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Usually I see the universal (small) cocartesian fibration $\mathcal{Z} \to Cat_\infty$ constructed in a relatively opaque fashion, such as via the unstraightening construction.

I've stumbled on what seems to be a relatively simple construction, and am looking for a sanity check that I'm not overlooking something:

Question: Is $\mathcal{Z}$ equivalent to the full subcategory of $(Cat_\infty)^{[1]}$ spanned by the arrows of the form $\mathcal{C}_{/X} \to \mathcal{C}$, with the structure map is induced by the target operation $(Cat_\infty)^{[1]} \to Cat_\infty$?

The rationale is as follows. For each small category, we have a series of fully faithful maps

$$ \mathcal{C} \subseteq \mathcal{P}(\mathcal{C}) \simeq RFib(\mathcal{C}) \subseteq (Cat_\infty)_{/\mathcal{C}} $$

with the first two maps being natural in $\mathcal{C}$.

The composite inclusion, I believe, sends an object $X$ to the right fibration $\mathcal{C}_{/X} \to \mathcal{C}$.

The mapping $\mathcal{C} \mapsto (Cat_\infty)_{/\mathcal{C}}$ should classify the target map $(Cat_\infty)^{[1]} \to Cat_\infty$. While $RFib(\mathcal{C}) \subseteq (Cat_\infty)_{/\mathcal{C}} $ is not natural, it nonetheless seems very plausible that $\mathcal{C} \mapsto RFib(\mathcal{C})$ should be be the classifying map for the fibration $RFib \to Cat_\infty$, where $RFib \subseteq (Cat_\infty)^{[1]}$ is the full subcategory spanned by right fibrations.

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    $\begingroup$ There is an approach to the universal (co)cartesian fibration in the quasicategory model not using straightening / unstraightening which appears in Nguyen's thesis. The left/right fibration version appears in Cisinski's book. The underlying idea is elegant, though it's slightly more complicated than one might think because of considerations having to do with strictness of the functors involved. It may or may not be related to your approach. $\endgroup$
    – Tim Campion
    Feb 6 at 5:45
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Yes, this ought to be true. One way to prove this would be to produce an equivalence to the universal cocartesian fibration defined in Lurie's Kerodon. Here is an outline of such a proof.

N.B. I have't yet worked out all the details in steps 3 and 4 below, so read this answer with a grain of salt.

The proof proceeds in four steps:

  1. We first define a Kan-enriched category $\mathbf{Rep}$, and show that its homotopy coherent nerve is equivalent (over the quasi-category of quasi-categories) to the quasi-category described in your question.

  2. We next show that the forgetful functor $\mathbf{Rep} \to \mathbf{qCat}$ is sent by the homotopy coherent nerve functor to a cocartesian fibration.

  3. We next define a morphism of cocartesian fibrations from the homotopy coherent nerve of $\mathbf{Rep} \to \mathbf{qCat}$ to the universal cocartesian fibration defined in Lurie's Kerodon.

  4. Finally, we show that this morphism of cocartesian fibrations is an equivalence on fibres, and hence an equivalence of cocartesian fibrations.

1. We begin by defining the Kan-enriched category $\mathbf{Rep}$. Its objects are pairs $(A,a)$, where $A$ is a quasi-category and $a$ is an object of $A$. We define a morphism from such an object $(A,a)$ to another $(B,b)$ to be a commutative square of quasi-categories $\require{AMScd}$ \begin{CD} A/a @>>> B/b \\ @VVV @VVV\\ A @>>> B \end{CD} where the vertical maps are the projections. Moreover, we define the hom Kan complexes of $\mathbf{Rep}$ by the following pullback squares of Kan complexes, where $\mathbf{qCat}$ denotes the usual Kan-enriched category of quasi-categories. \begin{CD} \mathrm{Hom}_{\mathbf{Rep}}((A,a),(B,b)) @>>> \mathrm{Hom}_\mathbf{qCat}(A/a,B/b)\\ @V V V @VV V\\ \mathrm{Hom}_\mathbf{qCat}(A,B) @>>> \mathrm{Hom}_\mathbf{qCat}(A/a,B) \end{CD} Composition in $\mathbf{Rep}$ is defined in terms of the composition in $\mathbf{qCat}$.

Observe that the left-hand vertical map in the above pullback square is a Kan fibration, since the right-hand vertical map is. These left-hand vertical maps define the action on homs of a simplicial functor $\mathbf{Rep} \to \mathbf{qCat}$, which is given on objects by $(A,a) \mapsto A$.

It follows from HTT.4.2.4.4 that a certain explicit functor from the homotopy coherent nerve of $\mathbf{Rep}$ to the quasi-category you define in your question is an equivalence over $N(\mathbf{qCat})$. (I would be happy to explain this in more detail, but for now let me press on).

2. We now show, using HTT.2.4.1.10, that the homotopy coherent nerve functor $N$ sends the forgetful functor $\mathbf{Rep} \to \mathbf{qCat}$ to a cocartesian fibration. Since this forgetful functor is a Kan fibration on homs, it is sent by $N$ to an inner fibration.

