aa r X i v : . [ m a t h . A T ] J un On complicial homotopy monoids
Ryo Horiuchi
For a Kan complex K and a vertex v ∈ K , we have the notion of the n -thhomotopy group π n ( K, v ). This notion has been playing a big role in geometry.In this paper, for a weak complicial set X in the sense of [4], a vertex x ∈ X and n ≥
1, we construct a monoid τ n ( X, x ) which is a generalization of homotopygroup.A stratified simplicial set is a pair of a simplicial set and a subset of itssimplices with certain conditions. In [4] Verity constructed a model structure onthe category of stratified simplicial sets, in which bifibrant objects are preciselyweak complicial sets, and showed that any Kan complex can be viewed as aweak complicial set. That is to say, the model structure is a generalization ofthat for ∞ -groupoids. Therefore it should be natural to try to generalize thenotion of homotopy groups of Kan complexes to weak complicial sets.The intuition used in this short article is that, as in a Kan complex everysimplex is “invertible”, in a weak complicial set every thin simplex is “invert-ible”. We see that under this intuition we can simply apply to weak complicialsets the analogous construction of simplicial homotopy group. Assuming the reader is familiar with simplicial sets, we recall some notationsabout weak complicial sets from [4].
Definition 2.1 ([4]) . A pair ( X, tX ) is a stratified simplicial set if • X is a simplicial set, • tX is a set of simplices in X such that dX ⊂ tX and X ∩ tX = ∅ ,where dX denotes the set of degenerate simplices in X .Let ( X, tX ) and ( Y, tY ) be stratified simplicial sets. A stratified map f :( X, tX ) → ( Y, tY ) is a simplicial map f : X → Y such that f ( x ) ∈ tY for all x ∈ tX . This may be called marked simplicial set in some literature. However this word markedsimplicial set seems to be used for a different notion as well. We follow the nomenclature of[4] to avoid confusion.
1e call X the underlying simplicial set of ( X, tX ) and elements in tX itsthin simplices. However, for the simplicity, we often write X for a stratifiedsimplicial set ( X, tX ) omitting tX . Also we denote the category of stratifiedsimplicial sets and stratified maps by Strat. Example 2.2.
Every simplicial set X defines stratified simplicial sets ( X, dX )and ( X, S n ≥ X n ). Each assignment gives rise to a functor Simp → Strat,which is left (resp. right) adjoint to the forgetful functor, where Simp denotesthe category of simplicial sets and simplicial maps.As the standard simplicial sets and their horns play a role, in particularthey give the definition of quasi-category, in the theory of simplicial sets, weneed the following specific stratified simplicial sets. For stratified simplicial sets(
X, tX ) and (
Y, tY ), we say that (
X, tX ) is a regular stratified simplicial subsetof (
Y, tY ) if X ⊂ Y as simplicial sets and tX = X ∩ tY . Definition 2.3 ([4]) . Let n be a natural number and k ∈ [ n ] . • The standard thin n -simplex ∆[ n ] t is the stratified simplicial set whoseunderlying simplicial set is the standard simplicial set ∆[ n ] and t ∆[ n ] t = d ∆[ n ] ∪ { Id [ n ] }• The k -complicial n -simplex ∆ k [ n ] is the stratified simplicial set whose un-derlying simplicial set is the standard simplicial set ∆[ n ] and t ∆ k [ n ] = d ∆[ n ] ∪ { α ∈ ∆[ n ] |{ k − , k, k + 1 } ∩ [ n ] ⊂ Im( α ) }• The n − -dimensional k -complicial horn Λ k [ n ] is the regular stratifiedsimplicial subset of ∆ k [ n ] generated by the set of faces { δ i | i ∈ [ n ] \ k }• ∆ k [ n ] ′′ (resp. Λ k [ n ] ′ ) is the stratified simplicial set whose underlying sim-plicial set is the same as that of ∆ k [ n ] (resp. Λ k [ n ] ) and its thin simplicesare t ∆ k [ n ] (resp. t Λ k [ n ] ) with all its n − -simplices • ∆ k [ n ] ′ := ∆ k [ n ] ∪ Λ k [ n ] ′ . These stratified simplicial sets define the notion of weak complicial set, whichis the subject of this paper.
