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nLab > nLab General Discussions: Too strong statement?

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    • CommentRowNumber1.
    • CommentAuthorarsmath
    • CommentTimeJan 20th 2011
    • (edited Jan 20th 2011)
    Is the following passage true (from the (infinity, 1)-category page)? The only result I've seen is that you can turn nice cofibrantly generated model categories into simplicial model categories.

    This indeed turns out to be true: there is a precise sense in which every model category presents an (∞,1)-category.
    * given a model category A;
    * it becomes canonically an SSet-enriched category making it a simplicial model category A;
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2011
    • (edited Jan 20th 2011)

    Already every category with weak equivalences (see there) presents an (∞,1)(\infty,1)-category, and up to equivalence all (∞,1)(\infty,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.

    • CommentRowNumber3.
    • CommentAuthorarsmath
    • CommentTimeJan 20th 2011
    You mean the two different constructions give you the _same_ (infinity,1)-category?
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2011

    You mean the two different constructions give you the same (infinity,1)-category?

    All standard constructions give equivalent (∞,1)(\infty,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…)

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJan 20th 2011

    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.

    • CommentRowNumber6.
    • CommentAuthorarsmath
    • CommentTimeJan 20th 2011
    I was asking about a different construction. If you follow the link in "every model category presents an (∞,1)-category" in the original passage, then it goes to the locally presentable (infinity, 1) category page, which talks about Dugger's construction, not hammock localization. Does that give the same (infinity, 1)-category?
    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJan 20th 2011

    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.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 21st 2011

    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.

    • CommentRowNumber9.
    • CommentAuthorarsmath
    • CommentTimeJan 24th 2011
    Thanks, the revised version is much clearer.

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