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Infinity Groupoids RESEARCH PROPOSAL A MODEL FOR HOMOTOPY TYPES EVGENY MUSICANTOV Abstract. In the following we propose an alternative model for homotopy types. We dened a category 1Grp and a class of weak equivalences. We prove that the homotopy category of new objects is equivalent to the usual homotopy category of simplicial sets. We hope that, these new objects will provide a natural language to talk about homotopy types. 1. Defense I would like to dene a model for homotopy types (here also called 1-groupoids). I hope that the new model will provide a natural language to talk about and to dene homotopy types. 1.1. What does this model consists of: i I will dene a category 1Grp, whose objects will be called 1-groupoids ii and a class of morphisms Conn ⊂ 1Grp ( thought as of weak equivalences ) iii homotopy hypothesis: there is a functor y : Kan ! 1Grp such that after localizing at weak equivalences in both sides the functor becomes an equivalence of categories more precisely we have that following Theorem 1. There is a unique, up to a unique isomorphism, functor Y : sSets[W−1] ! 1Grp[Conn−1] such that the following diagram is commutative Y sSet[W−1] / 1Grp[Conn−1] e 7 π Y 0 Kan Moreover, the functor Y is an equivalence of categories. 1.2. Denition of the category 1Grp. We dened the category 1Grp as a full category of functor F un(sSetsf ; Grp). Denition 1. A strict functor op will be called an -groupoid if X : sSetsf ! Grp 1 (i) given a family fAigi2Λ of simplicial sets, the natural functor X(tAi) ! ΠiX(Ai) is an equivalence of groupoids. (ii) given a cobration A,! B, the functor X(B) ! X(A) is a bration of groupoids. 1 RESEARCH PROPOSAL A MODEL FOR HOMOTOPY TYPES 2 (ii0) given a trivial cobration A,! B, the functor X(B) ! X(A) is a trivial bration of groupoids. (iii) given a pushout diagram in simplicial sets A / B C such that at least one of the maps is a cobration, the induced map X(B [A C) ! X(B) ×X(A) X(C) is a connected morphism. ( which will be dened in a moment). We dened the category 1Grp as a full category of functor F un(sSetsf ; Grp) spanned by 1-groupoids. 1.3. Connected morphisms. Denition 2. A functor f : G ! G0 of groupoids is called connected if it has right 0 1 1 lifting property with respect to fπgr(;) ! πgr(∆ ); πgr(@∆ ) ! πgr(∆ )g. Remark 1. A functor f : G ! G0 of groupoids is connected if and only if it is surjective on objects, surjective on morphisms and each ber is connected. In particular, it follows that for an 1-groupoid X and a pushout diagram A / B C such that at least one of the maps is a cobration, the induced map π0(X(B [A C)) ! π0(X(B) ×X(A) X(C)) One could think about connected maps as a weaker version of equivalences. Example 1. BG ! ∗ is a basic example of connected morphism which is not an equivalence. 1.4. To motivate the requirement of connectedness in (iii). 1.we give the main example of 1-groupoid namely, Example 2. Let be a Kan complex. Let op be a functor K y(K): sSetsf ! Grp dened by y(K)(t) = πgr(Hom(t; K)). We have that y(K) is an 1-groupoid. here πgr is the Poincare groupoid and Hom is the internal home functor. 2. state the following Lemma 1. Let K f g L / K0 RESEARCH PROPOSAL A MODEL FOR HOMOTOPY TYPES 3 be a diagram of Kan sets, such that at least one of the maps f,g is a Kan bration. Then, the canonical map of groupoids 0 0 c : πgr(K ×K L) ! πgr(K) ×πgr (K ) πgr(L) is connected. 1.5. Examples. The following is a motivational example of an 1-groupoid. Example 3. Let be a Kan complex. Let op be a functor K y(K): sSetsf ! Grp dened by y(K)(t) = πgr(Hom(t; K)). We have that y(K) is an 1-groupoid. Let f : K ! L be a map in Kan. The map f induces a map y(f): y(K) ! y(L) in 1Grp. Hence, we have build a functor y : Kan ! 