Definition 2.4 ([4]) . A stratified simplicial set is called a weak complicial set if it has the right lifting property with respect to the following inclusions: • Λ k [ n ] ֒ → ∆ k [ n ] for n ≥ and k ∈ [ n ] , If we write Λ k [ n ] simp for the simplical horn, this stratified simplicial set Λ k [ n ] is differ-ent from both of (Λ k [ n ] simp , d Λ k [ n ] simp ) and (Λ k [ n ] simp , S n ≥ Λ k [ n ] simp ) in general. Theunderlying simplicial set of Λ k [ n ] is Λ k [ n ] simp . This is also called complicial set in some literature. But again we follow the nomenclaturein [4]. ∆ k [ n ] ′ ֒ → ∆ k [ n ] ′′ for n ≥ and k ∈ [ n ] . In [4], it is shown that every quasi-category can be vied as a weak complicialset. Moreover, in [5], it is shown that every strict ω -category can be viewed asa weak complicial set (via Street’s ω -nerve functor). Therefore weak complicialset is a common generalization of ( ∞ , ω -category.In particular weak complicial set is a generalization of ∞ -groupoid that ishomotopy theoretically equivalent to topological space, so we may take weakcomplicial sets as spaces in which (higher) cells are not necessarily invertible.In addition, Verity constructed in [4] a model structure on Strat, in which weakcomplicial sets are precisely the bifibrant objects. Hence we already have thehomotopy theory of weak complicial sets. Note that in op. cit. the weak equiv-alences of stratified simplicial sets is defined without using homotopy monoidswe are constructing in the next section.Before going to the next section, we recall the cartesian product of stratifiedsimplicial sets. Definition 2.5 ([4]) . Let X and Y be stratified simplicial sets. Then the carte-sian product X ⊛ Y of them is a stratified simplicial set whose underlying sim-plicial set is X × Y and a simplex ( x, y ) ∈ X ⊛ Y is thin : def ⇔ x ∈ tX and y ∈ tY . We use the notations recalled in the previous section and construct homotopymonoids referring to famous textbooks such as [1] and [3].
Definition 3.1 ([4]) . Let f, g : A → X be stratified maps of stratified simplicialsets. We write f ∼ g if there exits a map H : ∆[1] t → X such that A ⊛ ∆[0] A × d (cid:15) (cid:15) ∼ = / / A f (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ A ⊛ ∆[1] t H / / XA ⊛ ∆[0] A × d O O ∼ = / / A g @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) commutes. We may call this H a (simple) homotopy from f to g . Since our aim is toconstruct homotopy monoids, we may need the notion of relative homotopy aswell. Definition 3.2.
Let f, g : A → X be stratified maps of stratified simplicial setsand B → A an inclusion of stratified simplicial sets. Assume that f | B = g | B . e write f ∼ B g if f ∼ g with H : A ⊛ ∆[1] t → X and A ⊛ ∆[1] t H / / XB ⊛ ∆[1] t ?(cid:31) O O proj / / B f | B = g | B O O commutes. Lemma 3.3.
The relation ∼ is an equivalence relation for vertices in a weakcomplicial set.Proof. Let X be a weak complicial set. For any vertex x in X we can take theconstant 1-simplex at x that is thin, then ∼ is reflexive.Assume x ∼ y and y ∼ z with x, y, z are vertices of X . Then we havethin 1-simplices H from x to y and H ′ from y to z . These give rise to a mapΛ [2] → X which lifts to ∆ [2] → X since X is a weak complicial set. Thus weobtain a 1-simplex from x to z . As H and H ′ are thin, this is actually a mapfrom ∆ [2] ′ . Hence eventually we have a map ∆ [2] ′′ → X since X is a weakcomplicial set. This map gives a thin 1-simplex from x to z to show that ∼ istransitive.Let x ∼ y with a thin 1-simplex H . Then we have a map Λ [2] → X whichmaps 2-face to H and 1-face to the constant of x . Since X is a weak complicialset, this map defines a map ∆ [2] → X . Since both of the homotopy H and theconstant 1-simplex are thin, this map indeed is a map ∆ [2] ′ → X and again X is a weak complicial set, this lifts to a map ∆ [2] ′′ → X to give a thin 1-simplexfrom y to x .This generalizes to higher simplexes due to the cartesian closedness of weakcomplicial sets, which is proven in [4]. Lemma 3.4.
For stratified maps A → X with X a weak complicial set, therelation ∼ is an equivalence relation. Moreover, if B → A is an inclusionof stratified simplicial sets, the relation ∼ B is also an equivalence relation forstratified maps A → X which coincide each other on B .Proof. By theorem 75 in [4], the closure map hom(
A, X ) → hom( B, X ) is acomplicial fibration between weak complicial sets. Since vertexes in hom(
A, X )correspond to maps A → X , the lemma above proves this one.Then we can define homotopy monoids as follows. Definition 3.5.
