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I am currently working with Kan extensions, and I have found a neat little fact, and I'd like to know if there is some literatture on it, and maybe some names and characterization I could use.

Question

Given a functor $K : A \to B$, I consider its associated nerve functor $N_K$, associates to every object $b$ in $B$ the presheaf on $A$ $N_Kb = B(K\_,b)$.

The functor $N_K$ preserves the limits: This is nothing else than the universal property of the limits : If you plug-in a limit into $N_K$, then you are essentially asking for things that map into a limit, and that is a limit of things that map into the component (Some people call this the continuity of Hom-functor wrt its second variable). However, it is not true that $N_K$ preserves the colimits in general.

Question: is there a name for the functors $K$ such that $N_K$ preserves the colimits? And are there other characterizations of such functors?

An example: There exist functors such that the associated nerve preserves colimits, and a very important example of such function is the Yoneda embedding. Indeed, if assume that $B=\widehat{A}$ and that $K$ is the Yoneda embedding, then it is certainly the case that the associated nerve $N_K$ preserves the colimits. In fact, a bit of Yoneda lemma and unraveling tell you that this fact exactly says that colimits in the presheaf category are computed pointwise! I would like to recognize analogous situations from this one.

My motivation

As I said before, I encountered this question while working with Kan extensions. In my actual example, I have right Kan extensions, but I am dualizing everything here and write it as left Kan extensions, as they get us closer to the familiar land of Yoneda.

A characterization of Kan extensions

I consider two functors $K : A \to B$ and $F : A \to C$, where $C$ is a cocomplete category, so that the Kan extension $\operatorname{Lan}_KF$ exists and is pointwise. For the specific problem that I am working on, I want to characterize the Kan extension as a colimit of a cone that is encoded as a natural transformation between the associated nerve functors.

The left Kan extension then satisfies the universal property: For every $b$, the object $\operatorname{Lan}_KF(b)$ is the unique (up to iso) object of $C$ such that for all $c$ in $C$ there is a natural isomorphism $C(\operatorname{Lan}_KF(b),c) \simeq \widehat{A}(N_Kb,N_Fc)$. This is a standard characterization, that is both in Mac Lane's Categories for the working mathematician, and Riehl's Category in context.

Left Kan extensions along $K$ preserve colimits

Under the assumption that $N_K$ preserve the colimits, and with this characterization of left Kan extensions, I have an easy proof that $\operatorname{Lan}_KF$ preserves the colimits. Take a diagram $\mathcal D:I\to B$ which has a limit in $B$, then we have the following equalities, which are given by the universal property of colimits (cocontinuity of the hom-functor wrt its first variable) and by the characterization of the Kan extension. \begin{align*} C(\operatorname{Lan}_KF(\operatorname{colim}\mathcal D), c) &= \widehat{A}(N_K(\operatorname{colim}\mathcal D),N_Fc) \\ &= \widehat{A}(\operatorname{colim}N_K\circ\mathcal D, c) \\ &= \lim \widehat{A}(N_K\circ\mathcal D, c) \\ &= \lim C(\operatorname{Lan}_KF\circ\mathcal D, c) \\ &= C(\operatorname{colim} (\operatorname{Lan}_KF\circ\mathcal D),c) \end{align*} If you spell it out a little, this computation shows that $\operatorname{Lan}_KF(\operatorname{colim}\mathcal D)$ satisfies the universal property of the colimit, and thus this proves that $\operatorname{Lan}_KF$ preserves the colimits.

Back to the Yoneda example

If you look back at my previous example, where $K$ is the Yoneda embedding, then the left Kan extension along the Yoneda embedding is the universal map obtained by the fact that presheaves are the free cocompletion, and it is a known fact that this functor preserves colimits, in fact it is how this functor is characterized. I am working with variations of this result, and Kan extensions provide a powerful tool for that. But I would like to know if adding that little caveat that $N_K$ preserves colimits is a standard trick, and if there are other characterizations to make things more "Yoneda-y".

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As you say, colimits in $[\mathcal{A}^\textrm{op}, \textbf{Set}]$ are computed componentwise. (If $\mathcal{A}$ is essentially small then "colimits" can be unqualified; otherwise we should probably stick to small colimits.) So $N_K : \mathcal{B} \to [\mathcal{A}^\textrm{op}, \textbf{Set}]$ preserves colimits if and only if, for every object $a$ in $\mathcal{A}$, the functor $\mathcal{B} (K a, -) : \mathcal{B} \to \textbf{Set}$ preserves colimits. That is, if and only if every $K a$ is a tiny object in $\mathcal{B}$.

Notice that neither the morphisms of $\mathcal{A}$ nor the action of $K : \mathcal{A} \to \mathcal{B}$ on morphisms matters here. If you were asking about, say, whether $N_K : \mathcal{B} \to [\mathcal{A}^\textrm{op}, \textbf{Set}]$ is fully faithful it would be a different story. As such, this is not really a property of the functor $K : \mathcal{A} \to \mathcal{B}$ per se, and as far as I know there is no name this condition.

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  • $\begingroup$ Thanks a lot. I am not sure I agree with you that it is not a property of the functor $K$. If I understand what you say correctly, one could say that this correspond to the fact that $K$ factors through the inclusion of tiny objects in $\mathcal{B}$. Anyway, that's just terminology. Your answer was quite helpful $\endgroup$ Apr 9, 2021 at 8:46

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