Cubical rigidification, the cobar construction, and the based loop space
aa r X i v : . [ m a t h . A T ] D ec CUBICAL RIGIDIFICATION, THE COBAR CONSTRUCTION,AND THE BASED LOOP SPACE
MANUEL RIVERA AND MAHMOUD ZEINALIAN
Abstract.
We prove the following generalization of a classical result of Adams:for any pointed path connected topological space (
X, b ), that is not necessarilysimply connected, the cobar construction of the differential graded (dg) coalge-bra of normalized singular chains in X with vertices at b is weakly equivalent as adifferential graded associative algebra (dga) to the singular chains on the Moorebased loop space of X at b . We deduce this statement from several more generalcategorical results of independent interest. We construct a functor C (cid:3) c fromsimplicial sets to categories enriched over cubical sets with connections which,after triangulation of their mapping spaces, coincides with Lurie’s rigidificationfunctor C from simplicial sets to simplicial categories. Taking normalized chainsof the mapping spaces of C (cid:3) c yields a functor Λ from simplicial sets to dg cate-gories which is the left adjoint to the dg nerve functor. For any simplicial set S with S = { x } , Λ( S )( x, x ) is a dga isomorphic to Ω Q ∆ ( S ), the cobar construc-tion on the dg coalgebra Q ∆ ( S ) of normalized chains on S . We use these factsto show that Q ∆ sends categorical equivalences between simplicial sets to mapsof connected dg coalgebras which induce quasi-isomorphisms of dga’s under thecobar functor, which is strictly stronger than saying the resulting dg coalgebrasare quasi-isomorphic. Mathematics Subject Classification (2010). 55U40, 57T30, 16T15.
Keywords. rigidification, cobar construction, based loop space. Introduction
In order to compare two different models for ∞ -categories, Lurie constructs in[Lur09] a rigidification , or categorification , functor C : Set ∆ → Cat ∆ , where Set ∆ denotes the category of simplicial sets and Cat ∆ the category of simplicial categories(categories enriched over simplicial sets). For a standard n -simplex ∆ n the simplicialcategory C (∆ n ) has the set [ n ] = { , , ..., n } as objects and for any i, j ∈ [ n ] with i ≤ j the mapping space C (∆ n )( i, j ) is isomorphic to the simplicial cube (∆ ) × j − i − if i < j , ∆ if i = j , and empty if i > j . In particular, C (∆ n )(0 , n ) ∼ = (∆ ) × n − for n > n − n from 0 to n . Adams described in [Ada52] an algebraic construction,known as the cobar construction, that when applied to a suitable differential gradedcoassociative coalgebra model of a simply connected space X produces a differentialgraded associative algebra (dga) model for the based loop space of X . Adams’ con-struction is based on certain geometric maps θ n : I n − → P ,n | ∆ n | , where P ,n | ∆ n | is the space of paths in the topological n -simplex | ∆ n | from vertex 0 to vertex n ,satisfying a compatibility equation that relates the cubical boundary to the simpli-cial face maps and the Alexander-Whitney coproduct. The definition of C (∆ n )(0 , n )resembles the construction of Adams’ maps θ n and it suggests that behind Adams’ constructions there is a space level story.In this article we describe explicitly the relationship between Lurie’s functor C and Adams’ cobar construction. As a consequence we obtain a generalization ofthe main theorem of [Ada52] to path connected spaces with possibly non-trivialfundamental group. To achieve this, we factor the functor C through a functor C (cid:3) c : Set ∆ → Cat (cid:3) c from the category of simplicial sets to the category of categoriesenriched over cubical sets with connections. If we apply the functor of normalizedcubical chains (over a fixed commutative ring k ) to the mapping spaces of C (cid:3) c weobtain a functor Λ : Set ∆ → dgCat k from simplicial sets to dg categories satisfyingthe following properties. The functor Λ is the left adjoint of the dg nerve functor de-scribed by Lurie in [Lur11]. Moreover, if S is a 0-reduced simplicial set, i.e. S = { x } ,then Λ( S )( x, x ) is a dga isomorphic to Ω Q ∆ ( S ), the cobar construction on the dgcoalgebra Q ∆ ( S ) of normalized simplicial chains with Alexander-Whitney coproduct.From the properties of C (cid:3) c described in the above paragraph we deduce thatΛ( S )( x, x ) and Q ∆ ( C ( S )( x, x )) are weakly equivalent as dga’s, where Q ∆ ( C ( S )( x, x ))is considered as a dga obtained by taking normalized simplicial chains on the simpli-cial monoid C ( S )( x, x ). In fact, Q ∆ ( C ( S )( x, x )) is a dg bialgebra (with Alexander-Whitney coproduct) but we are not concerned with the dg coalgebra structurein this article. From these results, it follows that if f : S → S ′ is a map be-tween 0-reduced simplicial sets such that C ( f ) : C ( S ) → C ( S ′ ) is a weak equiva-lence of simplicial categories (these maps are called categorical equivalences ) then Q ∆ ( f ) : Q ∆ ( S ) → Q ∆ ( S ′ ) is a map of connected dg coalgebras which induces aquasi-isomorphism of dga’s after applying the cobar functor. Maps f : C → C ′ between connected dg coalgebras which induce a quasi-isomorphism of dg algebrasΩ f : Ω C → Ω C ′ after applying the cobar functor Ω are called Ω-quasi-isomorphisms.We apply the preceding discussion to the 0-reduced simplicial set Sing( X, b ) of sin-gular simplices on a path conencted space X with vertices at a fixed point b . Fromthe relationships between C and C (cid:3) , between C (cid:3) and the cobar functor Ω, and fromsome basic homotopy theoretic properties of C , we deduce that Ω Q ∆ (Sing( X, b )) isweakly equivalent as a dga to the singular chains on Ω Mb X , the topological monoidof Moore loops in X based at b . In [Ada52] Adams obtained a similar statement fora simply connected space X using different methods. Our statement does not as-sume X is simply connected and therefore extends Adams’ classical result. The keyhomotopy theoretic property of C that implies our result is the following space levelstatement which lies at the heart of Section 2.2 of [Lur09]: for any path connectedpointed space ( X, b ) there is a weak homotopy equivalence of simplicial monoids be-tween C (Sing( X, b ))( b, b ) and Sing(Ω Mb X ).We believe this extension of Adams’ result has not been observed in the literaturemainly because of the historical development of the cobar construction. We highlighttwo situations in which the simply connected hypothesis comes into play:1) In [Ada52], Adams constructs a map of dg algebras from the cobar constructionof the dg coalgebras of chains to the cubical singular chains on the based loop space. UBICAL RIGIDIFICATION, COBAR CONSTRUCTION, BASED LOOP SPACE 3
