---- > [!definition] Definition. ([[adjoint functor]]) > Let $\mathsf{C},\mathsf{D}$ be [[category|categories]], and let $\mathscr{F}: \mathsf{C} \to \mathsf{D}$ and $\mathscr{G}: \mathsf{D} \to \mathsf{C}$ be [[covariant functor|functors]]: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage[mathscr]{euscript} > \usepackage{amsfonts} > > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACzgAzYAGEAviDGl0mXPkIoAjOSq1GLNpx78hwACISxqmFADm8IqEEAnCFyRkQOCEmUgARjDBQkAZkf0zKyIHNy8AgDG1sAAYobSIDZ2rtTODtSe3n4B6sGh2pHRAOKGFGJAA > \begin{tikzcd} > \mathsf{C} \arrow[r, "\mathscr{F}", bend left] & \mathsf{D} \arrow[l, "\mathscr{G}", bend left] > \end{tikzcd} > \end{document} > ``` > > We say that $\mathscr{F}$ and $\mathscr{G}$ are **adjoint** (and we say $\mathscr{G}$ is **right-adjoint to $\mathscr{F}$** and **$\mathscr{F}$ is left-adjoint to $\mathscr{G}$**) if there are natural bijections $\text{Hom}_{\mathsf{C}}\big(X, \mathscr{G}(Y)\big) \xrightarrow{{\sim}} \text{Hom}_{\mathsf{D}}\big( \mathscr{F}(X), Y \big)$ for all objects $X$ of $\mathsf{C}$ and $Y$ of $\mathsf{D}$. > 'Naturality' here means that there is a [[natural transformation|natural isomorphism]] of [[bifunctor|bifunctors]] $\mathsf{C}^{\text{op} } \times \mathsf{D} \to \mathsf{Set}$: $\text{Hom}_{\mathsf{C}}\big(-, \mathscr{G}(-)\big)\xRightarrow{\sim} \text{Hom}_{\mathsf{D}}\big( \mathscr{F}(-), - \big),$ where $\text{Hom}$ is the [[hom functor]] (in two variables d). We write $\mathscr{F} \dashv \mathscr{G}$. > [!basicexample] Example. (The free functor is left-adjoint to the forgetful functor) > The construction of the [[free group]] on a given set is concocted so that giving a set-function from a set $A$ to a [[group]] $G$ 'is the same as' specifying a [[group homomorphism]] $F(A) \to G$: > > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRADEAKAQQEoQAvqXSZc+QigCM5KrUYs2AcUHCQGbHgJEyk2fWatEIboNkwoAc3hFQAMwBOEALZIATNRwQk0uQba2QagY6ACMYBgAFUU0JEHssCwALHBU7RxdEMhBPb2p9BSMAHUKYAA8sOBw4AAIAQmri+ns0RKxUkAdnNw8vTLz5QxAAK1MBIA > \begin{tikzcd} > F(A) \arrow[r, "\exists ! \varphi"] & G \\ > A \arrow[ru, "f"'] \arrow[u, "j"] & > \end{tikzcd} > \end{document} > > ``` > > [[free group#^3e62b9|This can be formalized as saying]] that the free group construction is a [[covariant functor|functor]] $\mathscr{F}:\mathsf{Set} \to \mathsf{Grp}$. Consider also the [[forgetful functor]] $\mathscr{G}:\mathsf{Grp} \to \mathsf{Set}$. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage[mathscr]{euscript} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACzgAzYAGUYOAL4gJpdJlz5CKAIzkqtRizace-IcADiAJzRSJ6mFADm8IqEFGIXJGRA4ISVSABGMMFCQAZld6ZlZEDm5eAQBjI2AAMTNZEAcnT2p3F2pffyCQzXDI3Vj4gzMKCSA > \begin{tikzcd} > \mathsf{Set} \arrow[r, "\mathscr{F}", bend left] & \mathsf{Grp} \arrow[l, "\mathscr{G}", bend left] > \end{tikzcd} > \end{document} > ``` > The universal property gives us a bijection $\{ \text{set-functions } A \to G \} \leftrightarrow \{ \text{group homomorphisms }F(A) \to G \}$that is, a bijection > $\text{Hom}_{\mathsf{Set}}\big( A, \mathscr{G}(G) \big) \xrightarrow{\sim} \text{Hom}_{\mathsf{Grp}}\big(\mathscr{F}(A), G\big).$ > for all sets $A$ and [[group|groups]] $G$. ---- #### - [ ] universal enveloping algebra gives an example ---- #### 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 > ```