# Left Kan extension along Yoneda of pullback-preserving functor preserving pullbacks

Let $F\colon C \to D$ be a functor. The Kan Extension of $y_D \circ F$ along $y_C$ yields a functor $F_!: Fun(C^{op},Set) \to Fun(D^{op},Set)$. Here, $y_C$ and $y_D$ denotes the respective Yoneda embeddings. It is well-known that if $C$ and $D$ have finite limits and $F$ preserves them, then so does $F_!$.

Question: Suppose that we only assume that $C$ has pullbacks and that $F$ preserves them (so some pullbacks exist in $D$, too). Is it true that $F_!$ preserves pullbacks as well?

(I'm willing to assume that both $C$ and $D$ have finite products, but in my case, the functor $F$ does certainly not preserve these.)

I went through Lurie's ∞-categorical version of the "well-known" part (6.1.5.2 in HTT), but I was not able to verify that the last step works without assuming that $F$ preserves products because this step involves some heavy machinery. I could give a different proof for this last step if $F_!$ preserved monomorphisms, but I believe that this is not the case in general.

## 2 Answers

If you are wiling to assume that $C$ has a terminal object $1 \in C$, which I assume is the case as you said all finite products, you can do the following:

(As it is not clear if you are interested in a $1$-categorical statement or an $\infty$-categorical statement I'll write the proof in a very formal style which should accomodate $\infty$-categories as well with a little bit of work)

Saying that $F:C \rightarrow D$ preserve fiber products means that $F_{/1}: C \rightarrow D_{/F(1)}$ preserves all finite limits (as it preserve pullback and the initial objects).

In particular the induced functor $Prsh(C) \rightarrow Prsh(D_{/F(1)})$ preserves all finite limits by your "well known fact".

But $Prsh(D_{/F(1)})$ identifies canonically with $Prsh(D)_{/y_D(F(1))}$ and $y_D(F(1))$ is $F_!(y_C(1))$ with $y_C$ being the terminal object of $Prsh(C)$.

In the end the functor: $F_!$ preserve all finite limits when seen as a functor from $Prsh(C)$ to $Prsh(D)_{/F_!(1)}$ but this means that $F_!$ preserve fiber product as asked.

I have ignored the fact that $D$ does not have all finite limits as, as far as I know, this has never been part of the assumption of your "well known fact".

Indeed, if $C$ has all finite limits and $F:C \rightarrow D$ preserve them, then in order to check that $F_!$ preserve finite limits one justs need to check that for all $d \in D$ the functor $X \mapsto Hom(d,F_! ( X))$ from $Prsh(C)$ to $Set$ preserves all finite limits.

This functor is the left Kan extention along the Yoneda embedings of $C$ of the functor $C \rightarrow Set$ which send $c$ to $Hom(d,F(c))$, which commutes to finite limits by definition.

So in the end the relevant fact is that if $C$ has finite limits $F:C \rightarrow Set$ preserves them then its left Kan extention $Prsh(C) \rightarrow Set$ also preserve finite limits. But existence of limits in $D$ never really played a role.

I'm adding a counter example to show that the assumption that $C$ has a terminal object cannot be completely relaxed.

Let $C$ be a group $G$, seen as a one object category. And let $D$ be the terminal category. With $F:C \rightarrow D$ the unique functor.

The category $C$ have no terminal object but it has all fibered product (because all map are isomorphisms), and the functor $F$ preserve all fibered product limits (in fact we have both that any functor out of $C$ and that any functor to $D$ preserves all fibered product)

$F_!$ is the functor from $G$-set to set which send any $G$-set to its quotient $X/G$.

$F_!$ send the terminal object to the terminal object but does not preserve product: if $G$ denotes $G$ with its multiplication action on itself then $F_! (G \times G) = G$ but $F_!(G)=\{*\}$.

So $F_!$ does not preserve fibered product.

Two remarks:

• I do not know if there are other condition than having a terminal object that allow this sort of result to work.

• This examples suggest that things might work better if one work with $\infty$-categories and $Prsh$ means space valued presheaves. I do not know the status of this.

