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

## 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.

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+1 for diagrams! –  Theo Johnson-Freyd Mar 22 '10 at 16:40
I'm not a categorist, but that Gleason paper you link to surprises me a little, because I would have expected that if an inclusion or forgteful-like functor were to have an adjoint, it would have a left adjoint. (Think of CpctHff into RegularTop, or AbGp into Gp ...) So something atypical seems to be going on, unless I've misunderstood (which might admittedly very well be the case) –  Yemon Choi Mar 22 '10 at 19:51
@Yemon Choi: Yes, I'm also more used to reflective subcategories than to coreflective ones. But Gleason does prove that Lconn is a coreflective subcategory (the inclusion functor has a right adjoint). Searching the web, I found some additional results on coreflective subcategories in general topology in H. Herrlich and G.E. Strecker, "Coreflective subcategories in general topology," Fund. Math. 73 (1972), pp. 199--218 (matwbn.icm.edu.pl/ksiazki/fm/fm73/fm73124.pdf). Mac Lane also gives an example from algebra: torsion abelian groups in abelian groups. –  user2734 Mar 22 '10 at 20:19
@Yemon Choi: There are plenty of right adjoints to forgetful functors. A nice heuristic is that often a left adjoint freely adds the forgotten structure (destroying any that was already there) and a right adjoint kills off everything that doesn't have that structure. An example is the functor that sends cartesian categories to symmetric monoidal ones: the left adjoint freely makes every object a comonoid, while the right adjoint sends a category C to the category of comonoids in C. I think it was Lawvere who called these 'fascist' functors, because they're 'far right' adjoints. –  Finn Lawler Jun 1 '10 at 19:56
Off topic, but we really need to have some way to use xypic for diagrams on this site. Though these are pretty impressive. –  Daniel Litt Jun 29 '10 at 19:27