---- > [!definition] Definition. ([[categorical cokernel]]) > Let $\mathsf{C}$ be a [[category]] with [[terminal object|zero object]]. Let $\varphi: X \to Y$ be a morphism in $\mathsf{C}$. Consider the [[subcategory]] of the [[coslice category]] $\mathsf{C}^{Y}$, obtained by keeping as objects only those morphisms $\alpha: Y \to L$ satisfying [^1] $\alpha \circ \varphi = \text{0}.$ > Should it exist, the [[terminal object|initial object]] in this new [[category]] — [[terminal objects are unique up to a unique isomorphism|defined up to unique isomorphism]] — is called the **cokernel** of $\varphi$ and denoted $\text{coker } \varphi$. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAA0QBfU9TXfIRQBGclVqMWbAJrdeIDNjwEiAJjHV6zVohAAZOXyWCio4eK1TdAHWs4YADxzAAxhADWMAE4ACLrfovNAALLG5xGCgAc3giUAAzLwgAWyQyEBwIJFEJbTYAuiDQwxBElOzqTKR1XKsQW0YQuhKy1MQcqsQAZh4EpLauyqzEGoYsMB0QKDo4YMiQTUlJ20csOBw4HwBCH1sIGm8xieAGhiauBZAGOgAjGAYABX5lIRAvLCjgnBb+tKHq6h3MBQJBddKWSbEcJcIA > \begin{tikzcd} > X \arrow[r, "\varphi"] \arrow[rr, "0", bend left] & Y \arrow[r, "\alpha"] \arrow[d] & L \\ > & \text{coker }\varphi \arrow[ru, "\exists ! \overline{\alpha}"', dashed] & > \end{tikzcd} > \end{document} > ``` > ^definition ---- #### [^1]: '$0 ---- #### 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 > ``` here denotes the [[terminal object|zero morphism]]. ---- #### References > [!backlink] > {CODE_BLOCK_PLACEHOLDER} > [!frontlink] > {CODE_BLOCK_PLACEHOLDER}