Unexpected interaction between limits and colimits

Question

Let $D$ be a limit-complete category. My vague question is: given two diagrams in $D$, what comparisons between them induce a morphism of their limits? I'm especially interested in the case that the comparison has something to do with colimits.

Here is a well-known case in which the comparison has nothing to do with colimits. Let $I,J$ be small categories, and let $X:I\to D$ and $Y:J\to D$ be functors. Given a functor $f:J\to I$ and a natural transformation $\alpha:X\circ f\to Y$, there is an induced morphism between their limits $\lim(f,\alpha):\lim(X)\to \lim(Y)$

But today I came across another case in which one gets such a map between limits. Let $D={\bf Set}$ be the category of sets. Suppose $I$ is the W-shaped category $M\to N\leftarrow P\to Q\leftarrow R$ in $D$. Suppose $J$ is the cospan $M\to S\leftarrow R$, and let $g:I\to J$ be the functor sending $M\mapsto M, R\mapsto R$ and $N,P,Q\mapsto S$. \begin{align} I:\hspace{.5in}M\to \fbox{$N\leftarrow P\to Q$}\leftarrow R\\\\ g\downarrow\hspace{.8in}\\\\ J:\hspace{.5in}M\xrightarrow{\hspace{.5in}} S\xleftarrow{\hspace{.5in}} R \end{align} Given a functor $X:I\to D$, let $Y=Lan_gX:J\to D$ be the left Kan extension of $X$ along $g$. To my (perhaps very naive) surprise, I get a morphism of limits in this case too, $\lim(X)\to \lim(Y)$. To be explicit, this map sends a tuple $(m,p,r)\in \lim(X)$ to the pair $(m,r)\in \lim(Y)$.

Question 1 (Sets): Is it true for any small categories $I, J$, and $X:I\to{\bf Set}$ that a functor $g:I\to J$ induces a function $\lim(X)\to \lim(Lan_gX)$?

Question 2 (General): Is it true for general $D$ that for any categories $I, J$, and $X:I\to D$ that a functor $g:I\to J$ induces a morphism $\lim(X)\to \lim(Lan_gX)$?

Question 3 (Vague): Is this part of any more general result about the interaction between limits and colimits?

Any references would be appreciated.

Answer 1

The existence of that sort of morphism between limits relates to some properties of the functor $g : I \rightarrow J$.

Given functors $h: I \rightarrow D$, $l: J \rightarrow D$ and a natural transformation $t : h \rightarrow lg$ we want to construct a morphism $Lim(h) \rightarrow Lim(l)$. Let $z = Lim(h)$, and let $s : z \rightarrow h$ be the universal cone. Suppose that for each object $x$ of $J$ a choice of an object $g^{-1}(x)$ in its fiber over $g$ is made. Then, for any $x$ we can define a morphism $z \rightarrow l(x)$ by $$z \xrightarrow{s_{g^{-1}(x)}} h(g^{-1}(x)) \xrightarrow{t_{g^{-1}(x)}} lg(g^{-1}(x)) = l(x)$$ If this family is a cone $z \rightarrow l$, then we get a morphism $z \rightarrow Lim(l)$ that we want. This family is a cone for example when for each arrow $f :x \rightarrow x'$ in $J$ there is a chain of arrows in $I$ $$g^{-1}(x) \xrightarrow{k_1} \xleftarrow{k'_1}\xrightarrow{k_2}...\xrightarrow{k_{n}}\xleftarrow{k'_n} g^{-1}(x')$$

such that $g(k'_i) = 1$ and $g(k_n)...g(k_2)g(k_1) = f$. For then we can establish $l(f)s_{g^{-1}(x)}t_{g^{-1}(x)} = s_{g^{-1}(x)}t_{g^{-1}(x)}$. For example for $n = 1$, omitting indexes for $s$ and $t$, this is shown by $$l(f)ts = lg(k_1)ts = th(k_1)s = ts = th(k'_1)s = lg(k'_1)ts = ts.$$

In your example you take $g^{-1}(M) = M$, $g^{-1}(R) = R$, and for $g^{-1}(S)$ you can take either $N$, $P$ or $Q$. Pretty much this is why the morphism between the limits exists.

Answer 2

I'd say the answer to Q1 is no in general. Take $I=\{1\}$ to be a discrete category on one object, $J=\{0\rightarrow 1\}$, and $g\colon I\rightarrow J$ the inclusion. Let $X\colon I\rightarrow \mathbf{Set}$ be a non-empty set. I think that $\operatorname{Lan}_gX$ is the inclusion of the empty subset $\varnothing\rightarrow X$ and there is no map $X=\lim X\rightarrow \varnothing=\lim(\varnothing\rightarrow X)$.

Answer 3

My vague question is: given two diagrams in $D$, what comparisons between them induce a morphism of their limits?

This is exactly the topic of Paré's paper Morphisms of Colimits: from Paths to Profunctors, who introduces the notion of total profunctor to express the data between two diagrams that induces a morphism between their colimits.