---- > [!definition] Definition. ([[categorical coproduct]]) > Tl;dr: take the [[categorical product]] definition and 'reverse most of the arrows' within the [[universal property]] characterizing it. > > Suppose $A$ and $B$ are two objects of a [[category]] $\mathsf{C}$. Just as [[categorical product|products]] are [[terminal object|initial objects]] in the [[two-object slice category]] $\mathsf{C}_{A,B}$ obtained by considering morphisms with common *source*, **coproducts** are [[terminal object|initial objects]] in the [[two-object coslice category]] $\mathsf{C}^{A,B}$ obtained by considering morphisms with common *target*. Let us spell out this [[universal property]]... > > A **coproduct** $A \amalg B$ of $A$ and $B$ will be an object of $\mathsf{C}$, along with two morphisms $A \xmapsto{\iota_{A}} A \amalg B$ and $B \xmapsto{\iota_{B}}A \amalg B$, that satisfies the following [[universal property]]: for all objects $Z$ in $\mathsf{C}$ and morphisms $A \xrightarrow{f_{A}} Z$, $B \xrightarrow{f_{B}}Z$-objects > > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQBBEAX1PU1z5CKMgCZqdJq3YAhHnxAZseAkXKliEhizaIQALR4SYUAObwioAGYAnCAFsk6kDghJRNbdL1WA+lxpGegAjGEYABQEVYRAbLFMACxx5aztHRDIXN0QPSR12PzluSm4gA > \begin{tikzcd} > & A \arrow[ld, "f_A"'] \\ > Z & \\ > & B \arrow[lu, "f_B"] > \end{tikzcd} > \end{document} > ``` > ) there exists a unique commutative diagram [^1] of the form > > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEQBfU9TXfIRRkATFVqMWbAELdeIDNjwEiARlKrx9Zq0QcABAB1DdALaMA5vtk8+SwUREatk3SABa3cTCgX4RUAAzACcIUyQyEBwIJCcJHTZjfBw6AH1OWxAQsKR1KJjEOO0pPSSIFNTZagY6ACMYBgAFfmUhEGCsCwALHDkg0PDC6mikAGZqYrdjbAtzEGq6hub7FT0sMGxYPqyBiOGC8fiSrPT5kHqwKDHiTOzBvJHEQ8m2QMqzmvqmloc9Du7etQLldEABaUY3ChcIA > \begin{tikzcd} > A \arrow[rd, "\iota_A"] \arrow[rrd, "f_A", bend left] & & \\ > & A \amalg B \arrow[r, "\sigma" description] & Z \\ > B \arrow[ru, "\iota_B"'] \arrow[rru, "f_B"', bend right] & & > \end{tikzcd} > \end{document} > ``` > that is, there exists a unique morphism $\sigma:A \amalg B \to Z$ such that this diagram commutes. > > We say a [[category]] **has coproducts** if this [[universal property]] has a solution for all objects $A,B$ in it. $\iota_{A}$ and $\iota_{B}$ are called **coprojections**. ^definition > [!definition] Definition. (Coproduct of a family of objects) > More generally, the **coproduct of a family $\{ X_{j} \}_{j \in J}$** will be an object $\coprod_{j \in J}X_{j}$ of $\mathsf{C}$ which, when put with 'coprojections' $\{ X_{j} \xrightarrow{\iota_{j}} \coprod_{j \in J}X_{j} \}_{j \in J}$, forms an [[terminal object|initial object]] in the [[multi-object coslice category]] $\mathsf{C}^{\{ X_{j} \}_{j \in J}}$. > > Let us spell out this [[universal property]]. Fix an object $Z$ of $\mathsf{C}$. Then there exists a $\mathsf{C}$-morphism $\coprod_{j \in J}X_{j} \xrightarrow{\sigma} Z$ through which any family $\{ X_{j} \xrightarrow{f_{j}} Z \}$ simultaneously and uniquely factors through — that is, there is a unique $\mathsf{C}$-morphism $\sigma$ making the diagram > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQANAfQCsQBfUuky58hFOVLFqdJq3YAdBQGMIaAE7QuwHgAIlWMLoBS-Xdz6Dh2PASIAmKTIYs2iEAC0BMmFADm8ESgAGaaALZIZCA4EEiSsq6KCvg49LwCQiChEBGIUTFIjgny7sHpNIz0AEYwjAAKIrbiIOpYfgAWOCA0NWBQSAC0AMzEVlnhcTQFiEUuJSBKMAAeWHA4cLoAhPoK2H5h9N78QA > \begin{tikzcd} > & \coprod_{j \in J} X_j \arrow[r, "\exists ! \sigma"] & Z \\ > X_j \arrow[ru, "\iota_j"] \arrow[rru, "f_j"', bend right] & & > \end{tikzcd} > \end{document} > ``` > > commute for all $j \in J$. ^definition-2 > [!basicexample] > - [[coproducts are disjoint unions in set]] > - [ ] And $\mathsf{Top}$ ([[TODO]]) > - [[categorical product#^warning|Based on this example]] we have that the coproduct $k_{1},k_{2} \in \mathbb{Z}$ under the [[relation]] $\leq$ is $\max(k_{1},k_{2})$. > - [ ] [[TODO]] coproducts are 'join' in the category of posets ^basic-example ---- #### [^1]: I.e., a unique morphism in $\mathsf{C}^{A,B}$ ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```