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    • CommentRowNumber1.
    • CommentAuthorMirco Richter
    • CommentTimeJan 19th 2013
    • (edited Jan 19th 2013)

    Let C be an (oo,1)-cat and X an object of it. Has someone already given a description of the over category C/X, in terms of categories enriched over Kan simplicial sets?

    I mean if we look at C as a category with Kan simplicial sets as hom’s, how is the over category defined in this picture?

    The hom-set definition relays on the definition of elements f:*hom(.,.), but in the higher context, there are not just elements f:Δ[0]homC(.,.) but there are also the’higher elements’ (higer morphisms of course) f:Δ[n]homC(.,.), so on a first sight this is not straight forward.

    • CommentRowNumber2.
    • CommentAuthorMirco Richter
    • CommentTimeJan 19th 2013
    • (edited Jan 19th 2013)

    If I see it right, the idea in over-(infinity,1)-category doesn’t apply 1:1 to the simplicial enriched setting.

    To see that, suppose C is an (oo,1)-category and X,Y,Y are objects of C. Suppose the objects of C/X are morphisms f1:YX. To speak of a morphism we need to know what f1homC(Y,X) means and in the simplicial category setting, it means

    f1homsSet(Δ[0],homC(Y,X))

    Since the latter is a set ’’ makes sense in the usual way.

    The problem is then the 2-morphism

    Y1Y2f1f2f1X

    What’s that?

    We have 2-morphisms f2homC(Y,X)) and f2homC(Y,X)) defined by

    f2homsSet(Δ[1],homC(Y,X))

    but that is different from the diagram above.

    • CommentRowNumber3.
    • CommentAuthorMirco Richter
    • CommentTimeJan 19th 2013
    • (edited Jan 19th 2013)

    Why not using the following:

    Objects of the slice (oo,1)-category are n-morphisms for any n, or to be more precise objects of C/X are all pairs (Y,fn), where YObj(C) is an object of C and fn:YX is an n-morphism fnhomC(Y,X), which means

    fnhomsSet(Δ[n],homC(Y,X)) for some nN.

    Now the arrows in the slice (oo,1)-category should be Kan simplicial sets.

    For any two objects Y,YC we have the categorical composition law

    :homC(Y,X)sSethomC(Y,Y)homC(Y,X)

    This way we get an arrow (Y,f)(Y,f), which is the natural transformation of simplicial sets t:homC(Y,X)homC(Y,X), restriced to the simplicial subset fn(Δ[n]) of homC(Y,X).

    However we need a Kan simplicial set (Y,f)(Y,f), but we can define that in dimension m by t:fn×Δ[m]f for t as above.

    This is like the inner hom.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 21st 2013
    • (edited Jan 21st 2013)

    Hard to tell for me what you are asking here.

    Maybe the quickest way to get a handle on -slices as Kan-complex enriched categories is this: if you obtain your Kan-complex enriched category from a model category, then the Kan-complex slice category is analogously given by the slice model structure.

    • CommentRowNumber5.
    • CommentAuthorMirco Richter
    • CommentTimeJan 21st 2013
    • (edited Jan 22nd 2013)

    Hmm I thought to proof that this a valid slice category presentation, I should apply Cordiers simplicial nerve functor and then see, if what we get is equivalent to his standard definition of the slice in terms of quasy categories.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 22nd 2013
    • (edited Jan 22nd 2013)

    What is “this” in your last message?

    The proof that the slice model structure presensent the slice -category (if we are slicing over a fibrant representative) is spelled out at slice model structure.

    Or are you after the description of the slice as a quasi-category? That is simply this: if 𝒞 is a quasi-category and X an object, then then n-cells of 𝒞/X are those maps Δn*Δ0𝒞 out of the join of simplicial sets that take the cone point Δ0 to X.

    • CommentRowNumber7.
    • CommentAuthorMirco Richter
    • CommentTimeJan 22nd 2013
    • (edited Jan 22nd 2013)

    this means my proposed definition in post #3, because of what I said in #2 …

    The last part of your last post is clear. This is part of HTT, but not what I’m after.

    I’m talking about:

    data:

    Category enriched over Kan simplicial sets (This is a prsentation of an (oo,1)-category)

    Question:

    Given an object of that category, what is the slice category over that object.

    Can’t see why we have to take model structures into consideration.

    Most likely there is a bug in our communication, somewhere…

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 22nd 2013

    I am not aware of a direct formula for a general Kan-complex enriched category.

    But I said: in case it happens to come to you as the full subcategory of fibrant-cofibrant objects of a simplicial model category, then the slicing is given simply as the ordinary slicing of the model structure over a fibrant representative objects.

    If your Kan-complex enriched category does not arise this way, you can in principle 1. apply the homotopy coherent nerve to obtain the coresponding quasi-category, then 2. produce its slice as above and then 3. turn that again into a Kan complex enriched category by the adjoint of the homotopy coherent nerve.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJan 22nd 2013

    Here’s a more abstract idea: slice categories are a special case of comma categories, which are a weighted limit. Now consider the corresponding weighted homotopy limit in the model category of simplicially enriched categories. This might not work, however, if the latter model category is insufficiently well-behaved — e.g. it is not a monoidal model structure, hence not model-wise enriched over itself, and so it’s not entirely clear what sort of “weighting” one would need to consider. But it’s perhaps a direction worth thinking about.