RESEARCH PROPOSAL A MODEL FOR HOMOTOPY TYPES
EVGENY MUSICANTOV
Abstract. In the following we propose an alternative model for homotopy types. We de ned a category ∞Grp 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 de ne a model for homotopy types (here also called ∞-groupoids). I hope that the new model will provide a natural language to talk about and to de ne homotopy types.
1.1. What does this model consists of: i I will de ne a category ∞Grp, whose objects will be called ∞-groupoids ii and a class of morphisms Conn ⊂ ∞Grp ( thought as of weak equivalences ) iii homotopy hypothesis: there is a functor y : Kan → ∞Grp 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] → ∞Grp[Conn−1] such that the following diagram is commutative
Y sSet[W−1] / ∞Grp[Conn−1] e 7 π Y 0
Kan Moreover, the functor Y is an equivalence of categories. 1.2. De nition of the category ∞Grp. We de ned the category ∞Grp as a full category of functor F un(sSetsf , Grp). De nition 1. A strict functor op will be called an -groupoid if X : sSetsf → Grp ∞ (i) given a family {Ai}i∈Λ of simplicial sets, the natural functor X(tAi) → ΠiX(Ai) is an equivalence of groupoids. (ii) given a co bration 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 co bration 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 co bration, the induced map
X(B ∪A C) → X(B) ×X(A) X(C) is a connected morphism. ( which will be de ned in a moment).
We de ned the category ∞Grp as a full category of functor F un(sSetsf , Grp) spanned by ∞-groupoids. 1.3. Connected morphisms.
De nition 2. A functor f : G → G0 of groupoids is called connected if it has right 0 1 1 lifting property with respect to {πgr(∅) → πgr(∆ ), πgr(∂∆ ) → πgr(∆ )}. 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 ∞-groupoid X and a pushout diagram A / B
C such that at least one of the maps is a co bration, 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 ∞-groupoid namely, Example 2. Let be a Kan complex. Let op be a functor K y(K): sSetsf → Grp de ned by y(K)(t) = πgr(Hom(t, K)). We have that y(K) is an ∞-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 ∞-groupoid. Example 3. Let be a Kan complex. Let op be a functor K y(K): sSetsf → Grp de ned by y(K)(t) = πgr(Hom(t, K)). We have that y(K) is an ∞-groupoid. Let f : K → L be a map in Kan. The map f induces a map y(f): y(K) → y(L) in ∞Grp. Hence, we have build a functor y : Kan → ∞Grp. Let us consider the following example.
Example 4. Let be an -groupoid and . Then A op , X ∞ A ∈ sSets X : sSetsf → Grp de ned by XA(t) = X(A × t), is also an ∞-groupoid. Here is another set of examples coming from stable homotopy theory.
Example 5. Let C ∈ Ch≤0(Z) be non-positively graded chain complex. op De ne C˜ ∈ F un(sSets , Grp) by C˜(A) = π (Hom ≤0 ( A, C)). The func- f gr Ch (Z) Z tor C˜ is an ∞-groupoid. ( Let ZA ∈ 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 ∞-groupoid by looking at y(ZA), where ZA ∈ 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 ∈ F un(sSetsf , Grp) de ned by the truncation of the mapping spectra S˜(t) := τ [−1,0](St). The functor S˜ is an ∞-groupoid which is equivalent to y(Ω∞S). (In order to de ne S˜ the spectrum S does not have to be connective, yet the ∞-groupoid S˜ depends only on the connective truncation of S.) 1.6. To summarize. We gave a de nition of ∞Grp, the category of ∞-groupoids. We de ned a natural functor y : Kan → ∞Grp. The class of connected morphisms Conn ⊂ ∞Grp 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 → ∞Grp the canonical functor de ned 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 ∞-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) ∞ 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 ∞-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 ∞-category. Every K ∈ Kan should be an in nity category and if C and D are ∞-category there should be the category of functor CD. Finally, for every category Ho : ∞Cat → Cat. Thus every category de ned a functor Kan 7→ Cat. Question: is the functor ∞Cat → F un(Kan, Cat) is faithful? How to describe the image? These question can be stated rigorously, since ∞Cat 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 ∈ 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 ∈ S Homgeom(Loc(X), Loc(Y )) is an ∞-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. • Given a topos X and a homotopy type K the category HomGeom(X ,K) is a groupoid, called cohomology of X with coe cients in K. If X is has a constant shape than the functor K → HomGeom(X ,K) is representable by the shape.
