Coproduct

In category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Set theory, a disjoint union (or discriminated union) is a modified union operation which indexes the elements according to which set they originated In General topology and related areas of Mathematics, the disjoint union (also called the direct sum, free union, or coproduct) of In Abstract algebra, the free product of groups constructs a group from two or more given ones The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction The coproduct of a family of objects is essentially the "most general" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as Despite this innocuous-looking change in the name and notation, coproducts can be and typically are dramatically different from products.

1 Definition
2 Examples
3 Discussion
4 See also

Definition

The formal definition is as follows: Let C be a category and let {X_j : j ∈ J} be an indexed family of objects in C. In Mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index The coproduct of the set {X_j} is an object X together with a collection of morphisms i_j : X_j → X (called canonical injections although they need not be injections or even monic) which satisfy a universal property: for any object Y and any collection of morphisms f_j : X_j → Y, there exists a unique morphism f from X to Y such that f_j = f O i_j. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, if A is a Subset of B, then the inclusion map (also inclusion function, or canonical injection) is the In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism That is, the following diagram commutes (for each j):

The coproduct of the family {X_j} is often denoted

$X = \coprod_{j\in J}X_j$

$X = \bigoplus_{j \in J} X_j.$

Sometimes the morphism f may be denoted

$f=\coprod_{j \in J} f_j: \coprod_{j \in J} X_j \to Y$

to indicate its dependence on the individual f_j. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also

If the family of objects consists of only two members the product is usually written X₁ ∐ X₂ or X₁ ⊕ X₂ or sometimes simply X₁ + X₂, and the diagram takes the form:

The unique arrow f making this diagram commute is then correspondingly denoted f₁ ∐ f₂ or f₁ ⊕ f₂ or f₁ + f₂ or [f₁, f₂].

Examples

The coproduct in the category of sets is simply the disjoint union with the maps i_j being the inclusion maps. In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are In Set theory, a disjoint union (or discriminated union) is a modified union operation which indexes the elements according to which set they originated In Mathematics, if A is a Subset of B, then the inclusion map (also inclusion function, or canonical injection) is the Unlike direct products, 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. In Mathematics, one can often define a direct product of objectsalready known giving a new one 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, called the free product, is quite complicated. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Abstract algebra, the free product of groups constructs a group from two or more given ones On the other hand, in the category of abelian groups (and equally for vector spaces), the coproduct, called the direct sum, consists of the elements of the direct product which have only finitely many nonzero terms (this therefore coincides exactly with the direct product, in the case of finitely many factors). In Mathematics, the category Ab has the Abelian groups as objects and Group homomorphisms as Morphisms This is the prototype In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. As a consequence, since most introductory linear algebra courses deal with only finite-dimensional vector spaces, nobody really hears much about direct sums until later on. Linear algebra is the branch of Mathematics concerned with In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

In the case of topological spaces coproducts are disjoint unions with their disjoint union topologies. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In General topology and related areas of Mathematics, the disjoint union (also called the direct sum, free union, or coproduct) of That is it is a disjoint union of the underlying sets, and the open sets are sets open in each of the spaces, in a rather evident sense. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point). In Mathematics, a pointed space is a Topological space X with a distinguished basepoint x 0 in X. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Topology, the wedge sum (sometimes wedge product, though not to be confused with the Exterior product, which also shares this terminology is a "one-point

Despite all this dissimilarity, there is still, at the heart of the whole thing, a disjoint union: 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. In the mathematical subfield of Linear algebra, the linear span, also called the linear hull, of a set of vectors in a Vector

Discussion

The coproduct construction given above is actually a special case of a colimit in category theory. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts The coproduct in a category C can be defined as the colimit of any functor from a discrete category J into C. In Mathematics, especially Category theory, a discrete category is a category whose only Morphisms are the Identity morphisms It is the simplest Not every family {X_j} will have a coproduct in general, but if it does, then the coproduct is unique in a strong sense: if i_j : X_j → X and k_j : X_j → Y are two coproducts of the family {X_j}, then (by the definition of coproducts) there exists a unique isomorphism f : X → Y such that i_j = k_j f for each j in J. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

As with any universal property, the coproduct can be understood as a universal morphism. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism Let Δ: C → C×C be the diagonal functor which assigns to each object X the ordered pair (X,X) and to each morphism f:X → Y the pair (f,f). In Category theory, for any object a in any category C where the product a × a exists there exists the diagonal morphism In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry Then the coproduct X+Y in C is given by a universal morphism to the functor Δ from the object (X,Y) in C×C.

