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In his book Introduction to Infinity-Categories, Land in his Theorem 3.3.16 asserts an equivalence of $\infty$-categories where one of the categories $\mathrm{LFib}(\mathcal C)$ is the full subcategory of $(\mathrm{Cat}_\infty)_{/\mathcal C}$ on vertices that are left fibrations $?\to\mathcal C$ ($\mathcal C$ some $\infty$-category). My understanding of this material is that he is following Riehl & Verity’s 2018 article The comprehension construction. There, in Notation 6.1.12, is defined a category $\mathrm{coCart}(\mathrm{qCat})_{/\mathcal C}$, and my understanding is that Land’s $\mathrm{LFib}(\mathcal C)$ should be the full subcategory of $\mathrm{coCart}(\mathrm{qCat})_{/\mathcal C}$ on vertices which are left fibrations with target $\mathcal C$. The definition of $\mathrm{coCart}(\mathrm{qCat})_{/\mathcal C}$ makes use of the slice $\infty$-cosmos defined in Riehl-Verity’s Elements, Proposition 1.2.22 (here our $\infty$-cosmos is $\mathrm{qCat}$). Per that definition, it seems to me that morphisms between left fibrations $h:\mathcal E\to\mathcal C$ and $f:\mathcal F\to\mathcal C$ in $\mathrm{coCart}(\mathrm{qCat})_{/\mathcal C}$ are simply functors $\mathcal E\to\mathcal F$ over $\mathcal C$. But 1-simplices in $(\mathrm{Cat}_\infty)_{/\mathcal C}$ are the data of a morphism $g:\mathcal E\to\mathcal F$ and a 1-simplex in $\operatorname{Fun}(\mathcal E,\mathcal C)$ between $fg$ and $h$ which is an equivalence in $\operatorname{Fun}(\mathcal E,\mathcal C)$. So these don’t seem like the same categories to me. Can someone familiar with these sources help me reconcile these two definitions?

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First let me point out a small typo : it should be functors over $\mathcal C$ which send cocartesian edges to cocartesian edges (this doesn't matter for left fibrations, but for cocartesian fibrations it does).

For your main point, you are right that they are different quasi-categories: the simplicial sets are not isomorphic; but they're the same $\infty$-category, that is, the obvious forgetful functor from one to the other (so the one that uses the trivial $1$-simplex) is a categorical equivalence.

The fact that it is essentially surjective is obvious (the objects are the same), so the point is really about mapping spaces. But this is really a statement about pullbacks, namely the following:

Consider three quasi-categories $C,D,E$ with functors $p: C\to E, q:D \to E$. Suppose $q: D\to E$ is a cocartesian fibration. In this situation, the quasi-category $Fun_E(C,D)$ defined as a simplicial set by the (ordinary) pullback $Fun(C,D)\times_{Fun(C,E)}\{p\}$ is equivalent to the homotopy pullback, $Fun(C,D)\times_{Fun(C,E)}(Fun(C,E)^\simeq)_{/p}$

Note that the first one is the category of those functors $f :C\to D$ that are strictly over $E$, that is, with $q\circ f = p$, whereas $0$-simplices in the latter are a pair $(f, \sigma)$ where $f : C\to D$ is a functor and $\sigma: q\circ f \to p$ is an equivalence, and you can check that in fact it is what you're thinking of.

The reason for this equivalence is that $Fun(C,D)\to Fun(C,E)$ is also a cocartesian fibration, so in particular a fibration in the Joyal model structure, and so pull-back along it preserves weak equivalences. It then suffices to observe that the inclusion $\{p\}\to (Fun(C,E)^\simeq)_{/p}$ is an equivalence, because $Fun(C,E)^\simeq$ is a Kan complex.

The idea is that because $D\to E$ is a cocartesian fibration, you have "enough room" to "strictify" any equivalence $q\circ f\simeq p$ to an actual equality $q\circ f = p$ up to changing $f$.

(note that the latter, the one with $q\circ f\simeq p$, is the one that you would want to have as an $\infty$-category, because $=$ is too strict a notion. It just happens that in this case you can actually strictify)

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  • $\begingroup$ Thank you for this explanation! I omitted the statement about sending cocartesian edges to cocartesian edges because we’re talking about maps between left fibrations (so maybe it’s ok as written?). Literally speaking, the full faithfulness amounts to the fact that the ($\infty$-groupoid) core of the two pullbacks you wrote are equivalent, no? I see why the 0-simplices of $Fun(C,D)\times_{Fun(C,E)}(Fun(C,E)^\simeq)_{/p}$ coincide with 0-simplices of $\operatorname{map}_{(\mathrm{Cat}_\infty)_{/E}}(p,q)$, but I don’t know how to check it on higher simplices. $\endgroup$
    – Tomo
    Feb 14, 2022 at 4:57
  • $\begingroup$ I was wondering, maybe I can use HTT 5.5.5.12, which says that $\operatorname{map}_{(\mathrm{Cat}_\infty)_{/E}}(p,q)$ coincides with the homotopy-fiber at $p$ of the map $\operatorname{map}_{\mathrm{Cat}_\infty}(C,D)\to \operatorname{map}_{\mathrm{Cat}_\infty}(C,E)$ induced by $q$; as you point out, as $q$ is a cocartesian fibration, $Fun(C,D)\to Fun(C,E)$ is an isofibration, so wouldn’t this give the desired conclusion that $\operatorname{map}_{(\mathrm{Cat}_\infty)_{/E}}(p,q)\simeq Fun(C,D)^\simeq\times_{Fun(C,E)^\simeq}\{p\}$? $\endgroup$
    – Tomo
    Feb 14, 2022 at 4:57
  • $\begingroup$ Also, stupid question, but how do you actually define the functor from the full subcategory (on the left fibrations) of $\mathrm{coCart}(\mathrm{qCat})_{/E}$ to $(\mathrm{Cat}_\infty)_{/E}$ on $n$-simplices where $n>1$? I don’t know how to express $(\mathrm{Cat}_\infty)_{/E}$ as the homotopy-coherent nerve of a simplicial category, and I don’t have an explicit description of $n$-simplices of the homotopy-coherent nerve defining $\mathrm{coCart}(\mathrm{qCat})_{/E}$. $\endgroup$
    – Tomo
    Feb 14, 2022 at 5:35
  • $\begingroup$ You are right that for left fibrations, the condition on cocartesian edges is automatic, I just wanted to mention it in case :) Also, yes, if you already know what homotopy fibers are, that should be good. The thing with $(Fun(C,E)^\simeq)_{/p}$ was just a specific model for that homotopy fiber (albeit a practical one for intuition). For your last question, it will depend on the definitions but it shouldn't be hard : the point is that even for higher $n$'s you can forget from a strict thing to a non-strict thing. They seem to do this in RIehl-Verity 6.1.12. $\endgroup$ Feb 14, 2022 at 8:35
  • $\begingroup$ Let $C$ be a quasi-category. The problem I have is not to define the obvious functor $\mathrm{CoCart}(\mathrm{qCat})_{/C}\hookrightarrow\mathrm{qCat}_{/C}$ as in RV 6.1.12, it is to define a functor $Ng_*Q\to(\mathrm{Cat}_\infty)_{/C}$, where $Q$ denotes the full subcategory of the sliced $\infty$-cosmos $\mathrm{qCat}_{/C}$ on the left fibrations and $g$ denotes the groupoidal core functor, and $(\mathrm{Cat}_\infty)_{/C}$ denotes the usual overcategory of the $\infty$-category of $\infty$-categories. $\endgroup$
    – Tomo
    Feb 15, 2022 at 22:04

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