Editing Coproduct

{{Short description|Category-theoretic construction}}
{{About|coproducts in categories|"coproduct" in the sense of comultiplication|Coalgebra|additional substances that result from the manufacturing of another product|By-product}}

In [[category theory]], the '''coproduct''', or '''categorical sum''', is a construction which includes as examples the [[disjoint union]] of [[Set (mathematics)|sets]] and [[disjoint union (topology)|of topological spaces]], the [[free product]] of [[Group (mathematics)|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 within a given category.

== Definition ==
Let <math>C</math> be a [[Category (mathematics)|category]] and let <math>X_1</math> and <math>X_2</math> be objects of <math>C.</math> An object is called the coproduct of <math>X_1</math> and <math>X_2,</math> written <math>X_1 \sqcup X_2,</math> or <math>X_1 \oplus X_2,</math> or sometimes simply <math>X_1 + X_2,</math> if there exist morphisms <math>i_1 : X_1 \to X_1 \sqcup X_2</math> and <math>i_2 : X_2 \to X_1 \sqcup X_2</math> that satisfies the following [[universal property]]: for any object <math>Y</math> and any morphisms <math>f_1 : X_1 \to Y</math> and <math>f_2 : X_2 \to Y,</math> there exists a unique morphism <math>f : X_1 \sqcup X_2 \to Y</math> such that <math>f_1 = f \circ i_1</math> and <math>f_2 = f \circ i_2.</math> That is, the following diagram [[Commutative diagram|commutes]]:

[[Image:Coproduct-03.svg|280px|center]]

The unique arrow <math>f</math> making this diagram commute may be denoted <math>f_1 \sqcup f_2,</math> <math>f_1 \oplus f_2,</math> <math>f_1 + f_2,</math> or <math>\left[f_1, f_2\right].</math> The morphisms <math>i_1</math> and <math>i_2</math> are called {{em|[[canonical injection]]s}}, although they need not be [[Injective function|injections]] or even [[Monomorphism|monic]].

The definition of a coproduct can be extended to an arbitrary [[Indexed family|family]] of objects indexed by a set <math>J.</math> The coproduct of the family <math>\left\{ X_j : j \in J \right\}</math> is an object <math>X</math> together with a collection of [[morphism]]s <math>i_j : X_j \to X</math> such that, for any object <math>Y</math> and any collection of morphisms <math>f_j : X_j \to Y</math> there exists a unique morphism <math>f : X \to Y</math> such that <math>f_j = f \circ i_j.</math> That is, the following diagram [[Commutative diagram|commutes]] for each <math>j \in J</math>:

[[Image:Coproduct-01.svg|160px|center]]

The coproduct <math>X</math> of the family <math>\left\{ X_j \right\}</math> is often denoted <math>\coprod_{j\in J}X_j</math> or <math>\bigoplus_{j \in J} X_j.</math>

Sometimes the morphism <math>f : X \to Y</math> may be denoted <math>\coprod_{j \in J} f_j</math> to indicate its dependence on the individual <math>f_j</math>s.

== Examples ==
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.)

Given a [[commutative ring]] ''R'', the coproduct in the [[category of commutative algebras|category of commutative ''R''-algebras]] is the [[tensor product of algebras|tensor product]]. In the [[category of rings#R-algebras|category of (noncommutative) ''R''-algebras]], the coproduct is a quotient of the tensor algebra (see ''[[Free product of associative algebras]]'').

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).

The concept of disjoint union secretly underlies the above examples: 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.  This pattern holds for any [[variety (universal algebra)|variety in the sense of universal algebra]].

The coproduct in the category of [[Banach spaces]] with [[short map]]s is the [[Lp space|{{math|''l''<sup>1</sup>}}]] sum, which cannot be so easily conceptualized as an "almost disjoint" sum, but does have a [[unit ball]] almost-disjointly generated by the unit ball is the cofactors.<ref name=Ban1Cat>{{cite web|website=Annoying Precision|title=Banach spaces (and Lawvere metrics, and closed categories)|date=June 23, 2012|author=Qiaochu Yuan|url=https://qchu.wordpress.com/2012/06/23/banach-spaces-and-lawvere-metrics-and-closed-categories/}}</ref>

The coproduct of a [[poset category]] is the [[Join (mathematics)|join operation]].