Comparison of spectral sequences was the main technique used at the time to mea-sure how far a chain map is from being a quasi-isomorphism. In Adams’ setup, thehypotheses in Zeeman’s spectral sequence comparison theorem hold if the underlyingspace is simply connected and fail in general for spaces with non-trivial fundamentalgroup.2) The cobar construction is not invariant under quasi-isomorphisms of dg coal-gebras. Namely, there are quasi-isomorphisms of dg coalgebras f : C → C ′ forwhich Ω( f ) : Ω C → Ω C ′ is not a quasi-isomorphism of dg algebras. An explicitexample is described in Proposition 2.4.3 of [LoVa12]. However, the cobar construc-tion is invariant under quasi-isomorphisms of simply connected dg coalgebras, i.e.dg coalgebras C for which C ∼ = k and C = 0, as shown in Proposition 2.2.7 of[LoVa12]. Hence, Adams’ main statement regarding the relationship between thecobar construction and the based loop space also holds if we replace singular chainson a simply connected space X with the quasi-isomorphic dg coalgebra of simplicialchains associated to any simplicial set S with no non-degenerate 1-simplices whosegeometric realization is weakly homotopy equivalent to X . This generalization ofthis statement to spaces with non-trivial fundamental group fails.In the non-simply connected case we go around the use of spectral sequencesas described in 1) by turning the problem of showing that two dga’s are quasi-isomorphic into the more fundamental problem of showing that the two simplicialmonoids C (Sing( X, b ))( b, b ) and Sing(Ω Mb X ) are weakly homotopy equivalent. Thenby looking closely at the combinatorics we realize that the simplicial chain com-plex on C (Sing( X, b ))( b, b ) is weakly equivalent as a dga to the cobar construc-tion Ω Q ∆ (Sing( X, b )). We go around 2) by using the following observation: if
Set denotes the category of simplicial sets with a single vertex, then the functor Q K ∆ : Set → dgCoalg k defined by Q K ∆ ( S ) = Q ∆ (Sing( | S | , x )) sends weak homotopyequivalences of simplicial sets to Ω-quasi-isomorphisms of dg coalgebras. Notice that,in general, for any S ∈ Set the connected dg coalgebra of simplicial chains Q ∆ ( S ) isquasi-isomorphic to Q K ∆ ( S ) but not Ω-quasi-isomorphic. Hence, in order to preserveall the homological information of the based loop space, the chains functor should bealways precomposed with a Kan replacement functor and the notion of weak equiv-alences of dg coalgebras should be taken to be Ω-quasi-isomorphisms.We now say a few words regarding how the combinatorics in the construction of C is unraveled and how its cubical version C (cid:3) c is constructed. For any simplicial set S , Dugger and Spivak computed in [DS11] the mapping spaces C ( S )( x, y ) in terms ofnecklaces. A necklace is a simplicial set of the form T = ∆ n ∨ ... ∨ ∆ n k where in thewedge the final vertex of ∆ n i has been glued to the initial vertex of ∆ n i +1 ; a necklacein S from x to y is a map of simplicial sets f : T → S , where T is a necklace, and f sends the first vertex of T to x and the last vertex of T to y . For any necklace T one may associate functorially a simplicial cube C ( T ) and one of the main results in[DS11] is that C ( S )( x, y ) is isomorphic to the colimit of the simplicial sets C ( T ) overnecklaces T in S from x to y . It is tempting to replace the simplicial cubes C ( T )with standard cubical sets of the same dimension to obtain a cubical version of C .However, there are certain maps between necklaces that are not realized by mapsof cubical sets. For example the codegeneracy map s : ∆ → ∆ which collapses MANUEL RIVERA AND MAHMOUD ZEINALIAN the edge [1 ,
2] in ∆ yields a map between simplicial cubes C ( s ) : C (∆ ) → C (∆ )which does not correspond to a codegeneracy map between standard cubical sets.Nonetheless, C ( s ) corresponds to a co-connection morphism, whose definition isrecalled in section 2. Cubical sets with connections were introduced in [BH81] andcan be thought of as cubical sets with extra degeneracies. In section 3 we describeexplicitly the morphisms in the category of necklaces and then in section 4 we explainhow cubical sets with connections arise naturally from necklaces. We use the resultsin sections 3 and 4 and the description of C ( S )( x, y ) in terms of necklaces to define C (cid:3) c in section 5. In section 6 we show that C (cid:3) c gives rise to the functor Λ whichis the left adjoint of the dg nerve functor described by Lurie in [Lur11]. Finally,in section 7 we explain how Λ relates to the cobar construction and how to obtainan algebraic model for the based loop space of a path connected space. We alsoexplain how our results yield a model for the free loop space of a path connectedspace X . Namely, we deduce that the coHochschild complex of the dg coalgebra Q ∆ (Sing( X, b )) is quasi-isomorphic to the singular chains on the free loop space of X .Over a year since the results of this paper were posted on the Arxiv, two otherpreprints ([KaVo18] and [LG18]) discussing a cubical factorization of C also appeared.In [KaVo18], the authors use a cubical version of C to describe a cubical approachto Lurie’s theory of straightening and unstraightening. In [LG18], the author dis-cusses some homotopy theoretic properties of the category of categories enriched overcubical sets with connections using the framework of model categories. Acknowledgments.
We would like to thank Micah Miller who was a part of theearly stages of this project. We would also like to thank Thomas Nikolaus, RonnieBrown, Thomas Tradler, and Gabriel Drummond-Cole for conversations and theircomments. The first author acknowledges support by the ERC via the grant StG-259118-STEIN and the excellent working conditions at
Institut de Math´ematiques deJussieu-Paris Rive Gauche (IMJ-PRG). The second author was partially supportedby the NSF grant DMS-1309099 and would like to thank the
Max Planck Institutefor Mathematics for their support and hospitality during his visits.2.