• Nice! One quibble: it is not the case that if $F: C \to Set$ preserves finite limits then $F_!: Psh(C) \to Set$ preserves finite limits -- the hypothesis that $C$ has finite limits is really necessary. For instance let $C$ be the discrete category with two objects, and let $F$ send both to a one-element set. Then $Psh(C) = Set^2$, and $F_! (A,B) = A \amalg B$. The functor $F$ preserves finite limits, but $F_!$ fails to preserve the terminal object for instance. – Tim Campion Feb 24 at 17:53
• @TimCampion Of course, but I don't think I said that. $C$ was assumed to have pullback in the question, and I added the assumption that it has a terminal object, so all finite limits. – Simon Henry Feb 24 at 18:40
• Oh I see. I misunderstood your point about $D$ not needing finite limits. Sorry! – Tim Campion Feb 24 at 18:41

My previous answer left open the following:

Proposition: Let $$C$$ be a small $$\infty$$-category with all fiber products, let $$\mathcal{T}$$ be an $$\infty$$-topos and let $$F : C \rightarrow \mathcal{T}$$ be a functor preserving fiber products. Then the left Kan extention: $$\widehat{F} : \text{Prsh}(C) \rightarrow \mathcal{T}$$ also preserves fiber product.

Indeed in the previous answer I showed that the statement was true as soon as $$C$$ has a terminal object and I have given an example showing that the $$1$$-categorical version of the proposition is false. But the example I have given somehow suggested that the proposition might be true in the $$\infty$$-categorical settings. I recently needed an answer to this question so I thought about it, and I think I have a proof of the proposition above.

(To be honest, At this point I still have some doubt about the argument below, so I decided to spell it out very explicitly here in case someone care to comment on it: I would appreciate if someone could tell me if this appears somewhere in the literature, or if there is a simpler argument, or even just if he thinks it is correct...).

Proof: As $$C$$ has all fiber product, and $$F$$ preserves them, one has that for each object $$c\in C$$, the slice category $$C/c$$ has all finite limits, and the functor induced by $$F$$:

$$F/c : C/c \rightarrow \mathcal{T}/F(c)$$

preserves them. In particular the left Kan extention:

$$\widehat{F/c} : \text{Prsh}(C/c) \rightarrow \mathcal{T}/F(c)$$

preserve all finite limit by the results of Lurie. This functor is isomorphic to:

$$\widehat{F}/c : \text{Prsh}(C)/c \rightarrow \mathcal{T}/F(c)$$

It follows in particular that $$\widehat{F}$$ preserves all fiber product of the form $$X \times_c Y$$ where $$c$$ is a representable object. Note that up to this point, everything also works in the $$1$$-categorical case, and this is exactly how we concluded in the special case where $$C$$ has a terminal object. The idea is now to use the descent property to extend this to more general pullback.

Let $$V \rightarrow X$$ be any morphism in $$\text{Prsh}(C)$$, and write $$X$$ as the canonical colimits:

$$X = \underset{c \in Elt(X)}{\text{colim }} c$$

(where $$Elt(X)$$ is the category of elements of $$X$$ and I have identified objects of $$C$$ with their image by the Yoneda embedding)

For each elements $$c \rightarrow X$$, let $$V_c$$ be the fiber of $$V \rightarrow X$$ over $$c$$.

By descent, one has:

$$V = \underset{c \in Elt(X)}{\text{colim }} V_c$$

Now, as $$\widehat{F}$$ preserves all pullbacks whose bottom corner is representable, it follows that the the natural transforation: $$\widehat{F}(V_c) \rightarrow F(c)$$ is cartesian (in $$c$$), and hence, by descent in $$\mathcal{T}$$ it follows that all the maps $$\widehat{F}(V_c) \rightarrow F(c)$$ are pullback of the maps between the colimits:

$$\widehat{F}(V) \simeq \underset{c \in Elt(X)}{\text{colim }} \widehat{F}(V_c) \rightarrow \underset{c \in Elt(X)}{\text{colim }} F(c) \simeq \widehat{F}(X)$$

Which shows that $$\widehat{F}$$ also preserves all pullback of the form $$V \times_X c$$ as soon as $$c$$ is representable.

To conclude, one either use a again a similar (but simpler) colimit/descent argument in the variable $$c$$, or one simply write (using descent) that the functor $$\text{Prsh}(C)/X \rightarrow \mathcal{T}/\widehat{F}(X)$$ can be decomposed as a limits:

$$\text{Prsh}(C)/X \simeq \lim_{c \in Elt(X)} \text{Prsh}(C)/c \rightarrow \lim_{c \in Elt(X)} \mathcal{T}/F(c) \simeq \mathcal{T}/\widehat{F}(X)$$

(Note: one needs the fiber product preservation proved above to show this)

But as all the functors $$\text{Prsh}(C)/c \rightarrow \mathcal{T}/F(c)$$ preserves finite limits, their limits also preserve finite limits and hence for all $$X \in \text{Prsh}(C)$$ the functor $$\text{Prsh}(C)/X \rightarrow \mathcal{T}/\widehat{F}(X)$$ preserve all finite limits, which concludes the proof.