2. Definitions, examples, and statement of results 2.1. De nitions. Let sSets denote the category of simplicial sets and let Grp denote the 1-category of groupoids.
Let πgr : sSets → Grp denote the fundamental groupoid functor, de ned as follows. Given a T ∈ sSets, the objects of πgr(T ) are T0 and morphisms of πgr(T ) are generated by T1 subjects to the relations: d2σ ◦ d0σ = d1σ for each σ ∈ T2. 1 Let sSetsf denote the category of small simplicial sets. Next, we de ne the category ∞Grp of ∞-groupoids to be a full subcategory of op . F un(sSetsf , Grp) Remark 2. In the following de nition we refer to the usual Kan-Quillen model structure on sSets, and to the usual model structure on Grp. The functor πgr has a right adjoint, we denoted it by N. One has that with respect to the above model structures the pair of functors is a Quillen adjunction. πgr : sSets Grp : N De nition 3. A strict functor op will be called an -groupoid if X : sSetsf → Grp ∞ (i) for every family {Ai}i∈Λ of simplicial sets, the natural functor X(tAi) → ΠiX(Ai) is an equivalence of groupoids. (ii) for every co bration A,→ B, the functor X(B) → X(A) is a bration of groupoids. (ii0) for every trivial co bration A,→ B, the functor X(B) → X(A) is a trivial bration of groupoids. (iii) for every pushout diagram in simplicial sets
A / B
C such that at least one of the maps is a co bration, the induced map
X(B ∪A C) → X(B) ×X(A) X(C) is a connected morphism.2
To motivate the requirement of connectedness in (iii) we state the following Lemma 3. Let
1a simplicial set is said to be small if the cardinality of its non-degenerate simplexes is bounded ג xed cardinal by some 2here the bered product is the strict bered product of groupoids RESEARCH PROPOSAL A MODEL FOR HOMOTOPY TYPES 6
K
f g L / K0 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. In the above de nition, we used some terminology about functors between groupoids. The following is a deciphering taken form [Holl01].
De nition 4. A functor f : G → G0 of groupoids is called 0 1 • a bration if it has right lifting property with respect to πgr(∆ ) → πgr(∆ ) • a trivial bration if it is a bration and an equivalence of categories 0 1 • connected if it has right lifting property with respect to {πgr(∅) → πgr(∆ ), πgr(∂∆ ) → 1 πgr(∆ )} Remark 3. 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 ∞-groupoid X and a pushout diagram
A / B
C such that at least one of the maps is a co bration, the induced map
π0(X(B ∪A C)) → π0(X(B) ×X(A) X(C)) is a bijections. Remark 4. The above de nition of connected functors between groupoid is not pure categorical. Since, for example, we demand surjectivity on objects. How- ever, every essentially surjective functor of groupoids with connected bers can be suitably replaced by a connected functor. This is a common situation when one considers brant objects in a model cate- gory on something that should be considered as a 2-category, e.g. the above model category structure on the 1-category of groupoids. The same remark applies to brations and trivial brations of groupoids, where we also demanding surjectivity on objects. In sequel, I will try to replace those conditions by more purely ones. 2.2. Examples. The following is a motivational example of an ∞-groupoid. Example 7. Let be a Kan complex. Let op be a functor K y(K): sSetsf → Grp de ned by y(K)(t) = πgr(Hom(t, K)). We have that y(K) is an ∞-groupoid. Let us consider the following example. RESEARCH PROPOSAL A MODEL FOR HOMOTOPY TYPES 7
Example 8. Let be an -groupoid and . Then A op , X ∞ A ∈ sSets X : sSetsf → Grp de ned by XA(t) = X(A × t), is also an ∞-groupoid. Let Kan denote the full subcategory of Kan sets inside sSets. Let f : K → L be a map in Kan. The map f induces a map y(f): y(K) → y(L) in ∞Grp. Hence, we have build a functor y : Kan → ∞Grp. Here is another set of examples coming from stable homotopy theory.
