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Kan extensions are really useful in category theory as a means to construct an onslaught of interesting new functors (and adjunctions). For instance, the Nerve-Realization Paradigm can be understood as a left Kan extension along the Yoneda embedding. Left Kan extensions along some functor have a very natural definition in terms of being left adjoints to precomposition with said functor.

Now whilst reading higher algebra, and entering into the more technical $\infty$-operad chapters, it has become apparent that Lurie uses not only Kan extensions, but the more general notion of a $p$-Kan extension (aka relative Kan extensions), which seems quite hard to penetrate. Notably, references for this concept seem to be limited to Higher Topos Theory and Kerodon.

To formulate my question, let me recall the definition of a $p$-Kan extension (see here for more details):

Let $F \colon \mathscr{C} \to \mathscr{D}$ and $p \colon \mathscr{D} \to \mathscr{E}$ be functors of $\infty$-categories, and let $\mathscr{C}^0 \subset \mathscr{C}$ be a full subcategory (more generally, one should be able to consider a map $\mathscr{C}^0 \to \mathscr{C}$ for some category $\mathscr{C}^0$). For an object $x \in \mathscr{C}$, we say $F$ is $p$-left Kan extended from $\mathscr{C}^0$ at $x$ if the composite map
$$ (\mathscr{C}^0_{/x}) \star [0] \hookrightarrow (\mathscr{C}_{/x}) \star [0] \xrightarrow{\psi} \mathscr{C} \xrightarrow{F} \mathscr{D} $$ is a $p$-colimit diagram (see here) in the $\infty$-category $\mathscr{D}$. Here $\star$ denotes the join operation, and $\psi$ is the slice contraction morphism (see Kerodon, Tag 0188).

My question is then as follows:

Is there a more intuitive, conceptual way to understand this definition? More precisely put, the definition of left $p$-Kan extensions is model-dependent, as it relies on the specific construction of the slice contraction map $\psi$ (see Kerodon, Tag 0188). Therefore:

Is it possible to define left $p$-Kan extensions in a model-independent manner, akin to the approaches in Formalization of Higher Categories (link) or Elements of $\infty$-Category Theory (link)?

Thank you!

h3fr43nd
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  • Please ask one question at a time. – Ben Steffan Jan 25 '25 at 12:43
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    @BenSteffan It would be totally fine for people to address only one or some of these questions in an answer. Breaking apart every question into several individual Mathexchange posts didn't seem reasonable to me, as those questions are evidently closely related to each other. – h3fr43nd Jan 25 '25 at 12:47
  • It's your prerogative not to change anything but you're violating a side policy here. You should expect this question to be closed. – Ben Steffan Jan 25 '25 at 12:49
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    Okay thank you for pointing that out, I will try to adjust the question accordingly. – h3fr43nd Jan 25 '25 at 12:54
  • Taking left $p$-Kan extensions gives a left adjoint functor to a restriction functor. Isn't this essentially a model-independent characterization? – Thorgott Jan 25 '25 at 17:45
  • @Thorgott Could you spell that out in an answer maybe? But yes, if there is such an analogous adjunction just like for ordinary Kan extensions that would be a model independent definition, so I would be happy to see it :) – h3fr43nd Jan 25 '25 at 18:09

1 Answers1

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Though the definition of a relative Kan extension is relatively dense, I believe it can be made sense of as-is. Assuming the concept of an absolute Kan extension is understood, this is just the generalization replacing colimits with relative colimits, so it suffices to gain intuition for relative colimits. I have tried explaining those in this answer.

Alternatively, relative Kan extensions can also be understood by a universal property. To this end, suppose we are given a diagram $$\require{AMScd} \begin{CD} \mathcal{C}^0 @. \mathcal{D}\\ @V{i}VV @V{p}VV\\ \mathcal{C} @>>> \mathcal{D}^{\prime}, \end{CD}$$ where $i\colon \mathcal{C}^0\subseteq\mathcal{C}$ is a full subcategory inclusion and $p$ is a categorical fibration. Suppose that every functor $F_0\colon\mathcal{C}^0\rightarrow\mathcal{D}$ completing this square (in a strict sense) admits a $p$-left Kan extension along $i$, then the restriction functor $i^{\ast}\colon\mathrm{Map}_{\mathcal{D}^{\prime}}(\mathcal{C},\mathcal{D})\rightarrow\mathrm{Map}_{\mathcal{D}^{\prime}}(\mathcal{C}^0,\mathcal{D})$ admits a left adjoint given by $p$-left Kan extension. This is HTT, Proposition 4.3.2.17. The property of the restriction admitting a left-adjoint is now a statement you should be able to interpret in any model. (If you don't like assuming existence of all $p$-left Kan extensions, you can also spell out the universal property in an object-wise manner, see Kerodon, Proposition 7.3.6.1.)

