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| {{about|coproducts in categories|"coproduct" in the sense of comultiplication|Coalgebra}}
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| In [[category theory]], the '''coproduct''', or '''categorical sum''', is a category-theoretic construction which includes as examples the [[disjoint union|disjoint union of sets]] and [[disjoint union (topology)|of topological spaces]], the [[free product|free product of groups]], and the [[direct sum]] of [[Module (mathematics)|modules]] and [[vector space]]s. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a [[morphism]]. It is the category-theoretic [[Dual (category theory)|dual notion]] to the [[product (category theory)|categorical product]], which means the definition is the same as the product but with all [[morphism|arrows]] reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.
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| ==Definition==
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| The formal definition is as follows: Let ''C'' be a category and let {''X<sub>j</sub>'' : ''j'' ∈ ''J''} be an [[index set|indexed family]] of objects in ''C''. The coproduct of the family {''X<sub>j</sub>''} is an object ''X'' together with a collection of [[morphism]]s ''i<sub>j</sub>'' : ''X<sub>j</sub>'' → ''X'' (called ''[[canonical injection]]s'' although they need not be [[injective function|injections]] or even [[monomorphism|monic]]) which satisfy a [[universal property]]: for any object ''Y'' and any collection of morphisms ''f<sub>j</sub>'' : ''X<sub>j</sub>'' → ''Y'', there exists a unique morphism ''f'' from ''X'' to ''Y'' such that ''f<sub>j</sub>'' = ''f'' ∘ ''i<sub>j</sub>''. That is, the following diagram [[commutative diagram|commutes]] (for each ''j''):
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| [[Image:Coproduct-01.svg|160px|center]] | |
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| The coproduct of the family {''X<sub>j</sub>''} is often denoted
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| :<math> X = \coprod_{j\in J}X_j</math>
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| or
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| :<math>X = \bigoplus_{j \in J} X_j.</math>
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| Sometimes the morphism ''f'' may be denoted
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| :<math>f=\coprod_{j \in J} f_j: \coprod_{j \in J} X_j \to Y</math>
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| to indicate its dependence on the individual ''f''<sub>''j''</sub>.
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| If the family of objects consists of only two members the coproduct is usually written ''X''<sub>1</sub> ∐ ''X''<sub>2</sub> or ''X''<sub>1</sub> ⊕ ''X''<sub>2</sub> or sometimes simply ''X''<sub>1</sub> + ''X''<sub>2</sub>, and the diagram takes the form:
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| [[Image:Coproduct-03.svg|280px|center]]
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| The unique arrow ''f'' making this diagram commute is then correspondingly denoted ''f''<sub>1</sub> ∐ ''f''<sub>2</sub> or ''f''<sub>1</sub> ⊕ ''f''<sub>2</sub> or ''f''<sub>1</sub> + ''f''<sub>2</sub> or [''f''<sub>1</sub>, ''f''<sub>2</sub>].
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| == Examples ==
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| The coproduct in the [[category of sets]] is simply the '''[[disjoint union#Set theory definition|disjoint union]]''' with the maps ''i<sub>j</sub>'' 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 set|finitely]] many nonzero terms. (It therefore coincides exactly with the direct product in the case of finitely many factors.) | |
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| In the case of [[topological space]]s coproducts are disjoint unions with their [[disjoint union (topology)|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).
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| 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 [[linear span|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.
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| == Discussion ==
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| 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 any [[functor]] from a [[discrete category]] ''J'' into ''C''. Not every family {''X''<sub>''j''</sub>} will have a coproduct in general, but if it does, then the coproduct is unique in a strong sense: if ''i''<sub>''j''</sub> : ''X''<sub>''j''</sub> → ''X'' and ''k''<sub>''j''</sub> : ''X''<sub>''j''</sub> → ''Y'' are two coproducts of the family {''X''<sub>''j''</sub>}, then (by the definition of coproducts) there exists a unique [[isomorphism]] ''f'' : ''X'' → ''Y'' such that ''fi''<sub>''j''</sub> = ''k''<sub>''j''</sub> for each ''j'' in ''J''.
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| As with any [[universal property]], the coproduct can be understood as a universal 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''). 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''.
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| The coproduct indexed by the [[empty set]] (that is, an ''empty coproduct'') is the same as an [[initial object]] in ''C''.
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| 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''<sup>''J''</sup> → ''C''. The coproduct of the family {''X''<sub>''j''</sub>} is then often denoted by ∐<sub>''j''</sub> ''X''<sub>''j''</sub>, and the maps ''i''<sub>''j''</sub> are known as the '''[[inclusion map|natural injections]]'''.