== Discussion ==

The coproduct construction given above is actually a special case of a [[colimit]] in category theory. The coproduct in a category <math>C</math> can be defined as the colimit of any [[functor]] from a [[discrete category]] <math>J</math> into <math>C</math>. Not every family <math>\lbrace X_j\rbrace</math> will have a coproduct in general, but if it does, then the coproduct is unique in a strong sense: if <math>i_j:X_j\rightarrow X</math> and <math>k_j:X_j\rightarrow Y</math> are two coproducts of the family <math>\lbrace X_j\rbrace</math>, then (by the definition of coproducts) there exists a unique [[isomorphism]] <math>f:X\rightarrow Y</math> such that <math>f \circ i_j = k_j</math> for each <math>j \in J</math>.

As with any [[universal property]], the coproduct can be understood as a universal morphism. Let <math>\Delta : C\rightarrow C\times C</math> be the [[diagonal functor]] which assigns to each object <math>X</math> the [[ordered pair]] <math>\left(X, X\right)</math> and to each morphism <math>f : X\rightarrow Y</math> the pair <math>\left(f, f\right)</math>. Then the coproduct <math>X + Y</math> in <math>C</math> is given by a universal morphism to the functor <math>\Delta</math> from the object <math>\left(X, Y\right)</math> in <math>C\times C</math>.

The coproduct indexed by the [[empty set]] (that is, an ''empty coproduct'') is the same as an [[initial object]] in <math>C</math>.

If <math>J</math> is a set such that all coproducts for families indexed with <math>J</math> exist, then it is possible to choose the products in a compatible fashion so that the coproduct turns into a functor <math>C^J\rightarrow C</math>. The coproduct of the family <math>\lbrace X_j\rbrace</math> is then often denoted by 
: <math>\coprod_{j\in J} X_j</math>
and the maps <math>i_j</math> are known as the '''[[inclusion map|natural injections]]'''.

Letting <math>\operatorname{Hom}_C\left(U, V\right)</math> denote the set of all morphisms from <math>U</math> to <math>V</math> in <math>C</math> (that is, a [[hom-set]] in <math>C</math>), we have a [[natural isomorphism]] 
: <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
: <math>(f_j)_{j\in J} \in \prod_{j \in J}\operatorname{Hom}(X_j,Y)</math>
(a product in '''Set''', the [[category of sets]], which is the [[Cartesian product]], so it is a tuple of morphisms) to the morphism
: <math>\coprod_{j\in J} f_j \in \operatorname{Hom}\left(\coprod_{j\in J}X_j,Y\right).</math>
That this map is a [[surjection]] follows from the commutativity of the diagram: any morphism <math>f</math> is the coproduct of the tuple 
: <math>(f\circ i_j)_{j \in J}.</math>
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]] <math>C^\operatorname{op}</math> to '''Set''' is continuous; it preserves limits (a coproduct in <math>C</math> is a product in <math>C^\operatorname{op}</math>).

If <math>J</math> is a [[finite set]], say <math>J = \lbrace 1,\ldots,n\rbrace</math>, then the coproduct of objects <math>X_1,\ldots,X_n</math> is often denoted by <math>X_1\oplus\ldots\oplus X_n</math>. 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
: <math>X\oplus (Y \oplus Z)\cong (X\oplus Y)\oplus Z\cong X\oplus Y\oplus Z</math>
: <math>X\oplus 0 \cong 0\oplus X \cong X</math>
: <math>X\oplus Y \cong Y\oplus X.</math>
These properties are formally similar to those of a commutative [[monoid]]; a category with finite coproducts is an example of a symmetric [[monoidal category]].

If the category has a [[zero object]] <math>Z</math>, then we have a unique morphism <math>X\rightarrow Z</math> (since <math>Z</math> is [[terminal object|terminal]]) and thus a morphism <math>X\oplus Y\rightarrow Z\oplus Y</math>.  Since <math>Z</math> is also initial, we have a canonical isomorphism <math>Z\oplus Y\cong Y</math> as in the preceding paragraph.  We thus have morphisms <math>X\oplus Y\rightarrow X</math> and <math>X\oplus Y\rightarrow Y</math>, by which we infer a canonical morphism <math>X\oplus Y\rightarrow X\times Y</math>.  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 a [[additive category|semiadditive category]].

If all families of objects indexed by <math>J</math> have coproducts in <math>C</math>, then the coproduct comprises a functor <math>C^J\rightarrow C</math>.  Note that, like the product, this functor is ''covariant''.

== See also ==
* [[Product (category theory)|Product]]
* [[Limit (category theory)|Limits and colimits]]
* [[Coequalizer]]
* [[Direct limit]]

== References ==
{{reflist}}
* {{cite book | last=Mac Lane | first=Saunders | author-link=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 }}

== External links ==
* [https://web.archive.org/web/20080916162345/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 [https://web.archive.org/web/20081223001815/http://www.j-paine.org/ Jocelyn Paine].

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[[Category:Limits (category theory)]]