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Small A_∞-dg-categories and small dg-categories admit a model structure and the forgetful functor from dg-categories to A_∞-dg-categories is a right Quillen functor so that the resulting Quillen adjunction is actually a Quillen equivalence, which can be seen as a many-objects version of the Quillen equivalence between A_∞-dg-algebras and dg-algebras, as described, for example, by Berger and Moerdijk, “Axiomatic homotopy theory for operads”.
Is there a written source for this Quillen equivalence?
Good question. What I am aware of are generalizations of Berger-Moerdijk’s construction to (algebras over) coloured operads, hence to the case of an arbitrary but fixed sets of colors/objects. This is for instance in Berger-Moerdijk 06 (where the last example at the end one may think of as giving -categories with precisely two objects, I suppose) and more recently Caviglia 14.
@Urs: Of course, Quillen equivalence for a fixed set of objects follows from Berger and Moerdijk. With some additional effort one can patch these Quillen equivalences together to get a Quillen equivalence for all sets of objects, so I was wondering if somebody has done this before.
I understand that this is what you are asking. I said “What I am aware of…” meaning “…and nothing more”, implying that the answer to your question is “no”. But I’ll check.
Now I have heard back from one person who should know, and indeed the answer is: no, it seems this has not been written down yet.
It is not so easy to patch together these fixed object set model categories. This is similar to what happens in the Bergner model structure on simplicial categories and also in Simpson’s construction of the M-enriched Segal category model structure. In each case the fixed object set model structure is fairly easy to construct, but patching them together is both delicate and difficult. Simpson doesn’t actually construct his global model structure until page 446 of his book.
However, if you somehow can construct both these model structures and the Quillen adjunction between them, then the fact that it is a Quillen equivalence should follow easily from the fixed object set cases.
Something “close” to a model structure for -categories is in Lefèvre-Hasegawa 03, I suppose. (Again, I am not aware of anything beyond that in the literature.)
Great, thanks. I was wondering about this because I wanted to include such a result as an application in my joint paper with Jakob Scholbach about rectification of operads, but was unsure whether or not this is already written up by somebody else.
Somewhere on MO Bruno Valette remarks that the category of -categories taken at face value cannot be a model category, since it is lacking equalizers and hence does not have all small limits. (?)
@Urs but perhaps it is a category of fibrant objects instead? (like the category of Lie groupoids, for instance)
@Urs: Indeed, this is claimed in http://mathoverflow.net/questions/127028/model-structure-on-the-category-of-small-a-infty-categories-hocolims.
“not all equalizers exist”. This confuses me. It is the category of algebras over a small operad in a presentable category, hence presentable. See http://mathoverflow.net/questions/128145/cocompleteness-of-the-category-of-small-a-infty-categories. In particular it is complete and cocomplete.
Perhaps Bruno Valette is using some notion of weak -morphism?
@Chris: Could it be that the problem arises for functors that do not induce isomorphisms on objects? I certainly agree with you that for a fixed set of objects the category of A_∞-categories with this set of objects is locally presentable, in particular is complete and cocomplete, but I don’t see how one can express the category of all A_∞-categories in the same fashion.
No, I am almost certain this is a presentable category, at least if you restrict to small -categories.
I have emailed Bruno for clarification.
@Chris: That was also my impression, but I was confused by Bruno’s claim. Let us know what he thinks.
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