Example 9. Let C ∈ Ch≤0(Z) be non-positively graded chain complex. Let ZA ∈ Ch≤0(Z) be the free chain complex of abelian groups on the simplicial set A. We ≤0 denote by Hom ≤0 (·, ·) the internal home in Ch ( ). Ch (Z) Z op De ne C˜ ∈ F un(sSets , Grp) by C˜(A) = π (Hom ≤0 ( A, C)). The func- f gr Ch (Z) Z tor C˜ is an ∞-groupoid. Of course, we can get an equivalent ∞-groupoid by looking at y(ZA), where ZA ∈ 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 10. Let be a connective spectrum. Consider ˜ op S S ∈ F un(sSetsf , Grp) de ned by the truncation of the mapping spectra S˜(t) := τ [−1,0](St). The functor S˜ is an ∞-groupoid which is equivalent to y(Ω∞S). In order to de ne S˜ the spectrum S does not have to be connective, yet the ∞-groupoid S˜ depends only on the connective truncation of S. 2.3. Maps between ∞-groupoids. We de ne several classes of maps between ∞-groupoids to re ect the correspond- ing classes of maps in the category Kan. De nition 5. A map F : X → Y of ∞-groupoids is called (1) bration if for each t ∈ sSets the functor F (t) is a bration of groupoids. (2) connected if for each t ∈ sSets the functor F (t) is connected. Example 11. Let f : K → L be a bration between Kan complexes, y(f) is a bration in ∞Grp. Moreover, if f was a trivial bration to begin with, then y(f) is connected, actually y(f)(t) is equivalence for each t ∈ sSetsf . We think about connected morphisms between ∞-groupoid as weak equivalences and eventually we would like to localize the category ∞Grp with respect to them. Let us state the following Lemma for motivational purposes. Lemma 4. Let y : Kan → ∞Grp the canonical functor de ned 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.
Example 12. Let X be an ∞-groupoid and let g : A,→ B ∈ sSetsf be a co bra- tion, then ∗ B A is a bration of -groupoids. Moreover, if is happened gX : X → X ∞ g to be trivial co bration then the map ∗ is connected. gX 2.4. Equivalence of homotopy categories. Let W be the class of weak homotopy equivalence in sSets, and Conn be the class of connected morphisms in ∞Grp. Let π : Kan → sSets[W−1] be the composition of the inclusion Kan ,→ sSets and the localization sSets → sSets[W−1]. Finally, let Y 0 : Kan → ∞Grp[Conn−1] be the composition of y : Kan → ∞Grp and the localization ∞Grp → ∞Grp[Conn−1]. RESEARCH PROPOSAL A MODEL FOR HOMOTOPY TYPES 8
Proposition 1. There is a unique, up to a unique isomorphism, functor Y : sSets[W−1] → ∞Grp[Conn−1] such that the following diagram is commutative
Y sSet[W−1] / ∞Grp[Conn−1] e 7 π Y 0
Kan Moreover, the functor Y is an equivalence of categories. Remark 5. A more rigorous statement of the above is the following. De ne a groupoid G those object are pairs {F, ν} such that F : sSets[W−1] → ∞Grp[Conn−1] ν : F ◦ π → Y 0 is a isomorphism of functors, the morphisms in G((F, ν), (F 0, ν0)) are isomorphism of functors f : F → F 0 such that the appropriate diagram is commutative. The claim is that G is a contractible groupoid. 2.5. Model theoretic approach. The following I've learned form [Dugg99]. Let Man be a category of topological submanifolds of R∞, so Man is a small category. There is a Grothendieck topology on Man de ned by open coverings of topological manifolds. 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 ∞-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. Theorem 3 (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) ∞ 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 2. 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 ∞-groupoids objects are brant. With respect to that model structure on op , the F un(sSetsf , Grp) pair (Yd, Ob) is a Quillen equivalence. RESEARCH PROPOSAL A MODEL FOR HOMOTOPY TYPES 9
3. Proofs 3.1. Proof of Proposition 1. Let us recall that the functor Y 0 : Kan → ∞Grp[Conn−1] was de ned by the y following composition Kan / ∞Grp / ∞Grp[Conn−1] . We have the following
Claim 1. The functor Y 0 : Kan → ∞Grp[Conn−1] maps weak equivalences to isomorphisms.
Thus, the functor Y 0 descents to a functor y[W−1]: Kan[W−1] → ∞Grp[Conn−1]. Eventually, we would like to prove that y[W−1] is an equivalnce. Therefore, we would like to have a functor in the opposite direction. Towards this end, we de ne
De nition. Let K : F un(sSetsf , Grp) → sSets be a functor de ned by K(X)n := Ob(X(∆n)). The functor K maps ∞-groupoids to Kan sets. Thus, we have the functor (denoted by the same letter) K : ∞Grp → Kan.