Preliminaries
Denote by
Set the category of sets. For any small category C denote by Set C the category of presheaves on C with values in Set , so the objects of
Set C are func-tors C op → Set and morphisms are natural transformations between them. Forexample, if ∆ is the category of non-empty finite ordinals with order preservingmaps then
Set ∆ is the category of simplicial sets . We denote by ∆ n the stan-dard n-simplex , so ∆ n is obtained by applying the Yoneda emedding to [ n ], namely∆ n : [ m ] Hom ∆ ([ m ] , [ n ]). Recall that morphisms in the category ∆ are generatedby functions of two types: cofaces d i : [ n ] → [ n + 1], 0 ≤ i ≤ n + 1, and codegenera-cies s j : [ n ] → [ n − ≤ j ≤ n −
1. The Yoneda embedding yields simplicial setmorphisms between standard simplices Y ( d i ) : ∆ n → ∆ n +1 and Y ( s j ) : ∆ n → ∆ n − which we call coface and codegeneracy (simplicial) morphisms . We say a simplicialset S is 0-reduced if the set S is a singleton and we denote by Set be the fullsubcategory of the category Set ∆ of simplicial sets whose objects are 0-reduced sim-plicial sets. UBICAL RIGIDIFICATION, COBAR CONSTRUCTION, BASED LOOP SPACE 5
For any positive integer n , let n be the n -fold cartesian product of copies of thecategory = { , } which has two objects and one non-identity morphism. Denoteby the category with one object and one morphism. We will consider presheavesover the category (cid:3) c which is defined as follows. The objects of (cid:3) c are the categories n for n = 0 , , , ... . The morphisms in (cid:3) c are generated by functors of the followingthree kinds: cubical co-face functors δ ǫj,n : n → n +1 , where j = 0 , , ..., n + 1, and ǫ ∈ { , } ,defined by δ ǫj,n ( s , ..., s n ) = ( s , ..., s j − , ǫ, s j , ..., s n ) , cubical co-degeneracy functors ε j,n : n → n − , where j = 1 , ..., n , defined by ε j,n ( s , ..., s n ) = ( s , ..., s j − , s j +1 , ..., s n ) , and cubical co-connection functors γ j,n : n → n − , where j = 1 , ..., n − n ≥
2, definedby γ j,n ( s , ..., s n ) = ( s , ..., s j − , max( s j , s j +1 ) , s j +2 , ..., s n ) . Objects in the category
Set (cid:3) c are called cubical sets with connections and wereintroduced by Brown and Higgins in [BH81]. For any cubical set with connections K we have a collection of sets { K n := K ( n ) } n ∈ Z ≥ together with cubical face maps ∂ ǫj,n := K ( δ ǫj,n ) : K n +1 → K n , cubical degeneracy maps E j,n := K ( ε j,n ) : K n − → K n , and connections Γ j,n := K ( γ j,n ) : K n − → K n . For simplicity we often drop thesecond index in this notation and, for example, write ∂ j instead of ∂ j,n . Elementsof K n are called n -cells. The structure maps satisfy certain identities described in[BH81]. The standard n-cube with connections (cid:3) nc is the presheaf on (cid:3) c representedby n , namely, Hom (cid:3) c ( , n ) : (cid:3) opc → Set .For a fixed commutative unital ring k denote by Ch k the category of non-negativelygraded chain complexes over k . The tensor product over k defines on Ch k a symmet-ric monoidal structure. We have normalized chains functors Q ∆ : Set ∆ → Ch k and Q (cid:3) c : Set (cid:3) c → Ch k . The definition of Q ∆ is standard; we recall the definition of Q (cid:3) c following [Ant02]. First let C ∗ K be the chain complex such that C n K is the free k -module generated by elements of K n with differential ∂ : K n → K n − defined on σ ∈ K n by ∂ ( σ ) := P nj =1 ( − j ( ∂ j,n − ( σ ) − ∂ j,n − ( σ )). Let D n K be the submoduleof C n K which is generated by those cells in K n which are the image of a degeneracyor of a connection map K n − → K n . The graded module D ∗ K forms a subcomplexof C ∗ K . Define Q (cid:3) c ( K ) to be the quotient chain complex C ∗ K/D ∗ K .The functor Q ∆ : Set ∆ → Ch k lifts to a functor Q ∆ : Set ∆ → dgCoalg k , where dgCoalg k is the category of dg coalgebras over k , via the Alexander-Whitney con-struction as recalled in Section 7. There is a slight abuse of notation throughout thearticle: depending on the context Q ∆ ( S ) may be considered as a chain complex or asa dg coalgebra. For example, by Ω Q ∆ ( S ) we mean the cobar construction of Q ∆ ( S )considered as a dg coalgebra. MANUEL RIVERA AND MAHMOUD ZEINALIAN
The category
Set ∆ has a symmetric monoidal structure given by the cartesianproduct of simplicial sets. We will use the following (non-symmetric) monoidal struc-ture on Set (cid:3) c : for cubical sets with connections K and K ′ define K ⊗ K ′ := colim σ : (cid:3) nc → K,τ : (cid:3) mc → K ′ (cid:3) n + mc . Using the above monoidal structures we may define
Cat ∆ the category of smallcategories enriched over simplicial sets; these are called simplicial categories . Sim-ilarly denote by Cat (cid:3) c the category of small categories enriched over cubical setswith connections; these are called cubical categories with connections . We will alsoconsider the category dgCat k of small categories enriched over chain complexes over k ; these are called dg categories .The symbol ∼ = will always denote isomorphism and ≃ will mean weakly equivalent(in the derived sense) whenever there is a notion of weak equivalence in the underlyingcategory. Namely, we write A ≃ B if there is a zig-zag of weak equivalences between A and B . 3. The category of necklaces
We follow [DS11] for the next definitions and notation. A necklace T is a simplicialset of the form T = ∆ n ∨ ... ∨ ∆ n k where n i ≥ n i has been glued to the initial vertex of ∆ n i +1 . Each ∆ n i is called a bead of T . Since the beads of T are ordered and the vertices of each bead ∆ n i are orderedas well, there is a canonical ordering on the set V T of vertices of any necklace T .We denote by α T and ω T the first and last vertices of the necklace T . A morphism f : T → T ′ of necklaces is a map of simplicial sets which preserves the first and lastvertices. We say a necklace ∆ n ∨ ... ∨ ∆ n k is of preferred form if k = 0 or each n i ≥ T = ∆ n ∨ ... ∨ ∆ n k be a necklace in preferred form. Denote by b T the numberof beads in T . A joint of T is either an initial or a final vertex in some bead. Givena necklace T write J T for the subset of V T consisting of all the joints of T . For anytwo vertices a, b ∈ V T we write V T ( a, b ) and J T ( a, b ) for the set of vertices and jointsbetween a and b inclusive. Note that there is a unique subnecklace T ( a, b ) ⊆ T withjoints J T ( a, b ) and vertices V T ( a, b ). Denote by N ec the category whose objects arenecklaces in preferred form and morphisms are morphisms of necklaces. Note that
N ec is a full subcategory of
Set ∗ , ∗ ∆ = ∂ ∆ ↓ Set ∆ . Proposition 3.1.
Any non-identity morphism in
N ec is a composition of morphismsof the following type(i) f : T → T ′ is an injective morphism of necklaces and | V T ′ − J T ′ | − | V T − J T | = 1 (ii) f : ∆ n ∨ ... ∨ ∆ n k → ∆ m ∨ ... ∨ ∆ m k is a morphism of necklaces of the form f = f ∨ ... ∨ f k such that for exactly one p , f p : ∆ n p → ∆ m p is a codegeneracymorphism (so m p = n p − ) and for all i = p , f i : ∆ n i → ∆ m i is the identity map ofstandard simplices (so n i = m i for i = p )(iii) f : ∆ n ∨ ... ∨ ∆ n p − ∨ ∆ ∨ ∆ n p +1 ∨ ... ∨ ∆ n k → ∆ n ∨ ... ∨ ∆ n p − ∨ ∆ n p +1 ∨ ... ∨ ∆ n k is a morphism of necklaces such that f collapses the p -th bead ∆ in the domain tothe last vertex of the ( p − -th bead in the target and the restriction of f to all theother beads is injective. UBICAL RIGIDIFICATION, COBAR CONSTRUCTION, BASED LOOP SPACE 7
Proof.
