---- > [!definition] Definition. ([[equalizer]]) > > Let $\mathsf{I}$ be an 'indexing [[category]]' with two objects $\boldsymbol 1$ and $\boldsymbol 2$ and morphisms that look like this (the self-morphisms are identity): > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12AjCBqOAJ4BbHg2AAmAL4hJpdJlz5CKAIzkqtRizadR-YaIAEKmRphQA5vCKgAZgCcIQpGRA4ISNSC4wwUJADMrvTMrIgc7IxoABZ0MnIgDk4u1O6e1D5+SAC0QdQh2uG6MDhx1Ax0PgwACgp4BGz2WBbROPF2js6Irmnd5RAQaETiAOxktoxwMBoVVbXY9cogTS1tsh3JiF69XgwDQygAnGQ49iwzlTA1dUpsDDC2axSSQA > \begin{tikzcd} > \boldsymbol{2} \arrow[r, "\alpha", bend left] \arrow[r, "\beta"', bend right] \arrow[loop, distance=2em, in=215, out=145] & \boldsymbol 1 \arrow[loop, distance=2em, in=325, out=35] > \end{tikzcd} > \end{document} > ``` > > > Let $\mathsf{C}$ by any [[category]]. A [[covariant functor|functor]] ([[diagram]]) $\mathscr{K}:\mathsf{I} \to \mathsf{C}$ amounts to the choice of two objects $A_{1}=\mathscr{K}(\boldsymbol 1)$, $A_{2}=\mathscr{K}(\boldsymbol 2)$ in $\mathsf{C}$ and two 'parallel' morphisms from $A_{2}$ to $A_{1}$. [[categorical limit|Limits]] of such functors are called **equalizers**. > [!basicexample] > > Assume $\mathsf{C}=R\text{-}\mathsf{Mod}$ is the [[category]] of $R$-[[module|modules]] for some [[ring]] $R$; let $\varphi:A_{2} \to A_{1}$ be a [[linear map|homomorphism]], and choose $\mathscr{K}$ so $\mathscr{K}(\alpha)=\varphi$ and $\mathscr{K}(\beta)$ is the zero morphism. We claim the equalizer $\lim\limits_{{\longleftarrow}} \mathscr{K}$ is exactly the [[categorical kernel|kernel]] of $\varphi$. Indeed, if we take $\lambda_{\boldsymbol 2}$ to be the [[inclusion map]] $\iota: \ker \varphi \to A_{2}$ and $\lambda_{\boldsymbol 1}$ to be the zero map then the first defining bullet in [[categorical limit]] is, by construction and the definition of kernel of a module homomorphism, satisfied: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEB9AJhAF9S6TLnyEUARnJVajFmy7j+gkBmx4CRMuOn1mrRCAA6hnDAAeOYAGsYAJwAEfY-VtoAFln7SYUAObwiUAAzWwgAWyRuahwIJDIZPTZjfBw6EGoGOgAjGAYABWF1MRBbLF83HCVg0IjEeJikSQS5A2c6Vw8qkBDwyOjYxCbdFqNDF3cse2MAYyxbaanDFLSM7NyCtVE2UvLKvgo+IA > \begin{tikzcd} > A_2 \arrow[r, "\varphi"] & A_1 \\ > \text{ker }\varphi \arrow[u, "\iota"'] \arrow[ru, "\varphi \circ \iota = 0"'] & > \end{tikzcd} > \end{document} > ``` > Now assume $K$ is another object, endowed with morphisms $\mu_{\boldsymbol 1}$ and $\mu_{\boldsymbol 2}$ also satisfying that bullet's requirements. Commutativity enforces that $\mu_{\boldsymbol 2}=0$: > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage[mathscr]{euscript} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQBpEAX1PU1z5CKcqWLU6TVuwCCAfQBMPPiAzY8BIgrESGLNohDziPCTCgBzeEVAAzAE4QAtklEgcEJGUn72AHT8nZjkAgCMIRig4AE8ncMZgYm4QGkZ6UJhGAAUBDWEQeywLAAscZTtHF0Q3DyRtH2lDAKC5YDCIqNj44AVuZN4K5y8aWsR6vUaQZvocYrgAY3tgDm4ACjCYHHoASgBechSQNIzs3KF2QpKy7kpuIA > \begin{tikzcd} > & K \arrow[ld, "\mu_\boldsymbol{1}"'] \arrow[rd, "\mu_{\boldsymbol{2}}"] & \\ > A_2 \arrow[rr, "\mathscr{K}(\beta)=0"'] & & A_1 > \end{tikzcd} > \end{document} > ``` > > whence $K$ fits perfectly into the diagram for the universal property of kernels: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGkQBfU9TXfIRQBGclVqMWbAIIB9AEzdeIDNjwEi8sdXrNWiEHOFK+awUVHDxuqQYA6dnDAAeOYAGsYAJwAEXB-ReaAAWWNziMFAA5vBEoABmXhAAtkhkIDgQSADMOpL6IA4uWHA4cD4AhD4OEDTeDFhgMMAOyUyyDgBGEAxQcACeyd0MwPJcXCDUDHSdMAwACvzqQiBeWFHBOJMgDU1sUHRwwZEmIIkpadSZSKISemyt7V09fYPDoxM8CUmpiLfXiC0d1shTsgRCYS+Zx+lwyWUB1FmYCgOXSNgKxFO51+uThNzy93sdnwODo22mswWS3MBjWGy2XAoXCAA > \begin{tikzcd} > K \arrow[rd, "\exists ! \overline{\mu_\boldsymbol{2}}"', dashed] \arrow[r, "\mu_\boldsymbol{2}"] \arrow[rr, "0", bend left] & A_2 \arrow[r, "\varphi"] & A_1 \\ > & \text{ker }\varphi \arrow[u, "\iota"'] & > \end{tikzcd} > \end{document} > ``` > > ---- #### ---- #### 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 > ```