Let $\mathscr{C}$, $\mathscr{D}$, and $\mathscr{E}$ be (infinity) categories, and assume we're given a Cartesian diagram: $\require{AMScd}$ \begin{CD} \mathscr{C} \times_{\mathscr{E}} \mathscr{D} @>\operatorname{pr}_1>> \mathscr{D}\\ @V \operatorname{pr}_2 V V @VV \varphi V\\ \mathscr{C} @>>\psi> \mathscr{E} \end{CD} Now let $\mathbf{C}$ be some other cocomplete infinity category and let $F: \mathscr{D} \to \mathbf{C}$ be a functor.

Since $\mathbf{C}$ is cocomplete, we can consider the left Kan extension $\varphi_!(F)$ of $F$ along $\varphi$. By definition, $\varphi_!$ is the left adjoint of the restriction functor $$\varphi^*: \operatorname{Fun}(\mathscr{E},\mathbf{C}) \longrightarrow \operatorname{Fun}(\mathscr{D},\mathbf{C}),$$ where $\operatorname{Fun}(-,-)$ denotes the category of functors between two categories. Similarly, we can consider the left Kan extension $(\operatorname{pr}_2)_!(\operatorname{pr}_1)^*(F)$ of the pullback of $F$ to the fiber product along $\operatorname{pr}_2$.

By the universal property of left Kan extensions, there is a natural transformation $$\beta_F:(\operatorname{pr}_2)_!(\operatorname{pr}_1)^*(F) \longrightarrow \psi^* \varphi_!(F)$$ and I would like to know if there are reasonable conditions we can put on the categories and functors involved so that $\beta_F$ is an isomorphism. Probably this doesn't always hold, but my hope is that there are some nice situations in which it does.

  • $\begingroup$ I believe a natural isomorphism of this form is known not for pullback squares (“cartesian” in French parlance) but rather for comma squares (sometimes, but wrongly, called lax pullbacks). Actually this is one of the axioms for derivators. $\endgroup$
    – Zhen Lin
    Mar 5 at 6:21
  • $\begingroup$ @ZhenLin Thanks a bunch! Any chance you have a reference for this? I'm also slightly confused because aren't pullback squares special cases of comma squares? $\endgroup$
    – Exit path
    Mar 5 at 6:43
  • $\begingroup$ In the sense that in some degenerate cases, the pullback is also the comma category, yes. I don't have a reference. You can try chasing references from any source that proves that cocomplete $(\infty, 1)$-categories give rise to derivators. $\endgroup$
    – Zhen Lin
    Mar 5 at 6:52
  • $\begingroup$ @ZhenLin Great, thanks again for the clarification! $\endgroup$
    – Exit path
    Mar 5 at 7:05
  • $\begingroup$ It occurs to me that this must be proven in some form in the Riehl–Verity papers about ∞-cosmoi. But you will probably have to figure out how to express it in the language of Kan lifts... $\endgroup$
    – Zhen Lin
    Mar 5 at 7:09

1 Answer 1


I believe one set of conditions is for either $\varphi$ to be proper or $\psi$ to be smooth. The dual of this (using right Kan extensions rather than left Kan extensions) is proven by Cisinski in "Higher Categories and Homotopical Algebra", Theorem 6.4.13.

For completeness, a map between simplicial sets $p : X \to Y$ is said to be proper when pullback along any pullback of $p$ preserves final maps (right cofinal in Lurie's terminology). More precisely, $p$ is proper if for any pair of cartesian squares in the following form, if $f$ is final then $g$ is final: $\require{AMScd}$ \begin{CD} X'' @>g>> X' @>>> X\\ @V V V @V V V @V V p V \\ Y'' @>>f> Y' @>>> Y \end{CD}

A map is smooth just when its opposite is proper.

  • $\begingroup$ Nice, thanks! I believe this should also encompass the example of $\varphi$ being a coCartesian fibration as I wrote in the comments $\endgroup$
    – Exit path
    Mar 5 at 17:54
  • 4
    $\begingroup$ It would be helpful for the casual reader if you could add the definitions of "proper" and "smooth". I'm not familiar with the meanings of those words as applied to functors. $\endgroup$ Mar 5 at 19:00
  • $\begingroup$ @MikeShulman Good point! I've added such a definition. $\endgroup$ Mar 6 at 9:28

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