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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeAug 7th 2010

    Before asking this question, we must recall some notions:


    Fix an excellent (therefore symmetric monoidal closed, combinatorial, and all monomorphisms are cofibrations (there’s more, but we don’t need it now)) model category SS (note: SS behaves in almost every way like the category of simplicial sets with the cartesian monoidal structure. If you’re not familiar with the definition of excellent, pretend that SS is the category of simplicial sets).

    We say that an SS-enriched category CC is AA-filtered, or equipped with an AA-filtration, for a fixed poset AA if there exists a function r:Ob(C)Ar:Ob(C)\to A such that if r(X)r(Y)r(X)\nleq r(Y), Map C(X,Y)=Map_C(X,Y)=\emptyset where \emptyset is the initial object of SS.

    Any AA-filtered SS-category gives rise to a functor ACat SA\to Cat_S as follows: Let C aC_{\leq a} be the full subcategory of CC spanned by the set of objects XX such that r(X)ar(X)\leq a. This gives a filtered diagram of subcategories of CC, given by the assignment aC aa\mapsto C_{\leq a} (considered as a functor ACat SA\to Cat_S).


    Recall that the model structure on Cat SCat_S is combinatorial (and therefore cofibrantly generated and given as follows:

    Recall that there exists a small set of generators for Cof(S). Fix such a set of generators and call it kk.

    Define [1] ACat S[1]_A\in Cat_S for any object ASA\in S as the category with two objects a,ba,b such that Map(a,a)=1 SMap(a,a)=1_S, Map(b,b)=1 SMap(b,b)=1_S, Map(a,b)=AMap(a,b)=A, and Map(b,a)=Map(b,a)=\emptyset. Define [0][0] to be the SS-category with one object xx such that Map(x,x)=Map(x,x)=\emptyset.

    Define the set QQ to be the set containing the unique map Cat S[0]\emptyset_{Cat_S}\to [0], and all of the induced maps [1] A[1] B[1]_A\to [1]_B induced by generating cofibrations (AB)k(A\to B)\in k

    Then we define:

    Cof(Cat_S)=llp(rlp(Q))

    We define the class of weak equivalences as follows:

    A weak equivalence of SS-categories is an SS-enriched functor f:CDf:C\to D such that:

    • Map C(X,Y)Map D(fX,fY)Map_C(X,Y)\to Map_D(fX,fY) is a weak equivalence in SS

    • For every YDY\in D, there exists an object XCX\in C such that YY is SS-equivalent to F(X)F(X) (homotopy essential surjectivity)

    It’s well-known that these two classes define a model structure on Cat SCat_S.

    (Recall that a morphism gg of an SS-enriched category CC is called an equivalence if the corresponding map hghg in the Ho(S)-enriched category hChC is an isomorphism.)


    Let CC be a model category. Recall that a natural transformation in a diagram category C DC^D is called a weak equivalence (resp. projective fibration) if it is a weak-equivalence object-by-object (resp. a fibration object-by-object). The projective cofibrations are precisely those natural transformations with the left lifting property with respect to all natural transformations that are both fibrations and weak equivalences. If CC is left-proper combinatorial and DD is small, then C DC^D with the projective model structure is likewise.


    HTT Lemma A.3.5.9 states:

    For AA a poset and S an excellent model category, any AA-filtered SS-enriched category CC admits an SS-enriched functor f:CCf:C'\to C such that:

    • f is bijective on objects, and the morphisms Map C(X,Y)Map C(fX,fY)Map_{C'}(X,Y)\to Map_C(fX,fY) are trivial fibrations,
    • CC' is endowed with an induced AA-filtration by composition rfr\circ f,
    • the induced diagram aC aa\mapsto C'_{\leq a} is a projectively cofibrant object of the model category Cat S ACat_S^A

    This is given without proof, but there is a hint that one should use the small object argument. First of all, to show this, do we need to use the SS-enriched version of the small object argument? If so, could someone give me a reference that states and proves this form of the small object argument (see the nLab page for an informal description, but no actual statement or proof)?

    If not, could someone break down exactly how to prove this?

    • CommentRowNumber2.
    • CommentAuthorHarry Gindi
    • CommentTimeAug 8th 2010
    • (edited Aug 8th 2010)

    Nevermind, I got a response from Lurie by e-mail. I guess I’ll expand it out and write this up on an nLab page, since I went to the trouble of writing all of the background up! =)