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1. • CommentRowNumber1.
   • CommentAuthormrmuon
   • CommentTimeDec 12th 2018
   • PermaLink
   Author: mrmuon
   Format: MarkdownItexI hope that this is the right place to ask a question. I am trying to understand the hammock localization of a (plain) category with respect to a subcategory. I've looked at the original paper by Dwyer and Kan. Given any pair of objects, they define the Hom-set as a simplicial set in which the $k$-simplices are hammocks of width $k$. This defines a simplicial category. It appears to me that one can always cut a width $k$ hammock into $k$ width $1$ hammocks. In other words a $k$-simplex is a concatenation of $1$-simplices. I think that means that this simplicial space is just the nerve of a category, consisting of the width $0$ and $1$ hammocks. This implies that the hammock localization doesn't need to be treated as a simplicial category. It is just a 2-category in disguise. Is this correct? Is there some reason for preferring the simplicial category description? Eli

   I hope that this is the right place to ask a question.

   I am trying to understand the hammock localization of a (plain) category with respect to a subcategory. I’ve looked at the original paper by Dwyer and Kan. Given any pair of objects, they define the Hom-set as a simplicial set in which the $k$-simplices are hammocks of width $k$. This defines a simplicial category.

   It appears to me that one can always cut a width $k$ hammock into $k$ width $1$ hammocks. In other words a $k$-simplex is a concatenation of $1$-simplices. I think that means that this simplicial space is just the nerve of a category, consisting of the width $0$ and $1$ hammocks.

   This implies that the hammock localization doesn’t need to be treated as a simplicial category. It is just a 2-category in disguise.

   Is this correct? Is there some reason for preferring the simplicial category description?

   Eli

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 12th 2018
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   Author: Urs
   Format: MarkdownItexI am sure this is wrong in general, but I forget where to point to for a proof. Others here will know immediately (and then we should add this remark to the entry). But there are special conditions known when it does become true. One such condition is that your homotopical category admits the structure of a "[[category of fibrant objects]]". In that case indeed the 1-step hammocks are sufficient. This is discussed [here](https://ncatlab.org/nlab/show/category+of+fibrant+objects#DerivedHomSpaces).

   I am sure this is wrong in general, but I forget where to point to for a proof. Others here will know immediately (and then we should add this remark to the entry).

   But there are special conditions known when it does become true. One such condition is that your homotopical category admits the structure of a “category of fibrant objects”. In that case indeed the 1-step hammocks are sufficient. This is discussed here.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeDec 12th 2018
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   Author: Mike Shulman
   Format: MarkdownItexI think what goes wrong is that the simplicial set of hammocks of fixed *shape* (i.e. fixed length and choice of which arrows in the zigzag go backwards) is indeed the nerve of a category, but the simplicial set of hammocks of varying shapes is some kind of colimit of all of those, and the colimit of nerves isn't again a nerve. There is some discussion of the hammock localization from a point of view like this one in [[Homotopy Limit Functors on Model Categories and Homotopical Categories]]; this description of the hammock localization is on p103, and the rest of section 35 constructs a related 2-category (using a lax colimit instead, i.e. Grothendieck construction) whose homwise nerve is weakly equivalent to the hammock localization.

   I think what goes wrong is that the simplicial set of hammocks of fixed shape (i.e. fixed length and choice of which arrows in the zigzag go backwards) is indeed the nerve of a category, but the simplicial set of hammocks of varying shapes is some kind of colimit of all of those, and the colimit of nerves isn’t again a nerve. There is some discussion of the hammock localization from a point of view like this one in Homotopy Limit Functors on Model Categories and Homotopical Categories; this description of the hammock localization is on p103, and the rest of section 35 constructs a related 2-category (using a lax colimit instead, i.e. Grothendieck construction) whose homwise nerve is weakly equivalent to the hammock localization.

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeSep 13th 2021
   • (edited Sep 13th 2021)
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   Author: Dmitri Pavlov
   Format: MarkdownItexHas anything ever been written about hammock localizations in the 2-categorical context, with noninvertible 2-morphisms? Say, if we have a 2-category in which some 1-morphisms and 2-morphisms are marked as weak equivalences, there are some obvious candidates for the hammock localization of such a relative 2-category one can write down, and these can be interpreted as categories enriched in the Joyal model structure on simplicial sets. Naturally, one then ask whether this hammock-type construction produces the correct (∞,2)-localization of this relative 2-category. A canonical example is the 2-category of combinatorial model categories, left Quillen functors, and natural transformations, where a 1-morphism is a weak equivalence if it is a left Quillen equivalence, and a 2-morphism is a weak equivalence if its components on cofibrant objects are weak equivalences.

   Has anything ever been written about hammock localizations in the 2-categorical context, with noninvertible 2-morphisms?

   Say, if we have a 2-category in which some 1-morphisms and 2-morphisms are marked as weak equivalences, there are some obvious candidates for the hammock localization of such a relative 2-category one can write down, and these can be interpreted as categories enriched in the Joyal model structure on simplicial sets.

   Naturally, one then ask whether this hammock-type construction produces the correct (∞,2)-localization of this relative 2-category.

   A canonical example is the 2-category of combinatorial model categories, left Quillen functors, and natural transformations, where a 1-morphism is a weak equivalence if it is a left Quillen equivalence, and a 2-morphism is a weak equivalence if its components on cofibrant objects are weak equivalences.