If F is left adjoint to G, when does FG preserve limits? When do counits interchange with limits?

2010-03-22T10:45:42Z

Motivation

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:

$$ \begin{matrix} \operatorname{Lim}(T_1)& \stackrel{\text{limiting cone}}{\longrightarrow} & T_1\\ | & & |\\ \operatorname{Lim}(\alpha) & & \alpha\\ \downarrow & & \downarrow \\ \operatorname{Lim}(T_2)& \stackrel{\text{limiting cone}}{\longrightarrow} & T_2 \end{matrix} $$

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:

$$ \begin{matrix} \operatorname{Lim}(FGT)& \stackrel{\text{limiting cone}}{\longrightarrow} & FGT\\ | & & |\\ \operatorname{Lim}(\varepsilon T) & & \varepsilon T\\ \downarrow & & \downarrow \\ \operatorname{Lim}(T)& \stackrel{\text{limiting cone}}{\longrightarrow} & T \end{matrix} $$

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

$$ \begin{matrix} FG\operatorname{Lim}(T)& \stackrel{FG\tau}{\longrightarrow} & FGT\\ | & & |\\ \operatorname{Lim}(\varepsilon T) & & \varepsilon T\\ \downarrow & & \downarrow \\ \operatorname{Lim}(T)& \stackrel{\tau}{\longrightarrow} & T \end{matrix} $$

Since the naturality of $\varepsilon$ implies that for all $j\in \operatorname{obj}(J)$ the diagram $$ \begin{matrix} FG\operatorname{Lim}(T)& \stackrel{FG\tau_j}{\longrightarrow} & FGT(j)\\ | & & |\\ \varepsilon_{\mathrm{Lim}T}& & \varepsilon_{T(j)}\\ \downarrow & & \downarrow \\ \operatorname{Lim}(T)& \stackrel{\tau_j}{\longrightarrow} & T(j) \end{matrix} $$

is commutative, it follows that $\varepsilon_{\mathrm{Lim}T}$ can replace $\operatorname{Lim}(\varepsilon T)$ in the last but one diagram while keeping it commutative. By uniqueness, we get the nice equation
$$ \varepsilon_{\mathrm{Lim}T} = \operatorname{Lim}(\varepsilon T). $$ Note that it seems that all depends on $FG$ preserving $J$ limits.

Question

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?

Background

For solving an exercise from Mac Lane, I used some results from A. Gleason, ''Universally locally connected refinements,'' Illinois J. Math, vol. 7 (1963), pp. 521--531. 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
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.

Answer by Glen M Wilson for If F is left adjoint to G, when does FG preserve limits? When do counits interchange with limits?

2010-03-23T16:27:08Z

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)?

If F is left adjoint to G, when does FG preserve limits? When do counits interchange with limits? - MathOverflow

If F is left adjoint to G, when does FG preserve limits? When do counits interchange with limits?

Motivation

Question

Background

Answer by Glen M Wilson for If F is left adjoint to G, when does FG preserve limits? When do counits interchange with limits?