If F is left adjoint to G, when does FG preserve limits? When do counits interchange with limits? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T13:22:58Z http://mathoverflow.net/feeds/question/18988 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18988/if-f-is-left-adjoint-to-g-when-does-fg-preserve-limits-when-do-counits-intercha If F is left adjoint to G, when does FG preserve limits? When do counits interchange with limits? unknown (google) 2010-03-22T10:45:42Z 2010-07-13T20:22:20Z <h2>Motivation</h2> <p>Suppose that $F\colon X\to A$ is left adjoint to $G\colon A\to X$, and let $\varepsilon\colon FG\stackrel{.}{\to}I_A$ be the counit of the adjunction. Suppose also that $A$ is $J$-complete (for some category $J$), so that $\operatorname{Lim}$ is a functor $C^J\to C$, where for an arrow $\alpha\colon T_1\stackrel{.}{\to} T_2$ of $C^J$, $\operatorname{Lim}(\alpha)$ is the unique arrow of $A$ for which the following diagram is commutative:</p> <p><code>$$\begin{matrix} \operatorname{Lim}(T_1)&amp; \stackrel{\text{limiting cone}}{\longrightarrow} &amp; T_1\\ | &amp; &amp; |\\ \operatorname{Lim}(\alpha) &amp; &amp; \alpha\\ \downarrow &amp; &amp; \downarrow \\ \operatorname{Lim}(T_2)&amp; \stackrel{\text{limiting cone}}{\longrightarrow} &amp; T_2 \end{matrix}$$</code></p> <p>Let $T\colon J\to A$ be a functor. We have the natural transformation $\varepsilon T\colon FGT\stackrel{.}{\to} T$, and $\operatorname{Lim}(\varepsilon T)$ is the dotted line making the following diagram commutative:</p> <p><code>$$\begin{matrix} \operatorname{Lim}(FGT)&amp; \stackrel{\text{limiting cone}}{\longrightarrow} &amp; FGT\\ | &amp; &amp; |\\ \operatorname{Lim}(\varepsilon T) &amp; &amp; \varepsilon T\\ \downarrow &amp; &amp; \downarrow \\ \operatorname{Lim}(T)&amp; \stackrel{\text{limiting cone}}{\longrightarrow} &amp; T \end{matrix}$$</code></p> <p>If $FG$ preserves $J$-limits, and $\tau\colon \operatorname{Lim}(T)\stackrel{.}{\to}T$ is the lower limiting cone, then $FG\tau\colon FG\operatorname{Lim}(T)\stackrel{.}{\to}FGT$ is the upper limiting cone, and the above diagram becomes</p> <p><code>$$\begin{matrix} FG\operatorname{Lim}(T)&amp; \stackrel{FG\tau}{\longrightarrow} &amp; FGT\\ | &amp; &amp; |\\ \operatorname{Lim}(\varepsilon T) &amp; &amp; \varepsilon T\\ \downarrow &amp; &amp; \downarrow \\ \operatorname{Lim}(T)&amp; \stackrel{\tau}{\longrightarrow} &amp; T \end{matrix}$$</code></p> <p>Since the naturality of $\varepsilon$ implies that for all $j\in \operatorname{obj}(J)$ the diagram <code>$$\begin{matrix} FG\operatorname{Lim}(T)&amp; \stackrel{FG\tau_j}{\longrightarrow} &amp; FGT(j)\\ | &amp; &amp; |\\ \varepsilon_{\mathrm{Lim}T}&amp; &amp; \varepsilon_{T(j)}\\ \downarrow &amp; &amp; \downarrow \\ \operatorname{Lim}(T)&amp; \stackrel{\tau_j}{\longrightarrow} &amp; T(j) \end{matrix}$$</code></p> <p>is commutative, it follows that <code>$\varepsilon_{\mathrm{Lim}T}$</code> can replace $\operatorname{Lim}(\varepsilon T)$ in the last but one diagram while keeping it commutative. By uniqueness, we get the nice equation<br> $$\varepsilon_{\mathrm{Lim}T} = \operatorname{Lim}(\varepsilon T).$$ Note that it seems that all depends on $FG$ preserving $J$ limits.</p> <h2>Question</h2> <p>If $F\colon X\to A$ is left adjoint to $G\colon A\to X$ and $A$ has $J$-limits, when does $FG$ preserve $J$-limits? This is obviously true when $F$ preserves limits (for example, when there is also a left adjoint to $F$), but are there other interesting situations? </p> <h2>Background</h2> <p>For solving an exercise from Mac Lane, I used some results from A. Gleason, ''Universally locally connected refinements,'' Illinois J. Math, vol. 7 (1963), <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.ijm/1255644959" rel="nofollow">pp. 521--531</a>. In that paper, Gleason constructs a right adjoint to the inclusion functor $\mathbf{L\ conn}\subset \mathbf{Top}$ ($\mathbf{L\ conn}=$ locally connected spaces with continuous maps), and proves that the counit<br> of the product of two topological spaces is the product of the counits (Theorem C). This made me curious when do counits and limits interchange. </p> http://mathoverflow.net/questions/18988/if-f-is-left-adjoint-to-g-when-does-fg-preserve-limits-when-do-counits-intercha/19120#19120 Answer by Glen M Wilson for If F is left adjoint to G, when does FG preserve limits? When do counits interchange with limits? Glen M Wilson 2010-03-23T16:27:08Z 2010-03-23T16:27:08Z <p>Have you looked at the paper by B. Eckmann and P. J. Hilton entitled "Commuting Limits with Colimits" in the "Journal of Algebra", 11, 116-144 (1969)? </p>