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(Or more generally, what are some examples of Kan extensions which are not pointwise?)

By a total derived functor of a functor $F: C \to D$, I simply mean a (left or right) Kan extension of $F$ along some localization $C \to C[W^{-1}]$ at a class of arrows $W$. Usually when such a Kan extension exists, it comes from a functor $L: C \to C$ and a natural transformation $\eta: 1_C \implies L$ (or the other way around depending on handedness) such that on the image of $L$, $F$ already takes arrows in $W$ to isomorphisms, and the components of $\eta$ are in $W$ (Riehl calls this a deformation of the functor $F$, but I'm not sure how standard this terminology is). When such an $L$ exists, the total derived functor is simply $F \circ L$.

My question is: does a total derived functor ever exist without such a deformation / replacement functor $L$ existing? I presume the answer is yes, and I'd like to see some examples.

My motivation for this question actually comes from the fact that when a deformation exists, the Kan extension involved is pointwise, and in fact absolute (preserved by all functors). I'm actually just trying to understand pointwiseness of Kan extensions better by looking at examples of non-pointwise Kan extensions. There is an example of a non-pointwise Kan extension in Borceux's Handbook of Categorical Algebra, and it generalizes a bit, but I'd like to see more examples. Since the codomain category of a total derived functor rarely has many (co)limits, it's a good candidate to admit non-pointwise Kan extensions into it, and it seems one just has to get around the fact that these Kan extension are pointwise if they are computed as deformations.

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Yes. In fact, one such example comes from homotopical algebra:

Proposition. Let $\mathcal{C}$ be a small homotopical category and let $\gamma : \mathcal{C} \to \operatorname{Ho} \mathcal{C}$ be the localising functor. Then $\mathcal{C} (-, -) : \mathcal{C}^\mathrm{op} \times \mathcal{C} \to \mathbf{Set}$ admits a left Kan extension along $\gamma^\mathrm{op} \times \gamma : \mathcal{C}^\mathrm{op} \times \mathcal{C} \to \operatorname{Ho} \mathcal{C}^\mathrm{op} \times \operatorname{Ho} \mathcal{C}$, namely $\operatorname{Ho} \mathcal{C} (-, -) : \operatorname{Ho} \mathcal{C}^\mathrm{op} \times \operatorname{Ho} \mathcal{C} \to \mathbf{Set}$.

Bizarrely enough, this is a pointwise left Kan extension – so the proof is a direct calculation. If $\gamma : \mathcal{C} \to \operatorname{Ho} \mathcal{C}$ has a (necessarily fully faithful) right adjoint, then this can also be calculated using a right deformation. I think the converse is also true, but at any rate, right deformations usually do not exist.

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  • $\begingroup$ Mysterious pointwiseness strikes again! I was really hoping that this question would inevitably lead to non-pointwise extensions, but this is interesting too. $\endgroup$
    – Tim Campion
    Commented Oct 13, 2015 at 15:16
  • $\begingroup$ This computation comes down to showing that a morphism in $\mathrm{Ho}\mathcal{C}$ is the same thing as a triple of composable morphisms in $\mathrm{Ho}\mathcal{C}$ where the middle one has a lift to $\mathcal{C}$, modulo the equivalence relation generated by identifying two triples if the middle arrow of one factors through the middle arrow of the other in $\mathcal{C}$, compatibly with the other arrows. I think the hypotheses can be weakened to say that $\mathcal{C}$ is any category with a class of arrows such that $\mathrm{Ho}\mathcal{C}$ is still locally small, right? $\endgroup$
    – Tim Campion
    Commented Oct 13, 2015 at 19:19
  • $\begingroup$ Sure. This is only to ensure that the hom-functors land in $\mathbf{Set}$, of course. $\endgroup$
    – Zhen Lin
    Commented Oct 13, 2015 at 19:27
  • $\begingroup$ And I guess this pointwiseness isn't really mysterious at all, since $\mathsf{Set}$ is cocomplete. Although this answer is interesting, I'm going to hold out for something more interesting. I feel justified because (1) When people say "total derived functor" they usually mean something into some sort of homotopy category, not a cocomplete category like $\mathsf{Set}$ and (2) I was really interested in non-pointwise Kan extensions in the first place. $\endgroup$
    – Tim Campion
    Commented Oct 14, 2015 at 23:52

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