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| Letting Hom<sub>''C''</sub>(''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]]
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| :<math>\operatorname{Hom}_C\left(\coprod_{j\in J}X_j,Y\right) \cong \prod_{j\in J}\operatorname{Hom}_C(X_j,Y)</math> | |
| given by the [[bijection]] which maps every [[tuple]] of morphisms
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| :<math>(f_j)_{j\in J} \in \prod_{j \in J}\operatorname{Hom}(X_j,Y)</math>
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| (a product in '''Set''', the [[category of sets]], which is the [[Cartesian product]], so it is a tuple of morphisms) to the morphism
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| :<math>\coprod_{j\in J} f_j \in \operatorname{Hom}\left(\coprod_{j\in J}X_j,Y\right).</math>
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| That this map is a [[surjection]] follows from the commutativity of the diagram: any morphism ''f'' is the coproduct of the tuple
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| :<math>(f\circ i_j)_{j \in J}.</math>
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| 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''<sup>opp</sup> to '''Set''' is continuous; it preserves limits (a coproduct in ''C'' is a product in ''C''<sup>opp</sup>).
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| If ''J'' is a [[finite set|finite]] set, say ''J'' = {1,...,''n''}, then the coproduct of objects ''X''<sub>1</sub>,...,''X''<sub>''n''</sub> is often denoted by ''X''<sub>1</sub>⊕...⊕''X''<sub>''n''</sub>.
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| 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
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| :<math>X\oplus (Y \oplus Z)\cong (X\oplus Y)\oplus Z\cong X\oplus Y\oplus Z</math>
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| :<math>X\oplus 0 \cong 0\oplus X \cong X</math>
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| :<math>X\oplus Y \cong Y\oplus X.</math>
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| These properties are formally similar to those of a commutative [[monoid]]; a category with finite coproducts is an example of a symmetric [[monoidal category]].
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| If the category has a [[zero object]] ''Z'', then we have unique morphism ''X'' → ''Z'' (since ''Z'' is [[terminal object|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'''<sub>*</sub> (the category of [[pointed set]]s) it is a proper [[monomorphism]]. In any [[preadditive category]], this morphism is an isomorphism and the corresponding object is known as the [[biproduct]]. A category with all finite biproducts is known as an [[additive category]].
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| If all families of objects indexed by ''J'' have coproducts in ''C'', then the coproduct comprises a functor ''C''<sup>''J''</sup> → ''C''. Note that, like the product, this functor is ''covariant''.
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| ==See also==
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| *[[Product (category theory)|Product]]
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| *[[Limit (category theory)|Limits and colimits]]
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| *[[Coequalizer]]
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| *[[Direct limit]]
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| ==References==
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| * {{cite book | last=Mac Lane | first=Saunders | authorlink=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | edition=2nd | series=[[Graduate Texts in Mathematics]] | volume=5 | location=New York, NY | publisher=[[Springer-Verlag]] | year=1998 | isbn=0-387-98403-8 | zbl=0906.18001 }}
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| == External links ==
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| *[http://www.j-paine.org/cgi-bin/webcats/webcats.php Interactive Web page ] which generates examples of coproducts in the category of finite sets. Written by [http://www.j-paine.org/ Jocelyn Paine].
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| [[Category:Limits (category theory)]]
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Luke Bryan is really a superstar within the generating and the profession expansion very first second to his 3rd place record, And , is definitely the resistant. He broken to the picture in 2013 together with his crazy mix of lower-house availability, movie legend excellent appearance and words, is set t inside a significant way. The new recording Top on the region graph luke bryan pittsburgh tickets and #2 about the burst maps, buy luke bryan tickets (http://lukebryantickets.lazintechnologies.com) creating it the 2nd greatest first appearance during those times of 2014 for a nation artist.
The boy of your , knows persistence and willpower are important elements with regards to a successful profession- . His initially album, Continue to be Me, vip luke bryan tickets created the Top hits “All My Buddies Say” and “Country Gentleman,” although his work, Doin’ Factor, discovered the vocalist-three justin bieber meet and greet right No. 5 single men and women: In addition Getting in touch with Is actually a Wonderful Thing.”
Inside the slip of 2013, Tour: Luke Bryan And which in fact had an amazing selection of , including City. “It’s almost like you are obtaining a authorization to travel to the next level, claims those designers that have been an element of the Concert toursover in a bigger measure of performers.” It packaged as among the most successful trips in their 10-season historical past.
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