----- > [!proposition] Proposition. ([[right-adjoint functors commute with limits]]) > If $\mathscr{G}:\mathsf{D} \to \mathsf{C}$ has a [[adjoint functor|left-adjoint]] $\mathscr{F}:\mathsf{C} \to \mathsf{D}$ and $\mathscr{A}:\mathsf{I} \to \mathsf{D}$ is another [[covariant functor|functor]] (an '$\mathsf{I}$-shaped [[diagram]]'), then there is a canonical [[isomorphism]] $\mathscr{G}(\lim\limits_{{\longleftarrow}} \mathscr{A}) \xrightarrow{\sim} \lim\limits_{{\longleftarrow}} (\mathscr{G} \circ \mathscr{A})$ (*if* the [[categorical limit|limits]] exist, of course). ^proposition > [!note] Remark. > Note that it does not matter *what* the left-adjoint is, just that $\mathscr{G}$ has one. Mnemonic: RAPL (right-adjoints preserve limits). ^note > [!proof]- Proof Sketch. ([[right-adjoint functors commute with limits]]) > By definition, the [[categorical limit|limit]] of $\mathscr{A}$ is the [[terminal object|final]] subject to fitting in commutative diagrams > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage[mathscr]{euscript} > \usepackage{amsfonts} > \begin{document} > > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQAdDxrAWy569ccAPrBgAggHNGMAGY56AJyUQA7gF8NAAi696OABZwAxkuABBDSA2l0mXPkIpypYtTpNW7PQeNnLDQAKAEkAShs7EAxsPAIiACY3DwYWNkRODn0jU3MrIIApCI0PGCgpeCJQOVVeJFcQHAgkMk80n256XgAjKHoREJAaRnpumEYABQc45xAlLClDHEjq2vqaJqQktu8MgS7e-oKVkBqIOsRWzcRt1N3M7P884K4mNEN6COHR8anYp3Y80WyxKGiAA > \begin{tikzcd} > & \lim\limits_{{\longleftarrow}} \mathscr{A} \arrow[ld, "\lambda_I"'] \arrow[rd, "\lambda_J"] & \\ > \mathscr{A}(I) \arrow[rr, "\mathscr{A}(\alpha)"'] & & \mathscr{A}(J) > \end{tikzcd} > \end{document} > ``` > (with hopefully evident notation). Applying $\mathscr{G}$, we get commutative diagrams (in $\mathsf{C}$) > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage[mathscr]{euscript} > \usepackage{amsfonts} > > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQAdDgW3pwAs4AYwBOwAOIBfABRdGWbnIW44AfWDA5BAOaMYAMxz0RIiAHdJkgARdeA4WICCkgJQhJpdJlz5CKcqTE1HRMrOy2fIKiEtZcQlgiQjY8kQ7AztIAkm4eXth4BEQATIHBDCxsiJwp9tFScQlJEbVOMgBSOcEwUNrwRKD6ptxIASA4EEhkIRXhNVFiUrIcjPTcAEZQ9KrZIDQrazCMAAreBX4gIlja-DjuniCDEMOIo+NIJdNhVc3zMUsr602qg6dwGQ0mNDeiA+5S+1TsvykyXiiWSCLSGS4TDQ-Hobj29AOx1OvnYl2ut0klEkQA > \begin{tikzcd} > & \mathscr{G}(\lim\limits_{{\longleftarrow}} \mathscr{A}) \arrow[ld, "\mathscr{G}(\lambda_I)"'] \arrow[rd, "\mathscr{G}(\lambda_J)"] & \\ > \mathscr{G} \circ \mathscr{A}(I) \arrow[rr, "\mathscr{G} \circ \mathscr{A}(\alpha)"'] & & \mathscr{G}\circ \mathscr{A}(J) > \end{tikzcd} > \end{document} > ``` > > and hence, by the [[universal property]] defining the limit of $\mathscr{G} \circ \mathscr{A}$: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage[mathscr]{euscript} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAB12GsBbT7nrjgB9YMH4EA5gxgAzHHQBOiiAHcAvuoAEACk486OABZwAxouABxbZ1NZFprfsMnzwAILqAlCHWl0mLj4hCgADKQATFS0jCxszsZmFtZO7HYOqQaJbp46AJI+fgHYeAREEZHR9MysiBzsWa7J6rb2jglNHuo6AFKF-iAYJcFEZKFVsbX1jUlW3fy8C4I4ImISYNJyCspqmpkus56F0TBQkvBEoLIqPEjhIDgQSGQxNfENB27Welx0PABGUDowgKIGoDDo-xgDAACoFSiEQIosJIjDhfANrhBboh7o8kBVXnE6h1Zt9+H9AcC+hirjdntR8YhCdVidNPs1Uul2h9shZcpxGGgjHQfODIdC4cMynVkaj0UUQFicQBmRlPXHUVlTTgwAAeWDgKy0AEIwSBuGApkC4EZTrSlfTEGqHhqXlCwFAkABaFWhRXKpAupmEj1e53+ijqIA > \begin{tikzcd} > & \mathscr{G}(\lim\limits_{{\longleftarrow}} \mathscr{A}) \arrow[d, "\exists !", dashed] \arrow[ldd, bend right] \arrow[rdd, bend left] & \\ > & \lim\limits_{{\longleftarrow}} (\mathscr{G} \circ \mathscr{A}) \arrow[ld, "\mathscr{G}(\lambda_I)"'] \arrow[rd, "\mathscr{G}(\lambda_J)"] & \\ > \mathscr{G} \circ \mathscr{A}(I) \arrow[rr, "\mathscr{G} \circ \mathscr{A}(\alpha)"'] & & \mathscr{G}\circ \mathscr{A}(J) > \end{tikzcd} > \end{document} > ``` > > Now, the morphisms (in $\mathsf{C}$) $\lim\limits_{{\longleftarrow}} (\mathscr{G} \circ \mathscr{A}) \to \mathscr{G} \circ \mathscr{A}(I)$ > determine, via the [[adjoint functor|adjunction identification]] $\text{Hom}_{\mathsf{C}}\big( X, \mathscr{G}(Y) \big) \xrightarrow{\sim} \text{Hom}_{\mathsf{D}}\big( \mathscr{F}(X), Y \big),$ > morphisms (in $\mathsf{D}$) $\mathscr{F}\big(\lim\limits_{{\longleftarrow}} (\mathscr{G} \circ \mathscr{A})\big) \to \mathscr{A}(I).$ > By the [[universal property]] defining $\lim\limits_{{\longleftarrow}} \mathscr{A}$ we get > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage[mathscr]{euscript} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAB12GsBbT7nrjgB9YMH4EA5gxgAzHHQBOiiAHcAvuoAEnHnRwALOAGNFwAILqQ60uky58hFAAZSAJiq1GLNjvZ7DEzNLAAoASQBKa1sQDGw8AiI3d096ZlZEED8Ao1MLdRCAKSibO3jHIjJnVO8Mjn99XLMAMXVOACMsSRD+Xl7BHBExCTBpOQVlNU0tHobAvIBxbU5jLEVjbMag-IiOrpLPGChJeCJQWRUeJFcQHAgkMhAGOnaYBgAFewSnEEUugxw0XOl2u1DuSDcpRAFwgV0Qj3BiGSTxeb0+5USmT+kgBQOhIMQAGYwfdEDc0j5MpwYAAPLBwQZaACEIGo3DAdSgdDgBiOeJhcOJt1Jj1eYCgSAAtITnFCBUghYjkWKJUTZRR1EA > \begin{tikzcd} > & \mathscr{F}\big(\lim\limits_{{\longleftarrow}} (\mathscr{G} \circ \mathscr{A})\big) \arrow[d, "\exists !", dashed] \arrow[ldd, bend right] \arrow[rdd, bend left] & \\ > & \lim\limits_{{\longleftarrow}} \mathscr{A} \arrow[ld] \arrow[rd] & \\ > \mathscr{A}(I) \arrow[rr] & & \mathscr{A}(J) > \end{tikzcd} > \end{document} > ``` > Now, applying adjunction again the new morphism $\mathscr{F}\big( \lim\limits_{{\longleftarrow}} (\mathscr{G} \circ \mathscr{A}) \big) \to \lim\limits_{{\longleftarrow}} \mathscr{A}$ > determines a morphism d $\lim\limits_{{\longleftarrow}} (\mathscr{G} \circ \mathscr{A}) \to \mathscr{G}(\lim\limits_{{\longleftarrow}} \mathscr{A}).$ > Summarizing, we have obtained natural morphisms $\mathscr{G}(\lim\limits_{{\longleftarrow}} \mathscr{A}) \to \lim\limits_{{\longleftarrow}} (\mathscr{G} \circ \mathscr{A}) \text{ and } \lim\limits_{{\longleftarrow}} (\mathscr{G} \circ \mathscr{A}) \to \mathscr{G}(\lim\limits_{{\longleftarrow}} \mathscr{A}).$ > The claim is that they compose to the identity. ----- #### ---- #### 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 > ```