The question is the title.

In what follows, all $2$-categories and $2$-functors will be strict. Let $2-Cat$ denote the categories whose objects are $2$-categories and whose morphisms are $2$-functors. If $\mathcal{A}$ is a $2$-category, then $\pi_{0}(\mathcal{A})$ is the category whose objects are those of $\mathcal{A}$ and whose $1$-cells are obtained from those of $\mathcal{A}$ by identifying two $1$-cells if they are linked by a zigzag of $2$-cells in $\mathcal{A}$. (In other words, the $Hom$ sets of $\pi_{0}(\mathcal{A})$ are given by the $\pi_{0}$ of the $\underline{Hom}$ categories of $\mathcal{A}$.)

Dwyer-Kan equivalences were defined by Dwyer and Kan in the general setting of simplicial categories. In the realm of $2$-categories, a map $u : \mathcal{A} \to \mathcal{B}$ is a Dwyer-Kan equivalence if the two following conditions are satisfied:

$(i)$ For every objects $X$ and $Y$ of $\mathcal{A}$, the functor $\underline{Hom}_{\mathcal{A}}(X,Y) \to \underline{Hom}_{\mathcal{B}}(u(X),u(Y))$, induced by $u$, `is a weak equivalence (which means that its nerve is a simplicial weak equivalence or, which is equivalent, that this induced functor belongs to any basic localizer of $Cat$).

$(ii)$ The functor $\pi_{0}(\mathcal{A}) \to \pi_{0}(\mathcal{B})$, induced by $u$, is essentially surjective.

Note that these conditions imply that not only is $\pi_{0}(u)$ essentially surjective, but it is also an equivalence of categories.

Some people have apparently suggested that the localization of $2-Cat$ with respect to the class of Dwyer-Kan equivalences should give, up to equivalence, the category of $(\infty,1)$-categories. However, I have yet to find someone who could point out a proof in the literature or write a proof themselves when asked the question whether this result is more than folkloric belief. Could somebody provide something more concrete? Note that I have no definite clue whether this result is true or not. Arguments against its validity would be welcome without dismay.

  • $\begingroup$ I do not understand what is wrong with the LaTeX input. (There is no problem when I compile it.) Please someone tell me or edit the source accordingly, thanks! $\endgroup$ – Jonathan Chiche Dec 25 '11 at 9:26
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    $\begingroup$ Interesting question! I missread the question the first time around, so have deleted my non-answer. The content was just that the papers of Barwick and Kan (which can easily be found on the ArXiv) are related to this. $\endgroup$ – Chris Schommer-Pries Dec 25 '11 at 15:56
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    $\begingroup$ I think it's actually easier than what Kan and I do. There is a right Quillen equivalence $\mathrm{Cat}\to\mathrm{S}$ --- where $\mathrm{Cat}$ is equipped with its Thomason model structure and $\mathrm{S}$ is the usual relative category of simplicial sets ---, which is given by $\mathrm{Ex}^2$ of the nerve. This induces a functor $2-\mathrm{Cat}\to \mathrm{S}-\mathrm{Cat}$, which can be seen to be an equivalence of homotopy theories. $\endgroup$ – Clark Barwick Dec 25 '11 at 16:20
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    $\begingroup$ Thanks! I am aware of the Quillen equivalence which you mention, but I fail to see right now the details as to how it lifts to an equivalence between $2-Cat$ and the category of simplicial categories. When I saw your comment I first thought there was some result of which I was not aware, stating that a Quillen equivalence could be lifted up to the level of enriched categories under some assumptions, but as far as I know there is no known model structure on $2-Cat$ whose weak equivalences are Dwyer-Kan equivalences, so I am probably really missing something, and I am afraid it may be obvious. $\endgroup$ – Jonathan Chiche Dec 25 '11 at 18:23
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    $\begingroup$ I don't really mean to suggest that you use such a model structure on $2-\mathrm{Cat}$. The functor given by applying the nerve to each $\mathrm{Hom}$-category is already a weak equivalence-preserving functor $2-\mathrm{Cat}\to\mathrm{S}-\mathrm{Cat}$, and I claim that this is an equivalence of homotopy theories. To obtain a homotopy inverse, choose a homotopy inverse $\Gamma$ to the nerve [in the sense of the brilliant paper of Fritsch and Latch (MR0612870)] that is lax monoidal in which $\Gamma(X)\times\Gamma(Y)\to\Gamma(X\times Y)$ is an equivalence, and apply it to the mapping spaces. $\endgroup$ – Clark Barwick Dec 26 '11 at 1:17

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