---- > [!definition] Definition. ([[categorical kernel]]) > 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 [[slice category]] $\mathsf{C}_{X}$, obtained by keeping as objects only those morphisms $\alpha: K \to X$ satisfying [^1] $\varphi \circ \alpha = \text{0}.$ > > Should it exist, the [[terminal object|final object]] in this new [[category]] — [[terminal objects are unique up to a unique isomorphism|defined up to unique isomorphism]] — is called the **kernel** of $\varphi$ and denoted $\ker \varphi$. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGkQBfU9TXfIRQBGclVqMWbABrdeIDNjwEiAJjHV6zVohABNOXyWCio4eK1TdwLsAA6dnDAAeOYAGsYAJwAEXB-ReaAAWWFzc4jBQAObwRKAAZl4QALZIZCA4EEiiEtpsDowhdIYgSak51FlI6nlWIAF0QaGl5WmIGdWItQBGMGBQSADMGZY6IMStye2d2YhDmpLjDi5YcDhwPgCEPg4QNN4MWGAw9nZFwXTh1Ax0fQwACvzKQiBeWNHBOCA3x+NQdDgwSiUwq8yqc2EXAoXCAA > \begin{tikzcd} > K \arrow[r, "\alpha"] \arrow[rr, "0", bend left] \arrow[rd, "\exists ! \overline{\alpha}"', dashed] & X \arrow[r, "\varphi"] & Y \\ > & {}{\text{ker }\varphi} \arrow[u] & > \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}