# 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.

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 (Edit: see me second answer about this specific point).

• 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 '19 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 '19 at 18:40
• Oh I see. I misunderstood your point about $D$ not needing finite limits. Sorry! – Tim Campion Feb 24 '19 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.

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.

• Why do you have $V = colim V_c$? This is a statement in $Prsh(C)$, so I'm wondering why descent is helpful here. – Alexander Körschgen Aug 27 '19 at 10:53
• @AlexanderKörschgen : Regarding the counter-example in Set. When working in space taking homotopy orbits induce an equivalence of $\infty$-categories between space with a G-action and spaces over BG. An equivalence of category this preserves all limits, and so the homotopy orbit functors, which is just the composition of this equivalence with forgeting the map to BG reserve all contractible limits. Regarding Descent, I have to appologize I was refering to the fact that in an $\infty$-topos all colimits are effective and universal. I thought it was usual to call this "descent" [...] – Simon Henry Aug 27 '19 at 12:41
• [...] But I browse trough Lurie's 'Higher topos theory' to find a reference and I realized he never calls this descent in the book. The precise statement I'm refering to is for example condition (2) in theorem 6.1.0.6 of HTT. At the point you are referring I'm just using universality of colimits (so just the fact that as the category is locally Cartesian closed pullback preserve colimits, that part would actually still holds in set) but effectiveness is also used lated. (And thanks for looking at the proof ! ) – Simon Henry Aug 27 '19 at 12:45
• Ah found it: that use of the word "descent" is from C.Rezk preprint on higher toposes ( faculty.math.illinois.edu/~rezk/homotopy-topos-sketch.pdf ), (where he attribute it to Lurie actually) see 6.5 there. – Simon Henry Aug 27 '19 at 16:24
• (I'm wirting $F$ for $\widehat{F}$ ) one shows that $F(V_c)$ is indeed the pullback $F(c) \times_{F(X)} F(V)$ by observing that the natural transformation (in c) $F(V_c) -> F(c)$ is cartesian (because it is cartesian in $Prsh(C)$ and $F$ preserve pulblack whose bottom object is representable), and hence by descent (more precisely, effectiveness of colimits in $\mathcal{T}$ this time), each $F(V_c) -> F(c)$ is the pullback of the colimitis, i.e. of $F(V)->F(X)$ (because this is the colimit in $Prsh(C)$ and $F$ preserves colimits). That 's essentially what I tried to explain in the answer. – Simon Henry Aug 28 '19 at 17:07