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The bar construction can be used to construct cofibrant replacements in the transferred model structure, under some restrictions on the adjunction under consideration.
Is this discussed anywhere in the literature?
Dunno about the general case, but for the case of dg-modules what you are asking for might be the construction discussed in Barthel-May-Riehl 14
For functor categories it’s in my never-published preprint.
Thanks for the references!
An interesting question here is what is special about these two constructions that makes the bar construction produce a cofibrant replacement.
I guess one could postulate that the adjunction comes from some monad T.
Then it seems like we need a cofibrant replacement functor Q on the underlying category and a natural transformation TQ→QT with some obvious compatibility properties that ensure that the functor Q sends T-algebras to T-algebras.
This seems to be classical enough, but it doesn’t look like there is a reference for this…
Yes, I don’t know of one.
@Dmitri, is this relevant: http://nyjm.albany.edu/j/2013/19-5.html?
@DavidRoberts: Yes, this is quite interesting, thank you.
However, I don’t (yet?) see how to adapt it to my case: Proposition 10 in your paper produces a contractible object, whereas a cofibrant replacement must be weakly equivalent to the original object.
There is, of course, also a question how to make the hocolim cofibrant (with some obvious answers using weighted colimits).
By the way, the article http://ncatlab.org/nlab/show/two-sided+bar+construction#cofibrant_replacement rather boldly claims that the bar construction is a cofibrant replacement, which doesn’t seem right to me.
The bar construction of X is certainly a resolution of X, i.e., its homotopy colimit is canonically weakly equivalent to X, but additional effort (and assumptions) are needed to construct a cofibrant replacement out of it. If nobody objects, I would like to replace “cofibrant replacement” with “resolution” in the above article.
Well, it doesn’t actually say “is”, it says “may be regarded as”. (-: I think the intent was to use “cofibrant replacement” as an umbrella term for a well-behaved left resolution, including contexts more general than a model category (since we haven’t assumed any precise context at that point).
Perhaps it’s an idiosyncrasy of mine, but I reserve “bar construction” for operations which take an object in and yield an object in , and I use “bar complex” for the operation that produces a simplicial object in . The former is literally a cofibrant replacement in some important cases, e.g. diagrams of simplicial sets. (This seems to be insufficiently emphasised.)
Maybe my paper is more related to fibrant replacement: one gets a _co_simplicial resolution. One day I’ll figure out something it’s good for :-)
@Mike Shulman: Of course, but it seems like the word “resolution” would be more appropriate for this context.
@Zhen: IIRC Peter May uses “bar construction” and “simplicial bar construction” respectively.
@Dmitri: Personally, I find the word “resolution” too vague to convey the intended meaning. Maybe something like “left deformation” or “left approximation”?
@Mike Shulman: In their book Dwyer—Hirschhorn—Kan—Smith define left deformations in §39.2.(i) as a generalization of cofibrant replacements and left approximations in §41.1 as a generalization of left derived functors (both for the case of homotopical categories), but I don’t see how to fit the bar construction in such generality into their framework. It seems to me that one still has to throw some sort of cofibrant replacement in the picture.
On the other hand, “left resolution” seems to be quite appropriate here: at least for myself I define a left resolution of an object d∈D as an object c∈C whose image under the left derived functor of a given functor F: C→D is weakly equivalent to d. In our case, the functor F is the colimit functor from simplicial diagrams to the original category, and the bar construction, being simplicially homotopy equivalent to the original object, always satisfies the above property.
IIRC a left deformation is an endofunctor equipped with a weak equivalence to the identity, which is what the bar construction is.
@Mike Shulman: At least the way the nLab currently defines it, the bar construction sends objects to simplicial objects, so it definitely isn’t an endofunctor.
Oh, well, as in #13 I would disagree with that terminology.
Mike, are you saying you prefer to call it a ’simplicial bar construction’, what the nLab calls the bar construction?
That’s the terminology I learned from Peter May, at least.
I have no qualms about renaming it “simplicial bar construction”, but in that case it sounds like someone had then better write down what a “bar construction” is. Who shall it be? ;-)
The terminology I learned was that a bar construction is the geometric realization of a simplicial bar construction. I won’t argue for it especially strongly, but if a “bar construction” is a simplicial object then we need a name for its realization.
For what it’s worth, my proposal is to use “bar resolution” for the simplicial object and “bar construction” for its homotopy colimit.
Keep in mind that there is a general notion of two-sided bar construction where (in reasonable generality) is a monad in a bicategory and is a left -module, is a right -module. I tend to think of bar resolutions in the context of a special type of (simplicial) bar construction, one of the form which is used to actually resolve . (Maybe this is the only type of bar construction you are interested in at the moment.)
There are some other nuances of terminology in the article that I don’t want to press too hard at the moment (because they might be idiosyncratic), but I think what I was calling the bar construction there is a two-sided bar construction of the form where and the associated monad acts on on the right. This construction lives in the category of simplicial algebras (over the monad). The bar resolution , gotten by applying the forgetful functor to the bar construction, is the thing that actually supports the acyclic structure (or coalgebra over the decalage comonad), the thing which puts the “resolve” in “resolution.
That’s a good point; it’s really only a special case of the simplicial two-sided bar construction that can really be called a resolution (or its realization a cofibrant replacement or deformation).
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