Let X be a weak complicial set, x ∈ X a vertex and n ≥ .Then we define the n -th homotopy monoid τ n ( X, x ) to be the set of equivalenceclasses under ∼ ∂ ∆[ n ] of n -simplexes α in X such that ∆[ n ] α / / X∂ ∆[ n ] ?(cid:31) O O / / ∆[0] x O O ommutes. We are going to construct a monoid structure on this set. Consider two n -simplices α and β in X such that α | ∂ ∆[ n ] = β | ∂ ∆[ n ] is the constant at x . Wecan construct a stratified map Λ n [ n + 1] → X such that n − α , n + 1-face maps to β and other faces map to the constant x . Since X is a weakcomplicial set, this lifts to a map θ : ∆ n [ n + 1] → X . In particular we obtainan n -simplex d n ( θ ) with d n ( θ ) | ∂ ∆[ n ] is the constant at x . Remark 3.6.
Note that the non-degenerate n + 1-simplex Id [ n +1] in ∆ n [ n + 1]is thin. Thus the n + 1-simplex “between” α and β is thin. For example, when n = 1, we have the following picture: xx β / / d n ( θ ) ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ x, α ` ` ❆❆❆❆❆❆❆❆ where the n + 1-simplex surrounded by n -simplexes α , β and d n ( θ ) (and con-stants) is thin. Lemma 3.7.
With the notation above, the class [ d n ( θ )] is independent of thechoices of representatives of [ α ] and [ β ] and that of θ .Proof. Suppose α ∼ ∂ ∆[ n ] α ′ with a homotopy H and β ∼ ∂ ∆[ n ] β ′ with a homo-topy H ′ . Then, as we just saw, there are maps θ, θ ′ : ∆ n [ n + 1] → X . Then weconstruct a map ∆ n [ n + 1] ⊛ ∂ ∆[1] t → X which is θ (resp. θ ′ ) when restricted to ∆ n [ n + 1] ⊛ n [ n + 1] ⊛ ∂ ∆[1] t is the regular stratified simplicial subset of ∆[1] t whose underlyingsimplicial set is the boundary ∂ ∆[1].Also we construct a mapΛ n [ n + 1] ⊛ ∆[1] t → X using α , α ′ , H , β , β ′ and H ′ . More precisely, when restricted to Λ n [ n + 1] ⊛ n [ n + 1] ⊛
1) this map is the one that n − α (resp. α ′ ), n + 1-face maps to β (resp. β ′ ) and other faces map to the constant x .These maps give rise to a map(∆ n [ n + 1] ⊛ ∂ ∆[1] t ) ∪ (Λ n [ n + 1] ⊛ ∆[1] t ) → X. By lemma 72 in [4],(∆ n [ n + 1] ⊛ ∂ ∆[1] t ) ∪ (Λ n [ n + 1] ⊛ ∆[1] t ) → ∆ n [ n + 1] ⊛ ∆[1]is a left anodyne extension. Furthermore, looking at 1-simplices of ∆ n [ n + 1] ⊛ ∆[1] and those of ∆ n [ n + 1] ⊛ ∆[1] t , we see that∆ n [ n + 1] ⊛ ∆[1] = ∆ n [ n + 1] ⊛ ∆[1] t . n + 1] × ∆[1]. By definition ( α, β ) ∈ ∆[ n + 1] × ∆[1] is thin in ∆ n [ n + 1] ⊛ ∆[1] t (resp. in ∆ n [ n + 1] ⊛ ∆[1]) if and only if α is thin in ∆ n [ n + 1] and β is thin in∆[1] t (resp. in ∆[1]). Again by definition t ∆[1] t \ t ∆[1] = { Id [1] } and there isno thin 1-simplex in ∆ n [ n + 1] since n ≥ X is a weak complicial set, we obtain a map∆ n [ n + 1] ⊛ ∆[1] t → X to define a homotopy from d n ( θ ) to d n ( θ ′ ).Thus we can define multiplication on τ n ( X, x ) by [ α ][ β ] = [ d n ( θ )]. Theorem 3.8.
This multiplication gives rise to a monoid structure on τ n ( X, x ) .Proof. First we show that this multiplication is associative. Let α , β and γ represent elements in τ n ( X, x ). As above, we obtain an n + 1-simplex θ by α and β , an n + 1-simplex ψ by d n ( θ ) and γ , and an n + 1-simplex φ by β and γ .Referring to the remark above, we see that these data give rise to a mapΛ n [ n + 2] → X such that n − θ , n + 1-face to ψ , n + 2-face to φ and other faces to x . Since X is a weak complicial set, this lifts to a map u : ∆ n [ n + 2] → X . This shows that our multiplication is associative as follows:([ α ][ β ])[ γ ] = [ d n ( θ )][ γ ]= [ d n ( ψ )]= [ d n d n ( u )]= [ α ][ d n ( φ )]= [ α ]([ β ][ γ ]) , where we use the simplicial identity at the third “ = ” and use the definition ofour multiplication of [ α ] and [ d n ( φ )] at the fourth “ = ”.Note that the constant at x defines the unit e , then we obtain a monoidstructure on τ n ( X, x ). Remark 3.9.