Two comments on generality: You can replace the full subcategory inclusion $i$ by an arbitrary functor, but I don't know if this has been properly written down anywhere (absolute Kan extensions along arbitrary functors are in HTT/Kerodon, but the simultaneous generalization is in neither, though Lurie mentions that it works). Also, the model-specific assumption that $p$ is a categorical fibration can of course be dropped as soon as you switch to the characterization in terms of the universal property as every functor factors into an equivalence followed by a fibration.

Thorgott
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  • Thanks, this is exactly what I was looking for! :)

    Something else comes to mind upon seeing this characterization: If the functor $p$ is exponentiable (i.e. pullback along $p$ has a right adjoint), then one may define an internal hom-object $\text{Hom}{\mathcal{D}'}(-,\mathcal{D}) \to \mathcal{D}'$ in $\text{Cat}{/\mathcal{D}'}$. Does the above adjunction strenghten to a relative adjunction $i^* \colon \text{Hom}{\mathcal{D}'}(\mathcal{C},\mathcal{D}) \to \text{Hom}{\mathcal{D}'}(\mathcal{C}^0,\mathcal{D})$?

     – h3fr43nd Jan 26 '25 at 09:30
  • @h3fr43nd Sorry, how is the internal Hom-object defined? – Thorgott Jan 26 '25 at 11:48
  • If $p_* \colon \text{Cat}{/\mathcal{D}}\to \text{Cat}{/\mathcal{D}'}$ denotes the right adjoint to pullback along $p$ (i.e. $p^* \dashv p_$), then for $q \colon \mathcal{C} \to \mathcal{D}'$ we define $\underline{\text{Fun}}_{\mathcal{D}'}(\mathcal{C}, \mathcal{D}) = p_(p^*(q))$ (see "Formalization of Higher Categories" (linked in the question above) and there it is Definition 3.5.12). – h3fr43nd Jan 26 '25 at 11:58
  • @h3fr43nd Hmm, I'm not familiar with this construction, but at first glance it seems quite different from the relative functor categories $\mathrm{Map}{\mathcal{D}^{\prime}}(-,\mathcal{D})$. If you desire a more intrinsic characterization of these relative functor categories (though, of course, you can just write them down as a pullback explicitly), I believe they are the mapping categories in an appropriate $(\infty,2)$-categorical slice $(\mathrm{Cat}{\infty})_{/\mathcal{D}^{\prime}}$, though I'm out of my depth here. – Thorgott Jan 26 '25 at 13:38
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    The underlying $\infty$-category of the internal hom object is the same as the relative functor $\infty$-category in your answer (this is also proven at some point in the mentioned book). I will ponder this some more, but if it could be promoted to such a relative adjunction, then relative Kan extensions are literally just relative adjoints to precomposition in a relative context (which would be nice conceptually). – h3fr43nd Jan 26 '25 at 13:49
  • Sorry to bump this older question. If the Kan extension of $\mathcal{C}0\to\mathcal{D}$ along $\mathcal{C}_0\to\mathcal{C}$ is relative to some category $\mathcal{E},$ then for every $e\in\mathcal{E},$ I have a diagram of fibers $\mathcal{C}_0\times{\mathcal{E}}{e}\to\mathcal{C}\times_{\mathcal{E}}{e}\to\mathcal{D}\times_{\mathcal{E}}{e}.$ Would the relative Kan extension condition be roughly saying that these diagrams of fibers are all Kan extensions, varying functorially in the $e\in\mathcal{E}$? – Stahl Feb 06 '25 at 20:39
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    @Stahl This is not true. Take $\mathcal{C}=\Delta^1$ and $\mathcal{C}_0={0}$, then an arrow $\Delta^1\rightarrow\mathcal{D}$ is a $p$-left Kan extension if and only if it classifies a $p$-cocartesian edge $f\colon x\rightarrow y$, whereas your condition seems to be equivalent to $y$ being $p$-initial (as long as $f$ is a non-identity morphism). – Thorgott Feb 06 '25 at 21:54
  • Thanks for that, I see that it was too hopeful. Do you know if there are reasonable hypotheses on these categories which would make this pullback compatibility hold? The situation I'm looking at is something like the inclusion of morphisms of affine schemes into morphisms of stacks, mapping into the category of pairs $(X,\mathcal{F})$ of stacks $X$ together with a quasi-coherent sheaf $\mathcal{F}$ on the stack, all fibered over stacks (via the source or target maps), so that when I pullback, everything is relative to a particular source or base. – Stahl Feb 08 '25 at 00:16