Explaining the "free left fibration" functor for infinity categories

Question

This is a cross-post from here

I am reading A. Mazel-Gee's paper "All about the Grothendieck construction". In that paper he explains that the left adjoint ${\mathrm{Cat}_{\infty}}_{/\mathcal{C}}\to \mathrm{coCFib}(\mathcal{C})$ (from $\infty$-categories over $\mathcal{C}$ to cocartesian fibrations over $\mathcal{C}$) to the forgetful functor is the functor that sends $F:\mathcal{D}\to \mathcal{C}$ to the "free cocartesian fibration on F" $$\mathrm{Fun}([1],\mathcal{C})\times_{\mathcal{C}}\mathcal{D}\to\mathcal{C}$$ I am now wondering if there is a similar explicit description for the left adjoint ${\mathrm{Cat}_{\infty}}_{/\mathcal{C}}\to \mathrm{LFib}(\mathcal{C})$. This would be the composite of the previous functor with the reflexive localization $L:\mathrm{coCFib}(\mathcal{C})\to \mathrm{LFib}(\mathcal{C})$. Now by the results in the paper we have a commutative diagram of large $\infty$-categories $$\require{AMScd}\begin{CD}\mathrm{Fun}(\mathcal{C},\mathrm{Cat}_\infty) @>{(=)^{gpd}\circ -}>> \mathrm{Fun}(\mathcal{C},\mathcal{S})\\ @V{Gr}V{\simeq}V @V{Gr}V{\simeq}V \\ \mathrm{coCFib}(\mathcal{C}) @>{L}>> \mathrm{LFib}(\mathcal{C}),\end{CD}$$ where $\mathcal{S}$ is the $\infty$-category of spaces, $Gr$ denotes the Grothendieck construction and $(=)^{gpd}$ is the groupoidification functor.

This implies by the naturality of the Grothendieck construction that the fibers of $L(\mathcal{D}\to\mathcal{C})$ over $x$ identify with $(\mathcal{D}_x)^{gpd}$. But it is not straight-up groupoidification as that would take us to $\mathcal{S}_{/\mathcal{C}^{gpd}}$. If I understand correctly the description of the Grothendieck construction as a lax colimit then the functor L should be some kind of "free groupoidification of the fibers". But this is not as explicit as I would like : can we describe this process without referring to the functor by which the coCartesian fibration is classified ?

On the level of model categories, this is presented by the Quillen adjunction $${\mathrm{Set}_{\Delta}^+}_{/\mathcal{C}^\sharp} \leftrightarrows {\mathrm{Set}_\Delta}_{/\mathcal{C}}$$ between the functor forgetting the marked edges and the functor marking all edges ; the model structures are the marked one and the covariant one, respectively. Therefore the functor $L$ is given by a fibrant replacement of a coCartesian fibration $\mathcal{D} \to \mathcal{C}$ in ${\mathrm{Set}_\Delta}_{/\mathcal{C}}$. Do we have explicit such replacements ?

Answer 1

As suggested by David White, I emailed A. Mazel-Gee. Let me paraphrase his answer : We claim that given a cocartesian fibration $F:\mathcal{D}\to\mathcal{C}$, the free left fibration $LF:\mathcal{E}\to\mathcal{C}$ on $F$ is just given by inverting in $\mathcal{D}$ the morphisms sent to equivalences in $\mathcal{C}$. We will use Corollary 3.11 in this paper by Ayala & Francis. The natural map $\mathcal{D}\to \mathcal{E}$ is a map of coCartesian fibrations so we just have to check that the induced maps on fibers $\mathcal{D}_x\to\mathcal{E}_x$ for $x\in\mathcal{C}$ are localizations. But as I said in my original post, we have $\mathcal{E}_x=(\mathcal{D}_x)^{gpd}$ ; thus $\mathcal{D}\to \mathcal{E}$ is a localization. Now left fibrations reflect equivalences so a morphism in $\mathcal{D}$ get inverted in $\mathcal{E}$ if and only if it gets inverted in $\mathcal{C}$.

Answer 2

It's actually something you already know: It is the fibrewise groupoidification of the free cartesian fibration. The free cartssian fibration functor sends a functor $$p:A\to B\mapsto p': A\downarrow B\to B.$$ This is totally classical and due originally iirc to Ross Street. The thing to look for is the "slice 2-monad".