This monoid structure on τ n ( X, x ) is not necessarily a groupstructure. For example when n = 1 consider the following picture: xx x > > ⑦⑦⑦⑦⑦⑦⑦ x α ` ` ❅❅❅❅❅❅❅ If α is thin, this picture will be given by a map Λ [2] → X , then it will lift to amap ∆ [2] → X : xx x > > ⑦⑦⑦⑦⑦⑦⑦ β / / x α ` ` ❅❅❅❅❅❅❅ α ][ β ] = e . However, when α is not thin, we do not findits right inverse. The dual argument works for the left inverses and a similarargument works for higher n .However again, as [4, Example 16] shows that we can view Kan complexesas weak complicial sets. More precisely, for a Kan complex A , we obtain thestratified simplicial set ( A, S n ≥ A n ), which is a weak complicial set by defini-tion . We let th ( A ) denote the weak complicial set. Note that, by definition,all n -simplices with n ≥ ( A ) are thin. Corollary 3.10.
For a Kan complex A , its vertex a and n ≥ , its homotopygroup π n ( A, a ) and τ n (th ( A ) , a ) are the same as group. By the same observation we have the following as well.
Corollary 3.11.
Let m ≥ be a natural number, X be a weak complicial setwhose all k -simplices are thin with k ≥ m , and x ∈ X a vertex. Then τ n ( X, x ) is a group for n ≥ m .Proof. Let β be an n -simplices. By assumption it is thin. Thus it gives rises toa map Λ n − [ n + 1] → X which maps n + 1-face to β and others to the constantsat x . Since X is a weak complicial set, this map lifts to a map ∆ n − [ n + 1] → X which gives the left inverse of [ β ]. The dual arguments may give the right inverseof β .Example 57 of [4] shows that every quasi-category can be viewed as a weakcomplicial set via the functor (-) e from the category of quasi-categoies to thatof weak complicial sets. More precisely, for a quasi-category C , we obtain thestratified simplicial set ( C, dC ∪ S n ≥ C n ) and make its specific 1-simplices thinto obtain a stratified simplicial set C e . Theorem 56 in [4] shows that it is a weakcomplicial set. Note that, by construction, any n -simplex in C e with n ≥ Corollary 3.12.
Let C be a quasi-category and c ∈ C a vertex. Then thehomotopy monoid τ n ( C e , c ) has the group structure defined above when n ≥ . Note that in the first chapter of [2] Joyal has defined the fundamental cate-gory of a quasi-category and hence the fundamental monoid of a pointed quasi-category. More precisely, for a quasi-category C and a vertex c ∈ C , a category τ ( C ) is defined by the left adjoint of the nerve functor and called the fundamen-tal category of C . Then we may obtain the endomorphism monoid End τ ( C ) ( c ).It may be reasonable to compare τ ( C e , c ) and End τ ( C ) ( c ).Note also that so far we do not know whether higher homotopy monoidsare commutative in general or not, although as a classical result we know thathigher homotopy monoids for Kan complexes, which are homotopy groups, arecommutative. As is mentioned, this assignment defines a right adjoint functor to the forgetful functorand by definition the image of a weak complicial set under the forgetful functor is a Kancomplex. X , we define τ ( X ) to be the quotientset of X divided by the equivalence relation ∼ . Then, by definition, for aKan complex A , τ (th ( A )) = π ( A ), where π ( A ) denotes the set of connectedcomponents in A . So we may call τ ( X ) the the set of stratified connectedcomponents in X . This is a part of a project suggested by Lars Hesselholt when I was a studentsupervised by him. One of the aims of the project is to enlarge the theory ofhigher algebras so that we can study semirings in it. I appreciate him guidingmyself to this project.
References [1] Paul Goerss, Rick Jardine, Simplicial homotopy theory, Progress in Math-ematics, Birkh¨auser (1996)[2] A. Joyal, Theory of quasi-categories I. In preparation.[3] Peter May, Simplicial objects in algebraic topology, University of ChicagoPress, 1967[4] D. Verity, Weak complicial sets. I. Basic homotopy theory, Adv. Math. 219(2008), no. 4, 1081-1149,[5] D. Verity, Complicial Sets Characterising the Simplicial Nerves of Strict ωω