Claim 2. There is a canonical natural transformation c : id∞Grp → y ◦ K, such that each cX : X → y(K(X)) is connected. Let f : X → Y be a connected map in ∞Grp. Consider the following commuta- tive diagram:
c X X/ y(K(X))
f y(K(f))
cY Y / y(K(X)) ∼ It follows that, given t ∈ sSets, the map [t, K(f)] = π0(y(K(f))(t)) is a bijection. Thus, the map K(f): K(X) → K(Y ) is a weak equivalence, being a map between Kan sets and an isomorphism in Ho(sSets). Therefore, K maps connected morphisms in ∞Grp to weak equivalences in Kan. In particular, the functor K descents to K : ∞Grp[Conn−1] → Kan[W −1]. There is an evident isomorphism K ◦ y → idKan. This, together with connec- −1 −1 tivity of c : id∞Grp → y ◦ K, proves that the functor y[W ]: Kan[W ] → ∞Grp[Conn−1] is an equivalence. Next, we need to construct a functor form sSets to ∞Grp, which will be an equivalnce on the homotopy level. Toward this end, we choose a functorial brant replacement ν : sSets → Kan, (any replacement will do). Finally, we de ne Y := y[W−1] ◦ ν. It is immediate to verify that Y has the required properties.
3.2. Proof of Claim 1. Let h : K → L ∈ Kan be a weak equivalence. It is known that h has an inverse up to homotopy. Thus, it is enough to prove the following.
Claim. Let f, g : K → L be homotopic maps in Kan. We have that y(f) = y(g) in ∞Grp[W−1]. RESEARCH PROPOSAL A MODEL FOR HOMOTOPY TYPES 10
1 Proof. Fix a homotopy H : K → L∆ . We have that the two natural maps
∂1 1 y(L)∆ / y(L) are connected. The maps ∂ and ∂ have the same right inverse / 1 0 ∂0 s ∆1 y(L) / y(L) , namely, ∂1 ◦ s = ∂0 ◦ s = idy(L). Hence, ∂1 and ∂0 are equal 1 1 in ∞Grp[Conn−1]. Since y(L∆ ) =∼ y(L)∆ , we have that y(H) is a homotopy −1 between y(f) and y(g). Thus, y(f) = y(g) in ∞Grp[Conn ].
3.3. The natural transformation c : id∞Grp → y ◦ K. Let X ∈ ∞Grp, let us denote the Kan set K(X) by KX .
Claim 3. There is a unique family of maps ν(t): Ob(X(t)) → Ob(y(KX )(t)) parametrized by t ∈ sSetsf , with the following properties: (i) it compatible with structure maps, namely, for each g : t → s ∈ sSetsf , the following diagram commutes
ν(s) Ob(X(s)) / Ob(y(KX )(s))
Ob(X(g)) Ob(y(KX )(g)) ν(t) Ob(X(t)) / Ob(y(KX )(t))
n (ii) ν(∆ ) = idOb(X(∆n)), which makes sense by the de nition of K. Proof. The proof is an application of the properties (i) and (iii) in the de nition of ∞Grp.
By Lemma 5 below, the maps ν(t) can be extended to a maps in ∞Grp, functorial in X. Namely, ν extends to a natural transformation c : id∞Grp → y ◦ K.
Claim 4. Let X ∈ ∞Grp the above de ned map cX : X → y(K(X)) is connected.
Proof. follows from Lemma 5 below.
Lemma 5. Let X ∈ ∞Grp and let t ∈ sSetsf . Let f : p → q ∈ X(t).