We prove that any non-identity morphism of necklaces f : T → T ′ is acomposition of morphisms of type (i), (ii), and (iii) by induction on b T , the numberof beads of T . If b T = 1, then we must have b T ′ = 1 as well, so f is a morphism ofsimplicial sets between standard simplices which preserves first and last vertices. Itfollows that f is a composition of (simplicial) coface and codegeneracy morphisms.Cofaces and codegeneracies between standard simplices are morphisms of necklaces oftype (i) and of type (ii) or (iii), respectively. Assume we have shown the propositionfor b T ≤ k and suppose b T = k + 1. Let V T = { x , ..., x p } be the vertices of T and x i (cid:22) x i +1 . Let x j be the last vertex of the first bead of T , so T = T ( x , x j ) ∨ T ( x j , x p ) where T ( x , x j ) has one bead and T ( x j , x p ) has k beads. Let T f = T ′ ( f ( x ) , f ( x j )) ∨ T ′ ( f ( x j ) , f ( x p )). We have an injective morphism of necklaces t : T f → T ′ (notice that it is possible for T f = T ′ since f ( x j ) might not be ajoint of T ′ ). It follows that f = t ◦ ( g ∨ h ) where g : T ( x , x j ) → T ′ ( f ( x ) , f ( x j ))and h : T ( x j , x p ) → T ′ ( f ( x j ) , f ( x p )) are the morphisms of necklaces induced byrestricting f to T ( x , x j ) and T ( x j , x p ) respectively. By the induction hypothesiseach of g and h is a composition of morphisms of type (i), (ii), and (iii) and thisimplies that g ∨ h is a composition of such morphisms as well. In fact, we have g ∨ h = ( id T ′ ( f ( x ) ,f ( x j )) ∨ h ) ◦ ( g ∨ id T ( x j ,x p ) )and, clearly, the wedge of an identity morphism and a morphism which is a compo-sition of morphisms of type (i), (ii), and (iii) is again a morphism of such form.To conclude the proof we show that t : T f → T ′ is of the desired form. Moregenerally, let us prove that any non-identity injective morphism of necklaces t : R → R ′ is a composition of morphisms of type (i) by induction on the integer l ( R, R ′ ) := | V R ′ − J R ′ | − | V R − J R | . If l ( R, R ′ ) = 1 then t is of type (i). Assumewe have shown the claim for l ( R, R ′ ) = k . Suppose t : R → R ′ is injective and l ( R, R ′ ) = k + 1, then we have two cases: either (a) J R ′ = t ( J R ) or (b) J R ′ ⊂ t ( J R ).In case (a), it follows that both R and R ′ have the same number of beads, thus t = i ◦ j for inclusions of necklaces j : R → S , i : S → R ′ where S is the subnecklaceof R ′ spanned by t ( V R ) ∪{ v } and v is the smallest element of V R ′ − t ( V R ). Then j is oftype (i) and i is a composition of morphisms of type (i) by the induction hypothesis.For case (b), let t ( J R ) − J R ′ = { t ( x i ) , ..., t ( x i n ) } and consider the unique subnecklace S of R ′ defined by V S = t ( V R ) and J S = t ( J R ) − { t ( x i ) } . Then we have t = i ◦ j forinclusions of necklaces j : R → S , i : S → R ′ with j of type (i) and i a compositionof type (i) morphisms by the induction hypothesis. (cid:3) Remark 3.2.
Let us consider type (i) morphisms of the form f : T → ∆ p forsome integer p ≥
1. If b T = 1 then we have an injective map of simplicial sets f : ∆ p − → ∆ p which sends the first (resp. last) vertex of ∆ p − to the first (resp.last) vertex of ∆ p . The morphism f determines a ( p − p , i.e. an element of (∆ p ) p − . There are p + 1 non-degenerate elements in (∆ p ) p − ,however only p − f based on the constraint that f mustpreserve first and last vertices, namely, all the faces of the unique non-degenerateelement in (∆ p ) p except the first and last. If b T > v ∈ J T such that f ( v ) J T ′ . Moreover, since f is injective and | V T ′ − J T ′ | − | V T − J T | = 1,we have f ( J T − { v } ) = J ′ T and f ( V T ) = V T ′ . It follows that b T = 2 and theimage of f is a subnecklace T ′ ∨ T ′ of ∆ p starting and ending with the first andlast vertices of ∆ p , respectively, and containing all the vertices of ∆ p . Hence, wehave T ′ ∨ T ′ = ∆ p − i ∨ ∆ i for some 0 < i < p and each of these subnecklaces of ∆ p MANUEL RIVERA AND MAHMOUD ZEINALIAN corresponds to a unique term in the formula for the Alexander-Whitney diagonal Q ∆ (∆ p ) → Q ∆ (∆ p ) ⊗ Q ∆ (∆ p ) applied to the generator represented by the uniquenon-degenerate p -simplex in (∆ p ) p .4. The functor C (cid:3) c : N ec → Set (cid:3) c There is a functor C (cid:3) c : N ec → Set (cid:3) c which associates functorially to any ∆ n ∨ ... ∨ ∆ n k ∈ N ec a standard cube with connections of dimension n + ... + n k − k .The goal of this section is to define this functor carefully in a way which will beuseful later. We start by defining a functor P : N ec → Cat where
Cat is thecategory of small categories. Given a necklace T and two vertices a, b ∈ V T we maydefine a small category P T ( a, b ) whose objects are subsets X ⊆ V T ( a, b ) such that J T ( a, b ) ⊆ X and morphisms are inclusions of sets. For any necklace T ∈ N ec let P ( T ) = P T ( α, ω ) where α, ω ∈ V T are the first and last vertices of T . Let f : T → T ′ be a morphism in N ec , so f is a map of simplicial sets such that f ( α ) = α ′ and f ( ω ) = ω ′ where α, ω ∈ V T and α ′ , ω ′ ∈ V T ′ are the first and last vertices of T and T ′ , respectively. Notice that we have an inclusion J T ′ ⊆ f ( J T ). Thus f induces afunctor P f : P T ( α, ω ) → P T ′ ( α ′ , ω ′ ) defined on objects by P f ( X ) = f ( X ) and onmorphisms by the induced inclusion of sets. This yields a functor P : N ec → Cat .We might think of the objects of P ( T ) as strings of 0’s and 1’s as discussed below.This interpretation will yield a functor P which is naturally isomorphic to P . Wedefine a total order on the vertices of a necklace by setting a (cid:22) b if there is a directedpath from a to b . Proposition 4.1.
For any necklace T and any a, b ∈ V T such that a (cid:22) b , there isan isomorphism of categories φ T : P T ( a, b ) ∼ = N where N = | V T ( a, b ) − J T ( a, b ) | .Proof. Let V T ( a, b ) − J T ( a, b ) = { y , ..., y N } and y i (cid:22) y i +1 for i = 1 , ..., N − X of P T ( a, b ) (so J T ( a, b ) ⊆ X ⊆ V T ( a, b )) we define φ T ( X ) :=( φ T ( X ) , ..., φ NT ( X )) to be the object in the category N where, for 1 ≤ i ≤ N , wehave φ iT ( X ) = 1 if y i ∈ X and φ iT ( X ) = 0 if y i X . Given a morphism f : X → Y in P T ( a, b ) (so f is an inclusion of sets) we have an induced morphism φ T ( f ) : φ T ( X ) → φ T ( Y ) defined by φ T ( f ) := ( φ T ( f ) , ..., φ NT ( f )) where, for 1 ≤ i ≤ N , φ iT ( f ) : φ iT ( X ) → φ iT ( Y ) is the unique non-identity morphism in if φ iT ( X ) = 0and φ iT ( Y ) = 1, and φ iT ( f ) is an identity morphism otherwise. It is clear that thefunctor φ T : P T ( a, b ) → N is an isomorphism of categories. (cid:3) Consider the functor P : N ec → Cat defined on objects by P ( T ) = | V T − J T | and on morphisms f : T → T ′ by P ( f ) = φ T ′ ◦ P ( f ) ◦ φ − T : | V T − J T | → | V T ′ − J T ′ | .The above proposition implies that P is naturally isomorphic to P . In the followingproposition we describe explicitly the functor P ( f ) for morphisms f : T → T ′ oftype (i), (ii), and (iii) as in Proposition 3.1. Proposition 4.2.