Given an object $(A,a)$ of $\mathbf{Rep}$ and a morphism $F \colon A \to B$ of $\mathbf{qCat}$, there is a lift $(A,a) \to (B,Fa)$ in $\mathbf{Rep}$ given by the commutative square whose bottom map is $F \colon A \to B$, and whose top map is the functor $F/a \colon A/a \to B/Fa$ induced by $F$. To show that this morphism is cocartesian with respect to the forgetful functor, we must show that, for any object $(C,c)$ of $\mathbf{Rep}$, the commutative square of Kan complexes $\require{AMScd}$ \begin{CD} \mathrm{Hom}_{\mathbf{Rep}}((B,Fa),(C,c)) @>(F/a)^*>> \mathrm{Hom}_\mathbf{Rep}((A,a),(C,c))\\ @V V V @VV V\\ \mathrm{Hom}_\mathbf{qCat}(B,C) @>>F^*> \mathrm{Hom}_\mathbf{qCat}(A,C) \end{CD} is a homotopy pullback square. Since the vertical maps are Kan fibrations, it suffices to check this on fibres, i.e., that for every functor $G \colon B\to C$, the restriction map $$\mathrm{Hom}_B(B/Fa,G^*(C/c)) \to \mathrm{Hom}_A(A/a,(GF)^*(C/c))$$ is an equivalence of Kan complexes. But this follows from the Yoneda lemma, which shows that this map is equivalent to the identity on the right-hom space $\mathrm{Hom}_C^R(GFa,c)$. (Again, I can explain this in more detail, if you wish.)

Hence the forgetful functor $\mathbf{Rep} \to \mathbf{qCat}$ is sent by the homotopy coherent nerve functor to a cocartesian fibration.

3. In Kerodon, Lurie constructs a universal cocartesian fibration, which he denotes by $\mathcal{Q}\mathcal{C}_*^\mathrm{lax} \to \mathcal{Q}\mathcal{C}$. Here $\mathcal{Q}\mathcal{C}$ is the homotopy coherent nerve of the Kan-enriched category $\mathbf{qCat}$, and $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$ is defined as the "pith" of a certain ∞-bicategory. (See https://kerodon.net/tag/020S .) Explicitly, an $n$-simplex of $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$ is a simplicial functor $\mathfrak{C}(\Delta^{n+1}) \to \mathbf{sSet}$ whose value at $0$ is $\Delta^0$ and whose restriction to $\mathfrak{C}(\Delta^{\{1,\ldots,n+1\}})$ factors through $\mathbf{qCat}$. The universal cocartesian fibration sends such an $n$-simplex to the $n$-simplex of $\mathcal{Q}\mathcal{C}$ given by that restricted simplicial functor.

Thus an object of $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$ is the same thing as an object of $\mathbf{Rep}$, but a morphism $(A,a) \to (B,b)$ in $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$ is a functor $F \colon A \to B$ together with a morphism $g \colon Fa \to b$ in $B$. Such a morphism is cocartesian iff $g$ is an isomorphism in $B$.

Now comes the hard part, which is to define an explicit morphism of cocartesian fibrations over $N(\mathbf{qCat})$ from $N(\mathbf{Rep})$ to $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$. One can begin to construct such a morphism, call it $T$, in low dimensions by hand, but I haven't yet worked out how to give an explicit combinatorial description in all dimensions. On $0$-simplices, $T$ is the identity, i.e. sends $(A,a)$ to $(A,a)$. On $1$-simplices, $T$ sends a morphism $(F,\widetilde{F}) \colon (A,a) \to (B,b)$ in $N(\mathbf{Rep})$, i.e. a commutative square \begin{CD} A/a @>\widetilde{F}>> B/b \\ @VVV @VVV\\ A @>>F> B \end{CD} to the morphism $(F,\widetilde{F}(a)) \colon (A,a) \to (B,b)$ in $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$ whose second component $\widetilde{F}(a) \colon Fa \to b$ is the image under $\widetilde{F}$ of the final object of $A/a$. One can continue to define $T$ on $2$-simplices, but it becomes more involved, and I haven't yet figured out to describe $T$ for all $n$-simplices.

To see that $T$ preserves cocartesian morphisms, it suffices to observe that it sends the cocartesian lift $(F,F/a) \colon (A,a) \to (B,Fa)$ in $N(\mathbf{Rep})$ described in section 2 above to the morphism $(A,a) \to (B,Fa)$ in $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$ given by the functor $F \colon A \to B$ and the identity morphism on $Fa$ in $B$, which is cocartesian by what was said above.