1Grp. Let us consider the following example. Example 4. Let be an -groupoid and . Then A op , X 1 A 2 sSets X : sSetsf ! Grp dened by XA(t) = X(A × t), is also an 1-groupoid. Here is another set of examples coming from stable homotopy theory. Example 5. Let C 2 Ch≤0(Z) be non-positively graded chain complex. op Dene C~ 2 F un(sSets ; Grp) by C~(A) = π (Hom ≤0 ( A; C)). The func- f gr Ch (Z) Z tor C~ is an 1-groupoid. ( Let ZA 2 Ch≤0(Z) be the free chain complex of abelian groups on the simplicial ≤0 set A. We denote by Hom ≤0 (·; ·) the internal home in Ch ( ).) Ch (Z) Z Of course, we can get an equivalent 1-groupoid by looking at y(ZA), where ZA 2 sAb is the associated simplicial abelian group which is a Kan complex if one forgets the group structure. One can generalized the above example. Example 6. Let be a connective spectrum. Consider ~ op S S 2 F un(sSetsf ; Grp) dened by the truncation of the mapping spectra S~(t) := τ [−1;0](St). The functor S~ is an 1-groupoid which is equivalent to y(Ω1S): (In order to dene S~ the spectrum S does not have to be connective, yet the 1-groupoid S~ depends only on the connective truncation of S.) 1.6. To summarize. We gave a denition of 1Grp, the category of 1-groupoids. We dened a natural functor y : Kan ! 1Grp. The class of connected morphisms Conn ⊂ 1Grp is the class we want to localize in order to get a category equivalent to sSets[W−1]. To motivate localization at Conn let us state the following Lemma 2. Let y : Kan ! 1Grp the canonical functor dened above. Let f : K ! K0 be a bration in between Kan sets, then f is a trivial bration if and only if y(f) is connected. 1.7. Now we are ready to (re)state the result: the new model is equivalent to the standard one. 1.8. Model theoretic approach. A la Rezk-Dugger. Let Man be a category of topological manifolds, essentially small category . There is a natural Grothendieck topology on Man. RESEARCH PROPOSAL A MODEL FOR HOMOTOPY TYPES 4 One has the local model structure on the category of simplicial presheaves sP se(Man). Let us denote this model category by Shv(Man). One can further localize the model category Shv(Man) by homotopy equiva- lences on Man. Namely, one inverts all the projections M × I ! M, here I is the unit interval. The resulting model structure is referred as the I-local model struc- ture on Shv(Man) and denoted by Shv(Man)I . One can think of Shv(Man)I as a model for the 1-category of locally constant sheaves on Man. One has the following pair of adjoint simplicial functors ~· : sSets Shv(Man)I :Γ where A~(M) = A is the constant presheaf functor and Γ(F) = (R(F))(?) is the derived global sections functor. Here ? denoted a 0-dimensional manifold and R(F) is a functorial brant replacement of F. 1.8.1. This gives rise to evaluation of homotopy type on a manifold. On the other hand one can use the above. Theorem 2 (Dugger). The pair is a Quillen equiva- ~· : sSets Shv(Man)I :Γ lence. I am trying to apply similar ideas to the situation at hand. I would like to have a model structure on op such -groupoids are brant and connected F un(sSetsf ; Grp) 1 morphisms are weak equivalences. Towards this end, consider the following pair of adjoint functors op Yd : sSets F un(sSetsf ; Grp): Ob where Yd(S) is a level-wise discrete groupoid given by the Yoneda embedding n and (Ob(F ))n := Ob(F (∆ )). Conjecture 1. There is a model structure on op such that the class F un(sSetsf ; Grp) of weak equivalences is the class of connected morphisms and all the 1-groupoids objects are brant. With respect to that model structure on op , the F un(sSetsf ; Grp) pair (Yd; Ob) is a Quillen equivalence. 1.9. Questions and further. How to characterize an 1-category. Every K 2 Kan should be an innity category and if C and D are 1-category there should be the category of functor CD. Finally, for every category Ho : 1Cat ! Cat. Thus every category dened a functor Kan 7! Cat. Question: is the functor 1Cat ! F un(Kan; Cat) is faithful? How to describe the image? These question can be stated rigorously, since 1Cat has a model, complete Segal space for example. 1.9.1. Axiomatization of homotopy types. Given a well developed higher topos the- ory. • To every X 2 S (space, given as Kan set) we associate an topos (a hyper- complete topos), Loc(X), of locally constant sheaves on X, also equivalent to the comma topos S=X • Given X; Y 2 S Homgeom(Loc(X); Loc(Y )) is an 1-groupoid and the nat- ural map HomS (X; Y ) ! Homgeom(Loc(X); Loc(Y )) is an equivalence. RESEARCH PROPOSAL A MODEL FOR HOMOTOPY TYPES 5 • The imbedding S! RT opt−complete is fully faithful.
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