X\_2\; \backslash to\; Y$ such that $f\_1\; =\; f\; \backslash circ\; i\_1$ and $f\_2\; =\; f\; \backslash circ\; i\_2.$ That is, the following diagram commutes:
The unique arrow $f$ making this diagram commute may be denoted $f\_1\; \backslash sqcup\; f\_2,$ $f\_1\; \backslash oplus\; f\_2,$ $f\_1\; +\; f\_2,$ or $\backslash left;\; href="/html/ALL/s/\_1,\_f\_2\backslash right.html"\; ;"title="\_1,\; f\_2\backslash right">\_1,\; f\_2\backslash right$ 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 $\backslash left\backslash $ 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 $i\_j\; :\; X\_j\; \backslash to\; X$ such that, for any object $Y$ and any collection of morphisms $f\_j\; :\; X\_j\; \backslash to\; Y$ there exists a unique morphism $f\; :\; X\; \backslash to\; Y$ such that $f\_j\; =\; f\; \backslash circ\; i\_j.$ That is, the following diagram commutes for each $j\; \backslash in\; J$:
The coproduct $X$ of the family $\backslash left\backslash $ is often denoted $\backslash coprod\_X\_j$ or $\backslash bigoplus\_\; X\_j.$
Sometimes the morphism $f\; :\; X\; \backslash to\; Y$ may be denoted $\backslash coprod\_\; f\_j$ to indicate its dependence on the individual $f\_j$s.
Examples

The coproduct in thecategory 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 ''iinclusion 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 anyfunctor
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 $\backslash lbrace\; X\_j\backslash rbrace$ will have a coproduct in general, but if it does, then the coproduct is unique in a strong sense: if $i\_j:X\_j\backslash rightarrow\; X$ and $k\_j:X\_j\backslash rightarrow\; Y$ are two coproducts of the family $\backslash lbrace\; X\_j\backslash rbrace$, 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 ...

$f:X\backslash rightarrow\; Y$ such that $f\; \backslash circ\; i\_j\; =\; k\_j$ for each $j\; \backslash in\; J$.
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 $\backslash Delta\; :\; C\backslash rightarrow\; C\backslash times\; C$ 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 ...

$\backslash left(X,\; X\backslash right)$ and to each morphism $f\; :\; X\backslash rightarrow\; Y$ the pair $\backslash left(f,\; f\backslash right)$. Then the coproduct $X\; +\; Y$ in $C$ is given by a universal morphism to the functor $\backslash Delta$ from the object $\backslash left(X,\; Y\backslash right)$ in $C\backslash times\; C$.
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 $C^J\backslash rightarrow\; C$. The coproduct of the family $\backslash lbrace\; X\_j\backslash rbrace$ is then often denoted by
:$\backslash coprod\_\; X\_j$
and the maps $i\_j$ are known as the natural injections.
Letting $\backslash operatorname\_C\backslash left(U,\; V\backslash right)$ 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
:$\backslash operatorname\_C\backslash left(\backslash coprod\_X\_j,Y\backslash right)\; \backslash cong\; \backslash prod\_\backslash operatorname\_C(X\_j,Y)$
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
:$(f\_j)\_\; \backslash in\; \backslash prod\_\backslash operatorname(X\_j,Y)$
(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
:$\backslash coprod\_\; f\_j\; \backslash in\; \backslash operatorname\backslash left(\backslash coprod\_X\_j,Y\backslash right).$
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
:$(f\backslash circ\; i\_j)\_.$
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 ...

$C^\backslash operatorname$ to Set is continuous; it preserves limits (a coproduct in $C$ is a product in $C^\backslash operatorname$).
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 $J\; =\; \backslash lbrace\; 1,\backslash ldots,n\backslash rbrace$, then the coproduct of objects $X\_1,\backslash ldots,X\_n$ is often denoted by $X\_1\backslash oplus\backslash ldots\backslash oplus\; X\_n$. 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
:$X\backslash oplus\; (Y\; \backslash oplus\; Z)\backslash cong\; (X\backslash oplus\; Y)\backslash oplus\; Z\backslash cong\; X\backslash oplus\; Y\backslash oplus\; Z$
:$X\backslash oplus\; 0\; \backslash cong\; 0\backslash oplus\; X\; \backslash cong\; X$
:$X\backslash oplus\; Y\; \backslash cong\; Y\backslash oplus\; X.$
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 $X\backslash rightarrow\; Z$ (since $Z$ is terminal) and thus a morphism $X\backslash oplus\; Y\backslash rightarrow\; Z\backslash oplus\; Y$. Since $Z$ is also initial, we have a canonical isomorphism $Z\backslash oplus\; Y\backslash cong\; Y$ as in the preceding paragraph. We thus have morphisms $X\backslash oplus\; Y\backslash rightarrow\; X$ and $X\backslash oplus\; Y\backslash rightarrow\; Y$, by which we infer a canonical morphism $X\backslash oplus\; Y\backslash rightarrow\; X\backslash times\; Y$. 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 Setmonomorphism
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 $C^J\backslash rightarrow\; C$. Note that, like the product, this functor is ''covariant''.
See also

*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 ...

References

*External links

Interactive Web page

which generates examples of coproducts in the category of finite sets. Written b

Jocelyn Paine

{{Authority control Limits (category theory)