19
$\begingroup$

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.

$\endgroup$
7
  • $\begingroup$ 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! $\endgroup$ Dec 25, 2011 at 9:26
  • 1
    $\begingroup$ 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. $\endgroup$ Dec 25, 2011 at 15:56
  • 7
    $\begingroup$ 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. $\endgroup$ Dec 25, 2011 at 16:20
  • 1
    $\begingroup$ 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. $\endgroup$ Dec 25, 2011 at 18:23
  • 6
    $\begingroup$ 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. $\endgroup$ Dec 26, 2011 at 1:17

1 Answer 1

1
$\begingroup$

The paper The enriched Thomason model structure on 2-categories answers the question in the affirmative.

Specifically, it constructs a model structure on the category of 2-categories and proves that the functor from 2-categories to simplicial categories that preserves objects and applies the nerve functor followed by the double subdivision functor to every hom-category is a right Quillen equivalence.

Thus, 2-categories form another model for (∞,1)-categories.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.