Consider a category $C$ enriched in categories. Here we have a natural class $W$ of equivalences, namely those morphisms having inverses up to 2-isomorphisms. One can take the hammock localization $L^H(C,W)$ and get a simplicial category.

On the other hand one can take the maximal subgroupoid of each $hom$-category (or simply suppose that the hom-categories were groupoids from the start, this wouldn't change the above notion of equivalence), apply the nerve functor $hom$-wise and get another simplicial category.

Are these two simplicial categories Dwyer-Kan equivalent?

[I am also interested in a particular case of this question, which would take too long to expose here, which has the feature that the $hom$-categories are preorders, and hence the maximal subgroupoids are simply equivalence relations on the sets of morphisms (i.e. the sets of objects of the hom-categories). Yet I couldn't see by hand that the hom-simplicial sets of the hammock localization are homotopy discrete...]

  • 3
    $\begingroup$ If $C$ has one object and one 1-morphism, the first construction forgets about all the 2-morphisms but the second does not. $\endgroup$ – Eric Wofsey Mar 30 '14 at 23:06
  • 4
    $\begingroup$ More precisely, if we have an abelian group $A$, then we can form a 2-category $\mathbb{B}^2 A$ with a unique object $*$, unique 1-cell $\mathrm{id}_*$, and elements of $A$ as 2-cells. The hammock localisation cannot see the automorphisms of the 1-cell in this case, so the two simplicial categories are Dwyer–Kan equivalent if and only if $A$ is trivial. $\endgroup$ – Zhen Lin Mar 30 '14 at 23:50
  • $\begingroup$ Thank you for this example! You are right that the hammock localization cannot distinguish different 2-isomorphisms. Now if the hom-categories are preorders, the answer could still be affirmative. Any thoughts on this? $\endgroup$ – Peter Arndt Mar 31 '14 at 0:33
  • 2
    $\begingroup$ @PeterArndt There are some further necessary conditions. For instance, we need $\mathcal{C} [\mathcal{W}^{-1}]$ to be isomorphic to the category obtained by identifying isomorphic 1-cells. This fails for e.g. the 2-category with a unique object, a non-trivial involution, and 2-cells making the two 1-cells isomorphic. You could probably get around this by introducing some kind of path or cylinder object. $\endgroup$ – Zhen Lin Mar 31 '14 at 9:22
  • $\begingroup$ @Zhen Lin : This helped me. Thanks again! $\endgroup$ – Peter Arndt Mar 31 '14 at 9:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.