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Motivation
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. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ...
In mathematics, given a group G and two subgroups H and K of G, one can define the product of H and K, denoted by HK as the set of all elements of the form hk, for all h in H and k in K. In general HK is not...
In abstract algebra, it is possible to combine several rings into one large product ring. ...
In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...
Definition Let C be a category and let {Xi | i ∈ I} be an indexed family of objects in C. The product of the set {Xi} is an object X together with a collection of morphisms πi : X → Xi (called projections) which satisfy a universal property: for any object Y and any collection of morphisms fi : Y → Xi, there exists a unique morphism f : Y → X such that for all i ∈ I it is the case that fi = πi f. That is, the following diagram commutes (for all i): 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 various branches of mathematics, certain constructions are frequently 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. ...
 If the family of objects consists of only two members X, Y, the product is usually written X×Y, and the diagram takes a form along the lines of: The unique arrow h making this diagram commute is notated <f,g>.
Discussion The product construction given above is actually a special case of a limit in category theory. The product can be defined as the limit of any discrete subcategory in C. Not every family {Xi} needs to have a product, but if it does, then the product is unique in a strong sense: if πi : X → Xi and π'i : X' → Xi are two products of the family {Xi}, then (by the definition of products) there exists a unique isomorphism f : X → X' such that πi = π'i f for each i in I. 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, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...
An empty product (i.e. I is the empty set) is the same as a terminal object in C. In arithmetic, the empty product, or nullary product, is the result of multiplying no numbers. ...
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...
If I is a set such that all products for families indexed with I exist, then it is possible to choose the products in a compatible fashion so that the product turns into a functor CI → C. The product of the family {Xi} is then often denoted by ∏i Xi, and the maps πi are known as the natural projections. We have a natural isomorphism In category theory, a functor is a special type of mapping between categories. ...
In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...
(where MorC(U,V) denotes the set of all morphisms from U to V in C, the left product is the one in C and the right is the cartesian product of sets). In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ...
If I is a finite set, say I = {1,...,n}, then the product of objects X1,...,Xn is often denoted by X1×...×Xn. Suppose all finite products exist in C, product functors have been chosen as above, and 1 denotes the terminal object of C corresponding to the empty product. We then have natural isomorphisms In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...
In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...
These properties are formally similar to those of a commutative monoid. In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...
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