The coproduct indexed by the empty set (that is, an empty coproduct) is the same as an initial object in C. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members

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 → C. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories The coproduct of the family {X_j} is then often denoted by ∐_j X_j, and the maps i_j are known as the natural injections.

Letting Hom_C(U,V) denote the set of all morphisms from U to V in C (that is, a hom-set in C), we have a natural isomorphism

$\operatorname{Hom}_C\left(\coprod_{j\in J}X_j,Y\right) \cong \prod_{j\in J}\operatorname{Hom}_C(X_j,Y)$

given by the bijection which maps every tuple of morphisms

$(f_j)_{j\in J} \in \prod_{j \in J}\operatorname{Hom}(X_j,Y)$

(a product in Set, the category of sets, which is the Cartesian product, so it is a tuple of morphisms) to the morphism

$\coprod_{j\in J} f_j \in \operatorname{Hom}\left(\coprod_{j\in J}X_j,Y\right).$

That this map is a surjection follows from the commutativity of the diagram: any morphism f is the coproduct of the tuple

$(f\circ i_j)_{j \in J}.$

That it is an injection follows from the universal construction which stipulates the uniqueness of such maps. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. The naturality of the isomorphism is also a consequence of the diagram. Thus the contravariant hom-functor changes coproducts into products. In Mathematics, specifically in Category theory, Hom-sets ie sets of Morphisms between objects give rise to important Functors to the Category Stated another way, the hom-functor, viewed as a functor from the opposite category C^opp to Set is continuous; it preserves limits (a coproduct in C is a product in C^opp). In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the

If J is a finite set, say J = {1,. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. . . ,n}, then the coproduct of objects X₁,. . . ,X_n is often denoted by X₁⊕. . . ⊕X_n. Suppose all finite coproducts exist in C, coproduct functors have been chosen as above, and 0 denotes the initial object of C corresponding to the empty coproduct. We then have natural isomorphisms

$X\oplus (Y \oplus Z)\cong (X\oplus Y)\oplus Z\cong X\oplus Y\oplus Z$

$X\oplus 0 \cong 0\oplus X \simeq X$

$X\oplus Y \cong Y\oplus X$

These properties are formally similar to those of a commutative monoid; a category with finite coproducts is a symmetric monoidal category. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Mathematics, a monoidal category (or tensor category) is a category C equipped with a Bifunctor &otimes: C

If the category has a zero object Z, then we have unique morphism X → Z (since Z is terminal) and thus a morphism X ⊕ Y → Z ⊕ Y. Since Z is also initial, we have a canonical isomorphism Z ⊕ Y ≅ Y as in the preceding paragraph. We thus have morphisms X ⊕ Y → X and X ⊕ Y → Y, by which we infer a canonical morphism X ⊕ Y → X×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 while in Set_* (the category of pointed sets) it is a proper monomorphism. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which In Mathematics, a pointed set is a set X with a distinguished basepoint x 0 in X. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In any preadditive category, this morphism is an isomorphism and the corresponding object is known as the biproduct. In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category In Category theory and its applications to Mathematics, a biproduct is a generalisation of the notion of Direct sum that makes sense in any Preadditive A category with all finite biproducts is known as an additive category. In Mathematics, specifically in Category theory, an additive category is a Preadditive category C such that any finitely many objects A

Coproducts are actually special cases of colimits in category theory. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts The coproduct can be defined as the colimit of a discrete subcategory in C. In Mathematics, especially Category theory, a discrete category is a category whose only Morphisms are the Identity morphisms It is the simplest It follows that if coproducts exists in a given category (they need not) they are unique up to a unique isomorphism that respects the injections. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

If all families of objects indexed by J have coproducts in C, then the coproduct comprises a functor C^J → C. Note that, like the product, this functor is covariant.

Dictionary

-noun

Any of multiple products that are produced at the same time, or by the same process

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Contents

Definition

Examples

Discussion

See also

Dictionary

coproduct

-noun