How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T18:30:33Z http://mathoverflow.net/feeds/question/32791 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/32791/how-is-the-right-adjoint-f-to-the-inverse-image-functor-f-described-for-f How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$ vincenzoml 2010-07-21T11:40:57Z 2010-07-22T15:02:59Z <p>For $C,D$ small categories, and $f : C \to D$ a functor between them, there is a precomposition, or "inverse image", functor $f^* = (-) \circ f : Set^D \to Set^C$. It has a left and a right adjoint. What are their definitions, and in particular what is the right adjoint $f_*$? I couldn't find a definition in terms of functor categories, just "topological" ones.</p> http://mathoverflow.net/questions/32791/how-is-the-right-adjoint-f-to-the-inverse-image-functor-f-described-for-f/32808#32808 Answer by Michael A Warren for How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$ Michael A Warren 2010-07-21T15:28:21Z 2010-07-22T15:02:59Z <p>Given a functor $f:\mathcal{C}\to\mathcal{D}$ and any complete category $\mathcal{A}$ (e.g., take $\mathcal{A}=\text{Sets}$ to get the case you are asking about), there exists a right-adjoint $f_{*}:[\mathcal{C},\mathcal{A}]\to[\mathcal{D},\mathcal{A}]$ to the "inverse image functor" $f^{*}$ and this is given by taking right Kan extension. </p> <p>Explicitly, given a functor $X:\mathcal{C}\to\mathcal{A}$, the functor $f_{*}(X):\mathcal{D}\to\mathcal{A}$ is the right Kan extension of $X$ along $f$. This can be described explicitly using the limit formula $$f_{*}(X)(d)=\text{lim}_{d\to f(c)}X(c)$$ for $d$ an object of $\mathcal{D}$ (the action on arrows of $\mathcal{D}$ is then induced by the universal property of limits). The indexing category of the limit here is of course the comma category $(d\downarrow f)$.</p> <p>When $\mathcal{A}$ is cocomplete there is a corresponding left-adjoint $f_{!}\dashv f^{*}$ which is given by taking left Kan extension along $f$. This can be explicitly described by the colimit formula dual to the limit formula given above.</p> <p>(I should say that all of this is described very nicely in Mac Lane's book <em>Categories for the Working Mathematician</em>.)</p>