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In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. Basically, this means the definition is the same as the product but with all arrows reversed. Despite this innocuous-looking change in the name and notation, coproducts can be dramatically different from products. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...
In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. ...
The formal definition is as follows: Let C be a category and let {Xj | j ∈ J} be a indexed family of objects in C. The coproduct of the set {Xj} is an object X together with a collection of morphisms ij : Xj → X (called injections) which satisfy a universal property: for any object Y and any collection of morphisms fj : Xj → Y, there exists a unique morphism f from X to Y such that fj = f O ij. That is, the follow diagram commutes (for each j): In mathematics, an index set is another name for a function domain. ...
In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ...
In category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...
In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...
 The coproduct of the family {Xj} is often denoted Commutative diagram for coproduct. ...
Coproducts are actually special cases of colimits in category theory. The coproduct can be defined as the colimit of a discrete subcategory in C. It follows that if coproducts exists in a given category (they need not) they are unique up to a unique isomorphism that respect the injections. 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 of mathematics, like products and inverse limits. ...
In category theory, a discrete category is a category whose only morphisms are the identity morphisms. ...
In mathematics, the term up to xxxx is used to describe a situation in which members of an equivalence class can be regarded as a single entity for some purpose. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...
The coproduct indexed by the empty set (that is, an empty coproduct) is the same as an initial object in C. In mathematics, the empty set is the set with no elements. ...
In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there...
The coproduct in the category of sets is simply the disjoint union with the maps ij being the inclusions. 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.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. 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—so much for "dramatically different"). 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. In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ...
In set theory, a disjoint union or discriminated union is a set union in which each element of the resulting union is disjoint from each of the others; the intersection over a disjoint union is the empty set. ...
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. ...
In abstract algebra, the free product of groups constructs a group from two or more given ones. ...
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ...
The fundamental concept in linear algebra is that of a vector space or linear space. ...
In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...
In the case of topological spaces coproducts are disjoint unions on the underlying sets, and the open sets are sets open in each of the spaces, in a rather evident sense (see disjoint union (topology)). 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). Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In the mathematical field of topology a direct sum, direct disjoint sum or coproduct is an important universal construction for topological spaces. ...
In mathematics, a pointed space is a topological space X with a distinguised basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i. ...
An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...
In topology, the wedge sum is a one-point union of a family of topological spaces. ...
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 of a set of vectors is the set of all linear combinations of the vectors. ...
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