(i) Let
∂ , ∂ : X(t × ∆1) / X(t): s 1 0 o / be functors induced from
i , i : t / t × ∆1 : p 0 1 o /
There exist a unique natural transformation ηt : ∂1 → ∂0 such the ηt ◦ s = id. The transformation ηt is functorial in t ∈ sSets. 1 (ii) There exist F ∈ X(t × ∆ ) such that ηt(F ) = f. 1 (iii) Let F1,F2 ∈ X(t × ∆ ) be objects such that ηt(Fi) = f. Then, there exists 1 a map a : F1 → F2 ∈ X(t × ∆ ) such that ∂1(a) = idp and ∂0(a) = idq. RESEARCH PROPOSAL A MODEL FOR HOMOTOPY TYPES 11
3.4. Proof of Lemma 4. Lemma 6. Let F be a Kan set. We have that F is contractible if and only if for t every t ∈ sSets the fundamental groupoid πf (F ) is connected. Remark 6. This is Lemma 4 stated for f : F → ∗. Proof. (of Lemma 4) First, suppose that f is trivial bration, then each for every t ∈ sSets the functor y(f)(t) is an equivalence. Hence, y(f) is connected. Now, assume that y(f)(t) is connected for every t ∈ sSets. Let k0 ∈ K0 be a point and let F denote the ber f −1(k0). Then the ber (f t)−1(k0) of f t : Kt → K0t is t t t . It follows, by Lemma 3, that the canonical arrow 0t F πgr(F ) → πgr(K )×πgr (K ) ? t is connected. The groupoid 0t is connected as well, being a ber πgr(K ) ×πgr (K ) ? t of a connected functor. Thus, πgr(F ) is connected for every t ∈ sSets. Hence, F is contractible. A bration with contractible bers is a trivial bration.
Proof. (of the Lemma 6) First, we observe that the group π0(F ) is trivial. (∆1/∂∆1) Let f0 ∈ F . The group π1(F, f0) is trivial, as πgr(F ) is connected and hence there is a path between any loop around f0 and the constant path s(f0). Let n ≥ 2 and let t = ∆n/∂∆n. Consider ? =∼ ∂∆n/∂∆n ,→ ∆n/∂∆n = t. It induces the following pullback square
t t F ×F ? / F
f0 ? / F We have the following part of the long exact sequence
t t π1(F ) → π0(F ×F ?) → π0(F ) → π0(F ) t t ∼ Since the groups π0(F ) and π1(F ) are trivial, the group π0(F ×F ?) = πn(F, f0) is trivial. Hence F is contractible. 3.5. Proof of Lemma 5. It is enough to prove the Lemma for t = ?, since Xt is an ∞-groupoid on its own rights and functoriality in t is evident. (i) First, since s is an equivalence, in particular essentially surjective, thus, if such η exists it is unique. 1 Let F ∈ X(∆ ). Since ∂1 is a trivial bration, there exists a map φ : s(∂1(F )) → in 1 , such that . De ne . F X(∆ ) ∂1(φ) = id∂1(F ) η(F ) = ∂0(φ) 1 Let ψ : F1 → F2 a map in X(∆ ), and let φi : s(∂1(Fi)) → Fi be such ∂1(φi) = for . id∂1(Fi) i = 1, 2 There following diagram
ψ F1 / F2 O O
φ1 φ2
s∂1ψ s(∂1(F1)) / s(∂1(F2))
becomes commutative after applying ∂1. Since the functor ∂1 is an equivalence, the original diagram is commutative. Applying ∂0 to this diagram one get that η is a natural transformation. RESEARCH PROPOSAL A MODEL FOR HOMOTOPY TYPES 12
(ii) Consider the following functor of groupoids
1 1 ∼ ∂1 × ∂0 : X(∆ ) → X(∂∆ ) = X(∗) × X(∗) induced by ∂∆1 ,→ ∆1. 1 The functor ∂1 × ∂0 is a bration. Thus, there exists F ∈ X(∆ ) and a map
idp×f s(p) → F over p × p / p × q . We have that η(F ) = f. (iii) Since ∂1 is a trivial bration, there is a map a : F1 → F2 such that ∂1(a) = idp. Since η(F1) = η(F2), it follows that ∂0(a) = idq. References [DS95] B. Chorny, Brown representability for space-valued functors, http://arxiv.org/abs/0707.0904W. G. Dwyer and J. Spalinski, Homotopy the- ories and model categories, Handbook of algebraic topology (I. M. James, ed.), Elsevier Science B. V., 1995. [Dugg99] D. Dugger, Sheaves and homotopy theory, http://pages.uoregon.edu/ddugger/cech.html [GZ67] Pierre Gabriel, Michel Zisman, Calculus of Fractions and Homotopy Theory, Ergeb- nisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) [Holl01] Sharon Hollander, A Homotopy Theory for Stacks, http://arxiv.org/abs/math/0110247 [Rezk05] CHARLES REZK, TOPOSES AND HOMOTOPY TOPOSES, http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf