How to prove a 1-localization of a 1-category is already an $(\infty,1)$-localization?

Question

I don't even know this fits in here or in Mathematics Stack Exchange, but let me ask. I'm new to simplicial stuff, so a good reference would be quite helpful.

Let's say $C$ is a certain category, and that I have combinatorially constructed:

• a relative category $(D(C)\underset{\text{wide}}{\supseteq} W(C)) \in \mathbf{RelCat}$, and
• a functor $F_C\colon D(C) \to C$ that exhibits $C$ as the weak 1-localization of $(D(C),W(C))$.

By this, I mean that for every 1-category $E$, the precomposition functor $$ (F_C)^*\colon \operatorname{Fun}(C,E) \to \operatorname{Fun}(D(C), E) $$ is fully faithful, and its essential image consists of the functors $D(C)\to E$ that map $W(C)$ into the core (i.e., the largest subgroupoid) of $E$.

I have constructed $D(C)$ with meticulous care, and believe that $F_C$ exhibits $C$ as the localization of $(D(C), W(C))$ at the level of $(\infty,1)$-categories. However, I personally find it challenging to demonstrate. Below are some equivalent formulations of "localization at the level of $(\infty,1)$" or "$(\infty,1)$-localization", though I've not verified the proofs of the equivalences:

• Let $N(D(C),W(C)) \in \mathbf{sSet}^+$ denote the marked simplicial set obtained by marking the edges from $W(C)$ in the nerve $N(D(C))$. Let $N(C)^\natural$ stand for the cartesian marked simplicial set with respect to ${!}\colon N(C) \to \Delta[0]$: it is $N(C)$ with its isomorphisms from C marked. Then the induced map $F_C\colon N(D(C), W(C))\to N(C)$ is a weak equivalence in the category of marked simplicial sets $\mathbf{sSet}^+ = \mathbf{sSet}^+/\Delta[0]^\sharp$ with the cartesian model structure.
• The relative functor $F_C\colon (D(C), W(C))\to \min\nolimits^+ C$ is a weak equivalence in the category $\mathbf{RelCat}$ of relative categories endowed with the model structure Quillen equivalent to the model category $\mathbf{ssSet}$ of bisimplicial sets for complete Segal spaces. Here, $\min\nolimits^+ C$ denotes the relative category $(C\underset{\mathrm{wide}}{\supseteq}\{\mathrm{isomorphisms}\})$.
• Let $L^H(D(C),W(C))\in\mathbf{sCat}=\mathbf{sSet}\mbox{-}\mathbf{Cat}$ denote the hammock localization. Enrich $C$ simplicially with discrete simplicial hom-sets. The simplicial functor $L^H(D(C), W(C)) \to C$ induced by $F_C$ is a Dwyer-Kan equivalence.

I've been trying to prove the third formulation. The induced simplicial functor $F_C\colon L^H(D(C),W(C))\to C$ is strictly surjective on objects, but I found it difficult to prove that the functor has simplicial weak equivalences as hom-maps. I considered of contracting the simplicial set of hammocks into a discrete set of "strings" through a (transfinitely long) zig-zag of simplicial homotopies; however I have not succeeded in moving by homotopy the whole simplicial hom-set consistently with the defining colimits.

I have two things to ask:

1. Am I going on a wrong path? That is, is there an criterion for the desired claim that's easier to prove? (To note: $D(C)$ is seemingly not a model category, and $F_C$ is not a reflective localization) Or is it that I'm doing the right thing and that I just don't have the enough understanding of the concerning simplicial hom-sets?
2. Given a simplicial map from a non-Kan complex to a strictly discrete simplicial set, what would you try in order to prove that the map is a simplicial weak equivalence? A reference (including from textbooks or papers) would be very helpful.

Answer 1

A great reference for these types of questions is Cisinski's book Higher categories and homotopical algebra. Definition 2.2.8 on page 35 is for one-categorical localization, and Definition 7.1.2 on page 311 is for $\infty$-categorical localization. Chapter 7 discusses many ways to get at localizations, including via calculus of fractions, hammock localization, and bisimplicial sets (see also chapter 5). Existence of $\infty$-categorical localization (and, yet another way to think about it) is in Proposition 7.1.3. Furthermore, given a functor like your $F_C$, Proposition 7.1.9 gives a characterization for when it's an $\infty$-categorical localization. The subsequent results provide a whole yoga for checking those conditions. Since you write that you're a beginner, I strongly recommend reading this book.