Information AboutCoproduct |
| CATEGORIES ABOUT COPRODUCT | |
| limits in categories | |
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DEFINITION The formal definition is as follows: Let ''C'' be a category and let {''Xj'' : ''j'' ∈ ''J''} be an (for each ''j''): The coproduct of the family {''Xj''} is often denoted : or : Sometimes the morphism ''f'' may be denoted : to indicate its dependence on the individual ''f''''j''. If the family of objects consists of only two members the product is usually written ''X''1 ∐ ''X''2 or ''X''1 ⊕ ''X''2 or sometimes simply ''X''1 + ''X''2, and the diagram takes the form: The unique arrow ''f'' making this diagram commute is then correspondingly denoted ''f''1 ∐ ''f''2 or ''f''1 ⊕ ''f''2 or ''f''1 + ''f''2. EXAMPLES The coproduct in the Category Of Sets is simply the Disjoint Union with the maps ''ij'' being the Inclusion Map s. Unlike Direct Product 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 , 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 Finite ly 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- Dimension al vector spaces, nobody really hears much about direct sums until later on. In the case of Topological Space 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 s are sets ''open in each of the spaces'', in a rather evident sense. In the category of Pointed Space s, 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). 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. DISCUSSION The coproduct construction given above is actually a special case of a ''f'' : ''X'' → ''Y'' such that ''i''''j'' = ''k''''j'' ''f'' for each ''j'' in ''J''. As with any which assigns to each object ''X'' the Ordered Pair (''X'',''X'') and to each morphism ''f'':''X'' → ''Y'' the pair (''f'',''f''). 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''. 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''. 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 : given by the bijection which maps every Tuple of morphisms : (a product in Set, the Category Of Sets , which is the Cartesian Product , so it is a tuple of morphisms) to the morphism : That this map is a surjection follows from the commutativity of the diagram: any morphism ''f'' is the coproduct of the tuple : 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 ''C''opp to Set is continuous; it preserves limits (a coproduct in ''C'' is a product in ''C''opp). If ''J'' is a Finite set, say ''J'' = {1,...,''n''}, then the product of objects ''X''1,...,''X''''n'' is often denoted by ''X''1⊕...⊕''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 Isomorphism s : : : These properties are formally similar to those of a commutative Monoid ; a category with finite coproducts is a symmetric Monoidal Category .
Coproducts are actually special cases of Colimit s 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 respects the injections. 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''. SEE ALSO |
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