View source for Coproduct

''of a family of objects in a category''

A concept describing the (categorical analogues of the) construction of a direct sum of modules or a discrete union (bouquet) of sets in the language of morphisms. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c0264801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c0264802.png" />, be an indexed family of objects in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c0264803.png" />. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c0264804.png" />, together with morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c0264805.png" />, is called the coproduct of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c0264806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c0264807.png" />, if for any family of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c0264808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c0264809.png" />, there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648012.png" />. The morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648013.png" /> are called the imbeddings of the coproduct; the coproduct is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648015.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648016.png" /> in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648017.png" />. The morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648018.png" /> figuring in the definition of the coproduct is sometimes denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648019.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648020.png" />. The coproduct of a family of objects is defined uniquely up to an isomorphism; it is associative and commutative. The coproduct is the dual concept of the [[Product of a family of objects in a category|product of a family of objects in a category]].

The coproduct of the empty family of objects is the left zero (initial object) of the category. In an Abelian category, the coproduct is frequently called the direct sum of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648022.png" />, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648023.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648024.png" /> in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648025.png" />. In most categories of structured sets, the coproduct of a family of objects coincides with the free product of the family, and as a rule requires special description. Thus, in the category of groups, the coproduct is the free product of groups; in the category of modules it is the direct sum of modules; etc.

In a category with [[null morphism]]s, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648026.png" /> is a coproduct, there exist uniquely defined morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648027.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648029.png" />. In an Abelian category the coproduct and the product of a finite family of objects are one and the same.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.Sh. Tsalenko,   E.G. Shul'geifer,   "Fundamentals of category theory" , Moscow  (1974)  (In Russian)</TD></TR></table>



====Comments====
Also in not necessarily Abelian categories the coproduct of a family of objects is frequently called the sum of a family of objects or the direct sum of a family of objects. Often used notations are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026480/c02648032.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Popescu,   "Abelian categories with applications to rings and modules" , Acad. Press  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Adámek,   "Theory of mathematical structures" , Reidel  (1983)</TD></TR></table>