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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeMar 9th 2011
    • (edited Mar 9th 2011)

    Technical nonsense

    Recall that Θ 2\Theta_2 is the full subcategory of Str2catStr2-cat consisting of “free 2-categories on pasting diagrams”. Consider the category tSet:=Psh(Θ 2)tSet:=Psh(\Theta_2) (I’m calling this tSettSet to avoid having to write out Θ 2\Theta_2 every time).

    Consider the presheaf XX represented by the object [2]([2],[2])[2]([2],[2]). There exists a subpresheaf ΛX\Lambda\subset X, which is, roughly geometrically, Λ 1 2(Λ 1 2,Λ 1 2)\Lambda^2_1(\Lambda^2_1,\Lambda^2_1). Another way to think about this geometrically is simply as the pasting diagram that generates [2]([2],[2])[2]([2],[2]).

    Now, we may apply Cisinski’s machinery in the following way: Let LL be the subobject classifier of tSettSet, which, being injective, is fibrant and contractible for every Cisinski-type model structure. The object LL defines a separated interval object in the following canonical way: Since the terminal object, *\ast has exactly two subobjects *\emptyset \hookrightarrow \ast and **\ast \hookrightarrow \ast, we have exactly two morphisms {0},{1}:*L\{0\},\{1\}:\ast \hookrightarrow L, and their intersection is empty. Also, L*L\to \ast is the unique morphism to the terminal object, so this determines a separated interval. We let L\partial L be the subobject of LL induced from the coproduct {0}{1}L\{0\}\coprod \{1\}\to L.

    We choose the following homotopy data: Let MM be a small set of generating monomorphisms (in the sense that llp(rlp(M))llp(rlp(M)) is the class of monomorphisms), which exists by virtue of tSettSet being a presheaf category. Let SS be the set containing just the morphism ι:ΛX\iota:\Lambda\hookrightarrow X Then the data of (L,M,S)(L,M,S) may be completed to a homotopy structure on tSettSet by setting An L:=llp(rlp(C L(S,M)))An_L:=llp(rlp(C_L(S,M))), where C L(S,M)C_L(S,M) is the set of morphisms defined recursively as follows:

    C 0={A×L A×{ε}B×{ε}:ABM,ε=0,1}SC_0=\{A\times L \coprod_{A\times \{\varepsilon\}} B\times \{\varepsilon\}:A\hookrightarrow B \in M, \varepsilon=0,1\} \cup S

    For YY any set of morphisms of tSettSet, let

    C(Y)={A×L A×LB×L:ABY}C(Y)=\{A\times L \coprod_{A\times \partial L} B\times \partial L:A\to B \in Y\}

    Define C n=C(C n1)C_n=C(C_{n-1}) for n1n\geq 1, and let C L(S,M)= n0C nC_L(S,M)=\bigcup_{n\geq 0} C_n.

    It is a lemma of Cisinski that the class An L=llp(rlp(C L(S,M)))An_L=llp(rlp(C_L(S,M))) is a class of anodynes for the cylinder LL. Further, Cisinski’s lemma states that An LAn_L is the absolute smallest weakly saturated class of morphisms containing SS that is a compatible class of anodynes for the cylinder LL.

    By Cisinski’s big theorem in chapter 1.3 of Astérisque, we see that the homotopy structure (L,An L)(L,An_L) generates a combinatorial model structure on tSettSet, and that this is the minimal Cisinski model structure for which ι:ΛX\iota:\Lambda\hookrightarrow X is anodyne.


    The resulting model structure is a model for (oo,2)(oo,2)-categories and is on the nose equal to Joyal’s conjectural model structure for quasi-2-categories as presheaves on Θ 2\Theta_2.


    If we rerun the argument by replacing Θ 2\Theta_2 with Δ\Delta, XX with Δ 2\Delta^2, and Λ\Lambda with Λ 1 2\Lambda^2_1, it is known that we recover the Joyal model structure. If further, we add the outer 2-horn inclusions to SS, we recover the theory of Kan complexes. This, among other things, leads me to believe that we can think of the set SS above as encoding “operations” satisfying certain “identities” on fibrant objects. The inclusion ΛX\Lambda\hookrightarrow X that I chose above seems to encode composition of 1-morphisms, composition of 2-morphisms, and the interchange law.

    At least what I’m hoping is that the same magic that happens for quasicategories and Kan complexes works for quasi-2-categories. If it works (pretty big “if”!), I think it should be fairly straightforward to generalize the idea to (,n)(\infty,n)-categories modeled by Θ n\Theta_n for finite nn by induction.

    I’m posting this here to ask you guys if it sounds plausible. Does it?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2011

    Hi Harry,

    I can’t make a useful technical comment right now, but have a general comment, concerning for instance

    Joyal’s conjectural model structure for quasi-2-categories as presheaves on Θ 2\Theta_2.

    There has been a tremendous amount of activity recently by Julie Bergner, Charles Rezk and others on relating Theta-space-models for (,n)(\infty,n)-categories in all kinds of ways to all kinds of other models. I am not up to speed with this, though, only know that Julie and others told me about exciting developoments.

    I expect you can get a really good answer by emailing her or Charles Rezk (if he doesn’t see you on MO, already ;-)

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeMar 9th 2011

    Alright, I sent Julie an e-mail.