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Already every category with weak equivalences (see there) presents an (∞,1)-category, and up to equivalence all (∞,1)-categories arise this way.
The extra structure of a model category on top of the weak equivalences is just there to make the presentation easier to handle.
I though I had once put a rmark to that effect into the entry. But maybe not.
You mean the two different constructions give you the same (infinity,1)-category?
All standard constructions give equivalent (∞,1)-categories. Yes.
This is the statement of this theorem in the entry derived hom-space.
(The exposition of this clearly can do with some polishing…)
True, but I thought what arsmath was objecting to was the phrase “making it a simplicial model category.” Not every model category (as far as I know) can be given the structure of a simplicial model category on the same underlying category, although you can usually find a Quillen equivalent simplicial model category.
Well, I have edited the page (∞,1)-category to clarify the point I raised in #5, so that the section in question only refers to simplicial model categories. I know that from a non-simplicial model category one can construct the hom-spaces using framings, but those don’t have a strictly associative composition in general.
And yes, all the constructions should be the same.
the phrase “making it a simplicial model category.”
Oh, I wasn’t reading carefully. That is of course wrong. Thanks for fixing that.
Somebody should eventually clean up the discussion of the situation.
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