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This answer by Emily Riehl mentions a to-be-published proof of the fact that $\infty$-categorical (pointwise) Kan extensions exist when the target category admits (co)limits indexed by the relevant comma categories. Can this now be found anywhere (or does it follow easily from other things)? I didn't manage to find it in either the later papers by Riehl and Verity, nor their current book project. (As far as I could tell it is in Lurie's "Higher Topos Theory" also only done for fully faithful functors).

Moreover, what I am actually interested in is the existence of functorial pointwise extensions (as defined in Definition 13.5.7 of the above mentioned book), i.e. not just for a single functor but all of them at once. Is this also implied by the existence of sufficiently many (co)limits or even just the existence of a pointwise extension of each single diagram? And, again, is there a reference for this?

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  • $\begingroup$ Have you tried HTT 4.3.3? The existence follows from the stuff in there (you reduce to the case of an embedding), and the description as an adjoint (and hence functoriality) is 4.3.3.7. $\endgroup$ – Dylan Wilson Jul 31 at 20:11
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    $\begingroup$ It’s also in this paper in greater generality: arxiv.org/pdf/1809.05892.pdf $\endgroup$ – Dylan Wilson Jul 31 at 20:19
  • $\begingroup$ @DylanWilson Thanks, that paper answers much of the question (it wasn't really clear to me how HTT §4.3.3 implies existence)! However, one more thing is that the definition of functorial pointwise extensions used by Riehl and Verity I mentioned in the question is (at least a priori) stronger than just existence of an adjoint to the restriction (this implication is remarked upon directly after the definition). And I actually need this stronger (?) version as I would like to employ the Beck-Chevalley condition (Proposition 13.5.8 in the book). Do you know if this also follows? $\endgroup$ – Robin Stoll Aug 1 at 9:24
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Lurie's approach

First let me explain why this is already in HTT 4.3.3.

Recall that a (pointwise) left Kan extension of $F: A \to \mathcal{C}$ along an inclusion $i: A \to B$ is a functor $F: B \to \mathcal{C}$ such that, for each $b \in B$, $F(b)$ is a colimit of the diagram $A \times_B B_{/b} \to A \to \mathcal{C}$.

In general, if $g: A \to B$ is a functor between $\infty$-categories, and $\mathcal{C}$ another $\infty$-category. Then a left Kan extension of $F: A \to \mathcal{C}$ along $g$ is a functor $\overline{F}: (A\times [1]) \amalg_{A \times \{1\}} B \to \mathcal{C}$ which is a left Kan extension (in the previous sense) of its restriction to $A = A \times \{0\}$. (Notice that the data of $\overline{F}$ is precisely the data of a functor out of $B$ and a natural transformation from $F$ to its restriction to $A$ along $g$).

Let's unwind what that means. The value of $\overline{F}$ on anything of the form $(a, 0)$ is already determined, so the only question is what is the value of $\overline{F}$ at $b \in B$. Let $M = (A \times [1]) \amalg_{A \times \{1\}} B$. Then $\overline{F}(b)$ must be a colimit of the diagram $A \times_M M_{/b} \to A \to \mathcal{C}$. But the natural map $A \times_M M_{/b}$ is the same as $A \times_B B_{/b}$, so this is the usual notion of a pointwise Kan extesion.

Moreover: in the previous section, Lurie proves that Kan extensions along fully faithful functors exist and are unique provided the relevant colimits exist. So the Kan extension $\overline{F}$ exists if and only if the usual colimits over $A \times_B B_{/b}$ exist, by what we saw above. Whence the result.

Shah's approach

(This is logically unnecessary for the answer to your question, but worth advertising since it's a neat trick).

The actual proof of existence in HTT, in the fully faithful case, is a little difficult. Jay Shah has a nice approach which treats the fully faithful and general case at once and is very clean. The idea is to factor $g: A \to B$ as $A \to A\times_{B^{\{1\}}}B^{[1]} \to B$. The first map is a right adjoint (with left adjoint given by projecting to $A$) and we can always Kan extend along right adjoints: just compose with the left adjoint. The second map is a cocartesian fibration, and its fibers are precisely those diagrams of interest "$A \times_B B_{/b}$". So we are reduced to the following assertion: if $\pi: E \to B$ is a cocartesian fibration, then we may left Kan extend along $\pi$ provided the colimits along the fibers exist. This is not so bad: one forms the relative overcategory for a functor $F: E \to \mathcal{C}$, i.e. a cartesian fibration $E^{(F, B)/} \to B$ whose fiber over $b$ is the overcategory for $E_b \to E \to \mathcal{C}$. The assumption is that each fiber has an initial object. But the full subcategory of fiberwise initial objects inside of a cartesian fibration is a trivial Kan fibration over its image (this is not so hard to prove directly), and so we may choose a section. This completes the proof.

Functoriality

You ask about functoriality in the following form: fix an $\infty$-category $\mathcal{C}$ and a functor $g: A \to B$. Then (according to Riehl-Verity) $\mathcal{C}$ admits functorial pointwise left Kan extensions for $g$ if there is a left Kan extension of the evaluation map $A \times \mathcal{C}^A \to \mathcal{C}$ along $A \times \mathcal{C}^A \to B \times \mathcal{C}^A$.

By the existence result, we know that this Kan extension exists provided that, for each pair $(b, F)$, the colimit over $(A \times \mathcal{C}^A) \times_{B \times \mathcal{C}^A} (B \times \mathcal{C}^A)_{/(b, F)}$ exists. But the inclusion of $A \times_B B_{/b}$ into this category given by $(a, ga \to b) \mapsto ((a, F), ga \to B, F = F)$ is final (indeed, it has a left adjoint given by $((a, G), ga \to b, G \to F) \mapsto (a, ga \to B)$). Therefore, this colimit exists if and only if the corresponding colimit over $A \times_B B_{/b}$ exists.

In summary: if $\mathcal{C}$ admits colimits of shape $A \times_B B_{/b}$ for all $b$, then $\mathcal{C}$ admits 'functorial pointwise Kan extensions' in the sense of Riehl-Verity.

(It is also possible to deduce this functoriality formally from the existence of the left adjoint $g_!$ to the restriction $g^*$ but the above is probably simpler).

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  • $\begingroup$ Thanks, that helps a lot! One more question: Is it clear (or is it written down somewhere) that Kan extensions in the sense of Lurie (which are pointwise by definition) are pointwise extensions in the sense of Riehl-Verity? $\endgroup$ – Robin Stoll Aug 2 at 11:17
  • $\begingroup$ Oh sorry, I didn't realize there was an alternative definition for "pointwise". I tried looking at the Riehl-Verity definition, but unfortunately I'm not familiar with their setup so I couldn't make sense of it. Sorry! $\endgroup$ – Dylan Wilson Aug 2 at 14:45
  • $\begingroup$ No problem. Thank you anyway! $\endgroup$ – Robin Stoll Aug 2 at 17:56

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