4. It remains to show that the morphism of cocartesian fibrations $T$ (not yet fully) defined in section 3 is an equivalence on fibres. Let $A$ be a quasi-category. The fibre of the cocartesian fibration $N(\mathbf{Rep}) \to N(\mathbf{qCat})$ above $A$ is $N(\mathbf{Rep}_A)$, i.e. the homotopy coherent nerve of the full subcategory of $\mathbf{sSet}/A$ spanned by the projections $A/a \to A$, for $a \in A$. Whereas the fibre of Lurie's universal cocartesian fibration over $A$ is the left-hom quasi-category $\mathrm{Hom}^L_{N(\underline{\mathbf{qCat}})}(\Delta^0,A)$. (Here $\underline{\mathbf{qCat}}$ denotes the quasi-category enriched category of quasi-categories.) Both of these fibres are equivalent to $A$. Moreover, the induced map between these fibres is bijective on objects, and it will hopefully not be too much work to show that it is also an equivalence on homs, once the map $T$ above has been fully defined.

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    $\begingroup$ N.B. The simplicial category named $\mathfrak{C}(\Delta^n)$ in this answer (and in Kerodon) is the "co-dual" of the simplicial category that usually goes by this name (e.g. in HTT). This is to make sure that the morphisms in $\mathcal{Q}\mathcal{C}_*^\mathrm{lax}$ are what you want them to be. $\endgroup$ Feb 6 at 11:59
  • $\begingroup$ I think 2 needs no further explanation. The salient point as I understand is that $C/c$ is right fibrant and terminal object inclusions are right anodyne, so e.g. $\mathrm{Hom}_{\mathbf{qCat}/C}(G_!(B/Fa), C/c)$ is equivalent to the fiber of $C/c$ over $GFa$. $\endgroup$ Feb 6 at 21:49
  • $\begingroup$ I think I've worked through 1 (the fine details always trip me up with using 4.2.4.4). The idea is to construct $\mathbf{Rep}$ as a full subcategory of $(\mathbf{sSet}^+)^{[1]}$. By HTT 3.1.1.6, we can see $(C/c)^\natural \to C^\natural$ is a fibration between fibrant objects, and so gives an injectively fibrant diagram, so $\mathbf{Rep}$ is a full subcategory of the injectively fibrant-cofibrant objects. 4.2.4.4 shows taking the projectively fibrant-cofibrant objects models correctly models the exponential, but the categories projective and injective fibrant-cofibrant objects are equivalent. $\endgroup$ Feb 6 at 22:33
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    $\begingroup$ One thing I've been wondering is if there is a way to bypass the need to work things out on higher simplices. The classifying functor for Rep exists... so if the fibers over points and edges of $Cat_\infty$ are correct, can we says that's enough to show the classifying functor is essentially surjective and fully faithful? $\endgroup$ Feb 6 at 22:44
  • $\begingroup$ @UniversalCocartesianFibration Yes, that’s a good observation. Though I’m not confident that fibres over objects and morphisms suffice. $\endgroup$ Feb 7 at 0:26
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I think I've sketched out most of a proof.

The yoneda embedding $X \to PSh(X)$ is a natural transformation in $X$, and thus $X \to RFib(X)$ is natural, where $f : X \to Y$ acts as $f_!$. Consequently, we can apply the equivalence

$$ \widehat{Cat}_\infty^{Cat_\infty}(id, PSh) \cong (\widehat{Cat}_{\infty})^{cocart}_{/Cat_\infty}(Z, \int RFib(-)) $$

to induce a cocartesian functor $Z \to \int RFib(-)$ whose fibers are the fully faithful yoneda embedding. Since each $f_!$ preserves representables, this should imply that the full subcategory of $\int RFib(-)$ spanned by them is still a cocartesian fibration, and HTT 2.3.4.4 shows it's equivalent to $Z$.

So, now the goal is to show $\int RFib(-)$ is the full subcategory of $Cat_\infty^{[1]}$ spanned by the right fibrations.

By HTT 5.5.3.3, the cocartesian fibration $\int RFib(-)$ is a presentable fibration. Similarly, the cocartesian fibration $eval_1 : Cat_\infty^{[1]} \to Cat_\infty$ is a presentable fibration. (in particular, both are also cartesian fibrations, classified by a functor where morphisms act by pullback)

Since pullbacks of right fibrations are right fibrations, we have an inclusion $RFib(X) \to (Cat_\infty)_{/X}$ that is natural in $X$ where morphisms act by pullback, and so also a natural localization $(Cat_\infty)_{/X} \to RFib(X)$ where morphisms act by left adjoint to pullback.

The Grothendieck construction thus induces a cartesian functor $\int RFib(-) \to Cat_\infty^{[1]}$ whose fibers are the fully faithful inclusion of right fibrations in the slice category, and we argue again by HTT 2.3.4.4 that $\int RFib(-)$ is equvialent to the full subcategory of $Cat_\infty^{[1]}$ spanned by the right fibrations.

Thus, we have a sequence of fully faithful functors $Z \to \int RFib(-) \to Cat_\infty^{[1]}$ that respect the projection to $Cat_\infty$.

Remark The two embeddings are cocartesian and cartesian functors respectively, so they cannot be expected to correspond to a natural transformation. However, they should correspond to a lax natural transformation; there's probably some way to connect this to the description of $Z$ as a lax coslice category.

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