Let f : T → T ′ be a morphism in N ec and let N = | V T − J T | . (1) If f is of type (i) then P ( f ) : N → N +1 is a cubical co-face functor. (2) If f is of type (ii) then P ( f ) : N → N − is either a cubical co-connectionfunctor or a cubical co-degeneracy functor. (3) If f is of type (iii) then P ( f ) : N → N is the identity functor.Proof. For any morphism of necklaces f : T → T ′ we have J T ′ ⊆ f ( J T ). For f : T → T ′ of type (i) we prove below that if J T ′ ⊂ f ( J T ) then P ( T )( f ) is a cubicalco-face functor δ j,N and if J T ′ = f ( J T ) then P ( T )( f ) is a cubical co-face functor UBICAL RIGIDIFICATION, COBAR CONSTRUCTION, BASED LOOP SPACE 9 δ j,N . A morphism f : T → T ′ of type (ii) collapses two vertices v and w of T into avertex v ′ of T ′ and is injective on V T − { v, w } . We prove below that if v ′ J T ′ then P ( T )( f ) is a cubical co-connection functor γ j,N and if v ′ ∈ J T ′ then P ( T )( f ) is acubical co-degeneracy functor ε j,N . The proof for the third part of the propositionwill be straightforward.(1) Let f : T → T ′ be of type (i) and write { y ′ , ..., y ′ N +1 } = V T ′ − J T ′ where y ′ i (cid:22) y ′ i +1 . We have J T ′ ⊆ f ( J T ) since f is a morphism of necklaces. If J T ′ ⊂ f ( J T ) then there is v ∈ J T such that f ( v ) = y ′ j ∈ V T ′ − J T ′ forsome j ∈ { , ..., N + 1 } and f ( J T − { v } ) ⊆ J ′ T . Then for any object X in P ( T ), v ∈ J T ⊆ X so y j = f ( v ) ∈ f ( X ). Using the fact that f is injectiveand identifying objects X in P ( T ) with sequences of 0’s and 1’s via theisomorphism φ T : P ( T ) ∼ = N we see that P ( f ) : N → N +1 is given onobjects by P ( f )( s , ..., s N ) = ( s , ...., s j − , , s j , ..., s N )and on morphisms λ = ( λ , ..., λ N ) : ( s , ..., s N ) → ( s ′ , ..., s ′ N ) by P ( f )( λ ) = ( λ , ..., λ j − , id , λ j , ..., λ N ) . Thus P ( f ) is the cubical co-face functor δ j,N .If J T ′ = f ( J T ) then there exists exactly one j ∈ { , ..., N + 1 } such that f − ( y ′ j ) = ∅ . Then for any object X in P ( T ), y ′ j will never be an element of f ( X ). Using the fact that f is injective and identifying objects X in P ( T )with sequences of 0’s and 1’s via the isomorphism φ T : P ( T ) ∼ = N we seethat P ( f ) : N → N +1 is given on objects by P ( f )( s , ..., s N ) = ( s , ...., s j − , , s j , ..., s N )and on morphisms λ = ( λ , ..., λ N ) : ( s , ..., s N ) → ( s ′ , ..., s ′ N ) by P ( f )( λ ) = ( λ , ..., λ j − , id , λ j , ..., λ N ) . It follows that P ( f ) is the cubical co-face functor δ j,N .(2) Let f : T → T ′ be of type (ii) and write { y , ..., y N } = V T − J T where y i (cid:22) y i +1 and { y ′ , ..., y ′ N − } = V T ′ − J T ′ where y ′ i (cid:22) y ′ i +1 . There exists v ′ ∈ V T ′ such that f − ( v ′ ) = { v, w } for some v, w ∈ V T and | f − ( x ′ ) | = 1for all x ′ ∈ V T ′ − { v ′ } . Note that v and w are consecutive vertices in the p -th bead of T . We have two cases: either v ′ ∈ V T ′ − J T ′ or v ′ ∈ J T ′ .If v ′ ∈ V T ′ − J T ′ , then v, w ∈ V T − J T so we may write v = y j and w = y j +1 for some j ∈ { , ..., N − } . Hence, for any object X of P ( T ) we have that if X ∩ { y j , y j +1 } 6 = ∅ then v ′ ∈ f ( X ) and if X ∩ { y j , y j +1 } = ∅ then v ′ f ( X ).By identifying objects X in P ( T ) with sequences of 0’s and 1’s via the iso-morphism φ T : P ( T ) ∼ = N we see that P ( f ) : N → N − is given onobjects by P ( f )( s , ..., s N ) = ( s , ....s j − , max ( s j , s j +1 ) , s j +2 , ..., s N )and on morphisms λ = ( λ , ..., λ N ) : ( s , ..., s N ) → ( s ′ , ..., s ′ N ) by P ( f )( λ ) = ( λ , ..., λ j − , σ j,j +1 , λ j +2 , ..., λ N ) , where σ j,j +1 is the unique morphism max ( s j , s j +1 ) → max ( s ′ j , s ′ j +1 ) in thecategory . It follows that P ( f ) is the cubical co-connection functor γ j,N . If v ′ ∈ J T ′ , we may assume without loss of generality that w ∈ J T and v = y j ∈ V T − J T for some j ∈ { , ..., N } . Let X be any object of P ( T ).Every element of X − { y j } corresponds to a unique element in f ( X ) via P ( f ) (since f is of type (ii)) and if y j ∈ X then P ( f ) sends y j to the joint v ′ ∈ f ( X ). By identifying objects X in P ( T ) with sequences of 0’s and 1’svia the isomorphism φ : P ( T ) ∼ = N we see that P ( f ) : N → N − is givenon objects by P ( f )( s , ..., s N ) = ( s , ..., s j − , s j +1 , ..., s N )and on morphisms λ = ( λ , ..., λ N ) : ( s , ..., s N ) → ( s ′ , ..., s ′ N ) by P ( f )( λ ) = ( λ , ..., λ i − , λ i +1 , ..., λ N ) . It follows that P ( f ) is the cubical co-degeneracy functor ε j,N .(3) If f is of type (iii) then | V T | = | V T ′ | + 1 and the injectivity of f only failswhen it collapses two joints (the endpoints of the p -th bead ∆ ) to a joint in T ′ . Under the isomorphism φ T : P ( T ) ∼ = N this collapse does not have anyeffect since given an object X of P ( T ) the entries in the string φ T ( X ) of 0’sand 1’s only indicate which non-joint vertices of T are in X . It follows that P ( f ) : N → N is the identity functor. (cid:3) Remark 4.3.
Consider two morphisms of necklaces f : U → T and g : V → T .If f and g are both of type (i) and f = g then P ( f ) = P ( g ). If f and g areof both of type (ii) and f = g we may have P ( f ) = P ( g ). For example, let U = W ∨ ∆ m +1 ∨ ∆ n ∨ W ′ , V = W ∨ ∆ m ∨ ∆ n +1 ∨ W ′ , T = W ∨ ∆ m ∨ ∆ n ∨ W ′ , forany two necklaces W and W ′ . Consider the maps f = id W ∨ s m +1 ∨ id ∆ n ∨ id W ′ and g = id W ∨ id ∆ m ∨ s ∨ id W ′ , where s m +1 : ∆ m +1 → ∆ m and s : ∆ n +1 → ∆ n arethe last and first (simplicial) codegeneracy morphisms respectively. It follows that P ( f ) = P ( g ). The identification of these two morphisms after applying P shouldbe compared with the identification in the colimit defining the monoidal structure ofthe category of cubical sets with connections discussed in the next section. Finally,if f and g are of type (iii), then we always have P ( f ) = P ( g ). Corollary 4.4.
The functor P : N ec → Cat factors as a composition
N ec → (cid:3) c ֒ → Cat .Proof.
For any object T in N ec , P ( T ) = N is an object of (cid:3) c and, by Proposition4.2, for any morphism f in N ec , P ( f ) is a morphism in (cid:3) c . (cid:3) Hence, we may consider P as a functor from N ec to (cid:3) c . Finally, we define afunctor from the category of necklaces to the category of cubical sets as follows. Definition 4.5.
Define the functor C (cid:3) c : N ec → Set (cid:3) c to be the composition offunctors C (cid:3) c := Y ◦ P where Y : (cid:3) c → Hom
Cat (( (cid:3) c ) op , Set ) = Set (cid:3) c is the Yonedaembedding.Note that for any T in N ec , C (cid:3) c ( T ) is the standard cube with connections (cid:3) Nc where N = | V T − J T | . Remark 4.6.
All non-degenerate cells of C (cid:3) c ( T ) can be realized by injective maps ofnecklaces T ′ → T . More precisely, for every non-degenerate cell σ ∈ C (cid:3) c ( T ) n there UBICAL RIGIDIFICATION, COBAR CONSTRUCTION, BASED LOOP SPACE 11 is a necklace T σ , with | V T σ − J T σ | = n together with an injective map of necklaces ι σ : T σ → T such that the induced map of cubical sets with connections (cid:3) nc ∼ = C (cid:3) c ( T σ ) C (cid:3) c ( ι σ ) −−−−−→ C (cid:3) c ( T )corresponds to the cell σ . Notice T σ is not unique, since any other T ′ σ for which thereis a map T ′ σ → T σ of type (iii) also works.5. The cubical rigidification functor C (cid:3) c : Set ∆ → Cat (cid:3) c The goal of this section is to show that the functor C : Set ∆ → Cat ∆ defined byLurie factors naturally through categories enriched over cubical sets with connectionsvia a functor C (cid:3) c : Set ∆ → Cat (cid:3) c . More precisely, we construct functors C (cid:3) c : Set ∆ → Cat (cid:3) c and T : Cat (cid:3) c → Cat ∆ such that T ◦ C (cid:3) c is naturally isomorphic to C . Definition 5.1.
For any simplicial set S we define a category C (cid:3) c ( S ) enriched overcubical sets with connections. Define the objects of C (cid:3) c ( S ) to be the vertices of S ,i.e. the elements of S . For any x, y ∈ S define C (cid:3) c ( S )( x, y ) := colim T → S ∈ ( Nec ↓ S ) x,y C (cid:3) c ( T )where ( N ec ↓ S ) x,y is the category whose objects are morphisms f : T → S for some T ∈ N ec such that f ( α T ) = x and f ( ω T ) = y . For any x, y, z ∈ S the compositionlaw C (cid:3) c ( S )( y, z ) ⊗ C (cid:3) c ( S )( x, y ) → C (cid:3) c ( S )( x, z )is induced as follows. Note that given T → S ∈ ( N ec ↓ S ) x,y and U → S ∈ ( N ec ↓ S ) y,z , we obtain T ∨ U → S ∈ ( N ec ↓ S ) x,z . Then the composition C (cid:3) c ( U ) ⊗ C (cid:3) c ( T ) → C (cid:3) c (( T ∨ U )( α U , ω U )) ⊗ C (cid:3) c (( T ∨ U )( α T , ω T )) → C (cid:3) c ( T ∨ U )of morphisms of cubical sets with connections induces the desired composition lawafter taking colimits. Recall that ( T ∨ U )( α U , ω U ) denotes the unique subnecklaceof T ∨ U with joints J T ∨ U ( α U , ω U ) and vertices V T ∨ U ( α U , ω U ). It follows fromRemark 4.3 that the above composition passes to the colimit and yields a well definedcomposition rule. Finally, it is clear that C (cid:3) c ( S ) is functorial in S . Remark 5.2.
The set of n -cells in C (cid:3) c ( S )( x, y ) is (cid:16) G ( T → S ) ∈ ( Nec ↓ S ) x,y C (cid:3) c ( T ) n (cid:17) / ∼ where the equivalence relation is generated by ( t : T → S, σ ) ∼ ( t ′ : T ′ → S, σ ′ ) ifthere is a map of necklaces f : T → T ′ such that t = t ′ ◦ f and C (cid:3) c ( f )( σ ) = σ ′ .Here t : T → S and t ′ : T ′ → S are objects in ( N ec ↓ S ) x,y , and σ and σ ′ are n -cellsin C (cid:3) c ( T ) and C (cid:3) c ( T ′ ), respectively. Any non-degenerate n -cell [ t : T → S, σ ] ∈ C (cid:3) c ( S )( x, y ) n may be represented by a pair ( r : R → S, σ R ) where • R is a necklace with | V R − J R | = n such that there are no ( u : U → S ) ∈ ( N ec ↓ S ) x,y with | V U − J U | = n − f : R → U satisfying r = u ◦ f , and • σ R ∈ C (cid:3) c ( R ) n is the unique non-degenerate n -cell in C (cid:3) c ( R ). In fact, one can let R = T σ and r = t ◦ ι σ as in Remark 4.6. These representativesare not unique since we may have another representative ( r ′ : R ′ → S, σ R ′ ) if thereis a morphism of necklaces h : R → R ′ of type (iii) such that r ′ ◦ h = r . We write[ r : R → S ] for the equivalence class of the non-degenerate n -cell in C (cid:3) c ( S )( x, y )represented by ( r : R → S, σ R ). Let v be the j -th vertex in V R − J R . The face map ∂ j : C (cid:3) c ( S )( x, y ) n → C (cid:3) c ( S )( x, y ) n − is given by ∂ j [ r : R → S ] = [ ∂ j r : R v → S ]where R v is the subnecklace of R spanned by vertices V R − { v } and ∂ j r is therestriction of r to R v . The face map ∂ j : C (cid:3) c ( S )( x, y ) n → C (cid:3) c ( S )( x, y ) n − is givenby ∂ j [ r : R → S ] = [ ∂ j r : R ( α R , v ) ∨ R ( v, ω R ) → S ] where ∂ j r is the restriction of r to R ( α R , v ) ∨ R ( v, ω R ). Of course [ ∂ j r : R v → S ] and [ ∂ j r : R ( α R , v ) ∨ R ( v, ω R ) → S ]may be degenerate cells in C (cid:3) c ( S )( x, y ) n − even if [ r : R → S ] is non-degenerate.Let us recall Lurie’s construction of C : Set ∆ → Cat ∆ . Given integers 0 ≤ i ≤ j denote by P i,j the category whose objects are subsets of the set { i, i + 1 , ..., j } con-taining both i and j and morphisms are inclusions of sets. We have an isomorphismof categories P i,j ∼ = j − i − if i < j and P i,i ∼ = . For each integer n ≥ C (∆ n ) whose objects are the elements of the set { , ..., n } andfor any two objects i and j such that i ≤ j , C (∆ n )( i, j ) is the simplicial set N ( P i,j ),where N : Cat → Set ∆ is the nerve functor. If j < i , C (∆ n )( i, j ) is defined to beempty. The composition law in the simplicial category C (∆ n ) is induced by the mapof categories P j,k × P i,k → P i,k given by union of sets. The construction of C (∆ n )is functorial with respect to simplicial maps between standard simplices. Then thefunctor C : Set ∆ → Cat ∆ is defined by C ( S ) := colim ∆ n → S C (∆ n ). C is defined as a colimit in the category of simplicial categories. Dugger andSpivak computed in [DS11] the mapping spaces of C explicitly via necklaces. Moreprecisely, Proposition 4.3 of [DS11] states that there is an isomorphism of simplicialsets colim T → S ∈ ( Nec ↓ S ) x,y [ C ( T )( α T , ω T )] ∼ = C ( S )( x, y ) . We defined C (cid:3) c having this formula in mind. We do it this way, as opposed to firstdefining C (cid:3) c on standard simplices and then extending as a left Kan extension, toemphasize that maps of necklaces give rise to maps of cubical sets with connectionsand the relationship of this fact with Adams’ cobar construction, as we will explainlater on. The mapping spaces of the functor C (cid:3) c are cubical sets with connectionsconstructed by applying the Yoneda embedding to the category P ( T ) associatedto a necklace T and then taking a colimit, while the mapping spaces in C are sim-plicial sets obtained by applying the nerve functor to P ( T ) and then taking a colimit.Recall we have a triangulation functor | · | : Set (cid:3) c → Set ∆ defined on a cubicalset with connections K by | K | := colim (cid:3) nc → K N ( n ) ∼ = colim (cid:3) nc → K (∆ ) × n . Definea functor T : Cat (cid:3) c → Cat ∆ as follows. Given a category K enriched over Set (cid:3) c define T ( K ) to be the simplicial category whose objects are the objects of K andwhose mapping spaces are given by | K ( x, y ) | for any objects x and y in K . We havea composition law on T ( K ) induced by applying the functor | · | to the compositionlaw in K and using the fact that for cubical sets with connections K and K ′ we havea natural isomorphism | K ⊗ K ′ | ∼ = | K | × | K ′ | . In fact, since colimits commute we UBICAL RIGIDIFICATION, COBAR CONSTRUCTION, BASED LOOP SPACE 13 have the following isomorphisms of simplicial sets | K ⊗ K ′ | ∼ = | colim (cid:3) nc → K, (cid:3) mc → K ′ (cid:3) n + mc | ∼ = colim (cid:3) nc → K, (cid:3) mc → K ′ | (cid:3) n + mc | ∼ = colim (cid:3) nc → K, (cid:3) mc → K ′ (∆ ) × n + m ∼ = colim (cid:3) nc → K, (cid:3) mc → K ′ (∆ ) × n × (∆ ) × m ∼ = colim (cid:3) nc → K (∆ ) × n × colim (cid:3) mc → K ′ (∆ ) × m ∼ = | K | × | K ′ | . Proposition 5.3.
The functor C : Set ∆ → Cat ∆ is naturally isomorphic to thecomposition of functors Set ∆ C (cid:3) c −−→ Cat (cid:3) c T −→ Cat ∆ . Proof.
Let Y ( (cid:3) c ) ↓ (cid:3) Nc be the category whose objects are morphisms (cid:3) nc → (cid:3) Nc ofcubical sets with connections and whose morphisms are given by the correspondingcommutative triangles. Note | (cid:3) Nc | is the colimit in simplicial sets of the functor Y ( (cid:3) c ) ↓ (cid:3) Nc → Set ∆ that sends an object ( (cid:3) nc → (cid:3) Nc ) to N ( n ) ∼ = (∆ ) × n and amorphism in Y ( (cid:3) c ) ↓ (cid:3) Nc to the corresponding induced morphism between nerves.The identity morphism (cid:3) Nc → (cid:3) Nc is a terminal object in Y ( (cid:3) c ) ↓ (cid:3) Nc . Therefore, | (cid:3) Nc | = colim (cid:3) nc → (cid:3) Nc N ( n ) is given by the value of the functor on the identity mor-phism (cid:3) Nc → (cid:3) Nc , so | (cid:3) Nc | = N ( N ).Let S be a simplicial set. The objects of the simplicial categories T ( C (cid:3) c ( S )) and C ( S ) are the same, i.e. the elements of S . Since the triangulation functor | · | commutes with colimits, we have the following natural isomorphisms( T ( C (cid:3) c ( S )))( x, y ) ∼ = colim T → S ∈ ( Nec ↓ S ) x,y | C (cid:3) c ( T ) | ∼ = colim T → S ∈ ( Nec ↓ S ) x,y N ( | V T − J T | ) . Moreover, by Proposition 4.3 of [DS11] it follows that we have natural isomorphismscolim T → S ∈ ( Nec ↓ S ) x,y N ( | V T − J T | ) ∼ = colim T → S ∈ ( Nec ↓ S ) x,y [ C ( T )( α, ω )] ∼ = C ( S )( x, y ) . Hence, we have an isomorphism of simplicial categories T ( C (cid:3) c ( S )) ∼ = C ( S ) which isfunctorial on S . It follows that T ◦ C (cid:3) c and C are naturally isomorphic functors. (cid:3) The left adjoint
Λ :
Set ∆ → dgCat k of the DG nerve functor In section 1.3.1 of [Lur11] Lurie defines a functor N dg : dgCat k → Set ∆ , calledthe dg nerve , which is weakly equivalent to the left adjoint of the composite functorΓ : Set ∆ C −→ Cat ∆ Q ∆ −−→ dgCat k where Q ∆ is the functor obtained by applying the normalized chains functor Q ∆ : Set ∆ → Ch k on the mapping spaces. In this section we prove that the compositefunctor Λ : Set ∆ C (cid:3) c −−→ Cat (cid:3) c Q (cid:3) c −−−→ dgCat k , where Q (cid:3) c is the functor obtained by applying the normalized chains functor Q (cid:3) c : Set (cid:3) c → Ch k on the mapping spaces, is left adjoint to N dg .Recall Lurie’s definition of N dg . Let C be a dg category. For each n ≥
0, define N dg ( C ) n to be the set of all ordered pairs of sets ( { X i } ≤ i ≤ n , { f I } ), such that:(1) X , X , ..., X n are objects of the dg category C (2) I is a subset I = { i − < i m < i m − < ... < i < i + } ⊆ [ n ] with m ≥ f I is an element of C ( X i − , X i + ) m satisfying df I = X ≤ j ≤ m ( − j ( f I −{ i j } − f i j <...
The functor
Λ :
Set ∆ → dgCat k is left adjoint to N dg : dgCat k → Set ∆ .Proof. First, we show that for any standard simplex ∆ n and any dg category C thereis bijection θ n, C : dgCat k (Λ(∆ n ) , C ) ∼ = Set ∆ (∆ n , N dg ( C ))which is functorial with respect to morphisms in the category ∆. Given a dg functor F : Λ(∆ n ) → C we construct an n -simplex θ n, C ( F ) = ( { X , ..., X n } , { f I } )in N dg ( C ) n . The objects of Λ(∆ n ) are the integers 0 , , ..., n so we let X i = F ( i )for i = 0 , , ..., n . For every subset I = { i − < i < ... < i m < i + } ⊆ [ n ] define σ I to be the generator of the chain complex Λ(∆ n )( i − , i + ) = Q (cid:3) c ( C (cid:3) c (∆ n )( i − , i + ))represented by the non-degenerate element of ( C (cid:3) c (∆ n )( i − , i + )) m which is the onebead sub-necklace inside ∆ n consisting of the ( m + 1)-simplex with i − as first vertex, i + as last vertex, and i , ..., i m as non-joint vertices, in other words, σ I is representedby the ( m +1)-simplex inside ∆ n spanned by vertices i − , i , ..., i m , i + . It follows fromRemark 3.2 that dσ I = m X j =1 ( − j ( ∂ j σ I − ∂ j σ I ) = m X j =1 ( − j ( σ I −{ i j } − σ i j <...
