Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
  
  
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeMar 9th 2011
   • (edited Mar 9th 2011)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexTechnical nonsense ------------------------- Recall that $\Theta_2$ is the full subcategory of $Str2-cat$ consisting of "free 2-categories on pasting diagrams". Consider the category $tSet:=Psh(\Theta_2)$ (I'm calling this $tSet$ to avoid having to write out $\Theta_2$ every time). Consider the presheaf $X$ represented by the object $[2]([2],[2])$. There exists a subpresheaf $\Lambda\subset X$, which is, roughly geometrically, $\Lambda^2_1(\Lambda^2_1,\Lambda^2_1)$. Another way to think about this geometrically is simply as the pasting diagram that generates $[2]([2],[2])$. Now, we may apply Cisinski's machinery in the following way: Let $L$ be the subobject classifier of $tSet$, which, being injective, is fibrant and contractible for every Cisinski-type model structure. The object $L$ 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 $\{0\},\{1\}:\ast \hookrightarrow L$, and their intersection is empty. Also, $L\to \ast$ is the unique morphism to the terminal object, so this determines a separated interval. We let $\partial L$ be the subobject of $L$ induced from the coproduct $\{0\}\coprod \{1\}\to L$. We choose the following homotopy data: Let $M$ be a small set of generating monomorphisms (in the sense that $llp(rlp(M))$ is the class of monomorphisms), which exists by virtue of $tSet$ being a presheaf category. Let $S$ be the set containing just the morphism $\iota:\Lambda\hookrightarrow X$ Then the data of $(L,M,S)$ may be completed to a homotopy structure on $tSet$ by setting $An_L:=llp(rlp(C_L(S,M)))$, where $C_L(S,M)$ is the _set_ of morphisms defined recursively as follows: $$C_0=\{A\times L \coprod_{A\times \{\varepsilon\}} B\times \{\varepsilon\}:A\hookrightarrow B \in M, \varepsilon=0,1\} \cup S$$ For $Y$ any set of morphisms of $tSet$, let $$C(Y)=\{A\times L \coprod_{A\times \partial L} B\times \partial L:A\to B \in Y\}$$ Define $C_n=C(C_{n-1})$ for $n\geq 1$, and let $C_L(S,M)=\bigcup_{n\geq 0} C_n$. It is a lemma of Cisinski that the class $An_L=llp(rlp(C_L(S,M)))$ is a class of anodynes for the cylinder $L$. Further, Cisinski's lemma states that $An_L$ is the absolute smallest weakly saturated class of morphisms containing $S$ that is a compatible class of anodynes for the cylinder $L$. By Cisinski's big theorem in chapter 1.3 of Astérisque, we see that the homotopy structure $(L,An_L)$ generates a combinatorial model structure on $tSet$, and that this is the minimal Cisinski model structure for which $\iota:\Lambda\hookrightarrow X$ is anodyne. Conjecture ------- The resulting model structure is a model for $(oo,2)$-categories and is on the nose equal to Joyal's conjectural model structure for quasi-2-categories as presheaves on $\Theta_2$. Motivation -------------- If we rerun the argument by replacing $\Theta_2$ with $\Delta$, $X$ with $\Delta^2$, and $\Lambda$ with $\Lambda^2_1$, it is known that we recover the Joyal model structure. If further, we add the outer 2-horn inclusions to $S$, we recover the theory of Kan complexes. This, among other things, leads me to believe that we can think of the set $S$ above as encoding "operations" satisfying certain "identities" on fibrant objects. The inclusion $\Lambda\hookrightarrow X$ 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 $(\infty,n)$-categories modeled by $\Theta_n$ for finite $n$ by induction. I'm posting this here to ask you guys if it sounds plausible. Does it?

   Technical nonsense

   Recall that is the full subcategory of consisting of “free 2-categories on pasting diagrams”. Consider the category (I’m calling this to avoid having to write out every time).

   Consider the presheaf represented by the object . There exists a subpresheaf , which is, roughly geometrically, . Another way to think about this geometrically is simply as the pasting diagram that generates .

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

   We choose the following homotopy data: Let be a small set of generating monomorphisms (in the sense that is the class of monomorphisms), which exists by virtue of being a presheaf category. Let be the set containing just the morphism Then the data of may be completed to a homotopy structure on by setting , where is the set of morphisms defined recursively as follows:

   For any set of morphisms of , let

   Define for , and let .

   It is a lemma of Cisinski that the class is a class of anodynes for the cylinder . Further, Cisinski’s lemma states that is the absolute smallest weakly saturated class of morphisms containing that is a compatible class of anodynes for the cylinder .

   By Cisinski’s big theorem in chapter 1.3 of Astérisque, we see that the homotopy structure generates a combinatorial model structure on , and that this is the minimal Cisinski model structure for which is anodyne.

   Conjecture

   The resulting model structure is a model for -categories and is on the nose equal to Joyal’s conjectural model structure for quasi-2-categories as presheaves on .

   Motivation

   If we rerun the argument by replacing with , with , and with , it is known that we recover the Joyal model structure. If further, we add the outer 2-horn inclusions to , we recover the theory of Kan complexes. This, among other things, leads me to believe that we can think of the set above as encoding “operations” satisfying certain “identities” on fibrant objects. The inclusion 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 -categories modeled by for finite by induction.

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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 9th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi 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 $\Theta_2$. There has been a tremendous amount of activity recently by Julie Bergner, Charles Rezk and others on relating [[Theta-space]]-models for $(\infty,n)$-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 ;-)

   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 .

   There has been a tremendous amount of activity recently by Julie Bergner, Charles Rezk and others on relating Theta-space-models for -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 ;-)

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeMar 9th 2011
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexAlright, I sent Julie an e-mail.

   Alright, I sent Julie an e-mail.