Let S be a simplicial set and x, y ∈ S . A generator ξ of degree n in the chain complex Λ( S )( x, y ) is an equivalence class which may be representedby a non-degenerate n -cell σ in the cubical set with connections C (cid:3) c ( S )( x, y ). Since C (cid:3) c ( S )( x, y ) is defined as a colimit, the non-degenerate n -cell σ is itself an equivalenceclass [ r : ∆ n ∨ ... ∨ ∆ n k → S ], where ( r : ∆ n ∨ ... ∨ ∆ n k → S ) ∈ ( N ec ↓ S ) x,y , n + ... + n k − k = n and such that there is no ( u : ∆ m ∨ ... ∨ ∆ m l → S ) ∈ ( N ec ↓ S ) x,y with m + ... + m l − l < n together with a map of necklaces f : ∆ n ∨ ... ∨ ∆ n k → ∆ m ∨ ... ∨ ∆ m l UBICAL RIGIDIFICATION, COBAR CONSTRUCTION, BASED LOOP SPACE 15 satisfying r = u ◦ f . Moreover, any s : ∆ n ∨ ... ∨ ∆ n i ∨ ∆ ∨ ∆ n i +1 ∨ ... ∨ ∆ n k → S satisfying r ◦ π = s , where π : ∆ n ∨ ... ∨ ∆ n i ∨ ∆ ∨ ∆ n i +1 ∨ ... ∨ ∆ n k → ∆ n ∨ ... ∨ ∆ n k isthe map of simplicial sets which collapses the ( i + 1)-th bead in the domain necklaceto a point, also represents the equivalence class σ . This follows essentially fromProposition 4.2 (3).7. Rigidification and the cobar construction
In this section, we relate the functor C (cid:3) c : Set → Cat (cid:3) c to the cobar functorΩ : dgCoalg k → dgAlg k . More precisely, we prove that Ω Q ∆ ( S ), the cobar con-struction on the dg coalgebra of normalized chains on a simplicial set S with onevertex x , is isomorphic as a dga to Λ( S )( x, x ), where Λ is the functor obtained byapplying the normalized cubical chains functor on the mapping spaces of C (cid:3) c , ornaturally isomorphically, the left adjoint to the dg nerve functor, as described in theprevious section. Then we deduce a relationship between C : Set → Cat ∆ andΩ : dgCoalg k → dgAlg k : we show Ω Q ∆ ( S ) is naturally weakly equivalent (quasi-isomorphic) as a dga to Γ( S )( x, x ), where Γ : Set ∆ → dgCat is the functor obtainedby applying normalized chains to the mapping spaces of C .Let k be a fixed commutative ring. We may consider k as a graded k -moduleconcentrated on degree 0. A graded coassociative coalgebra ( C, ∆) over k is counitalif it is equipped with a degree 0 map ǫ : C → k , called the counit , such that( ǫ ⊗ id) ◦ ∆ = id = (id ⊗ ǫ ) ◦ ∆.We say a differential graded coassociative coalgebra (dg coalgebra, for short)( C, ∂, ∆) over a commutative ring k is connected if C ∼ = k . Given a connected dgcoalgebra ( C, ∂, ∆) which is free as a k -module in each degree, the cobar construc-tion of C is the differential graded associative algebra (Ω C, D ) defined as follows.Consider the graded k -module sC where C i = C i for i > C = 0 and s is theshift by −
1, i.e. ( sC ) i = C i +1 . Let ∆ = Id ⊗ ⊗ Id + ∆ ′ and for any c ∈ C write∆ ′ ( c ) = P c ′ ⊗ c ′′ . The underlying algebra of the cobar construction is the tensoralgebra Ω C = T sC = k ⊕ sC ⊕ ( sC ⊗ sC ) ⊕ ( sC ⊗ sC ⊗ sC ) ⊕ ... and the differential D is defined by extending D ( sc ) = − s∂c + P ( − deg c ′ sc ′ ⊗ sc ′′ asa derivation to all of Ω C . This construction yields a functor Ω : dgCoalg k → dgAlg k where dgAlg k is the category of augmented dg algebras over k .For any simplicial set S , the chain complex Q ′ ∆ ( S ) of unnormalized chains over k has a natural coproduct ∆ : Q ′ ∆ ( S ) → Q ′ ∆ ( S ) ⊗ Q ′ ∆ ( S ) given by∆( x ) = M p + q = n f p ( x ) ⊗ l q ( x )for any x ∈ Q ∆ ( S ) n , where f p denotes the front p -face map (induced by the map[ p ] → [ p + q ], i i ) and l q is the last q -face map (induced by the map [ q ] → [ p + q ], i i + p ). This coproduct is known as the Alexander-Whitney diagonal map.Moreover, this dg coalgebra structure passes to the normalized chain complex Q ∆ ( S ).Thus, we may consider Q ∆ as a functor Q ∆ : Set ∆ → dgCoalg k . In particular, Q ∆ ( S ) is a dg coalgebra which is free as a k -module in each degree. If S is 0-reduced, i.e. S = { x } , then Q ∆ ( S ) is counital and connected with counit mapgiven by the composition Q ∆ ( S ) ։ Q ∆ ( S ) = k [ x ] ∼ = −→ k . From now on all of thecoalgebras in this article will be assumed to be counital. Theorem 7.1.
Let S be a -reduced simplicial set with S = { x } . There is anisomorphism of differential graded algebras Λ( S )( x, x ) ∼ = Ω Q ∆ ( S ) .Proof. For each integer n ≥ ∂ : Q ′ ∆ ( S ) n → Q ′ ∆ ( S ) n − and thecoproduct ∆ : Q ′ ∆ ( S ) n → L p + q = n Q ′ ∆ ( S ) p ⊗ Q ′ ∆ ( S ) q can be written as sums ∂ = P ni =0 ( − i ∂ i and ∆ = P ni =0 ∆ i as usual. In particular, for σ ∈ S n , ∆ ( σ ) = min σ ⊗ σ and ∆ n σ = σ ⊗ max σ where min σ and max σ denote the first and last vertices of σ ,respectively. The truncated maps ∂ ′ = P n − i =1 ( − i ∂ i and ∆ ′ = P n − i =1 ( − i ∆ i alsodefine a differential graded coassociative coalgebra structure on Q ′ ∆ ( S ). Consider thedga Ω Q ′ ∆ ( S ) = Ω( Q ′ ∆ ( S ) , ∂ ′ , ∆ ′ ). First, we show Λ( S )( x, x ) = Q (cid:3) c ( C (cid:3) c ( S )( x, x )) ∼ =Ω Q ′ ∆ ( S ) / ∼ for some equivalence relation ∼ and then we construct an isomorphismΩ Q ′ ∆ ( S ) / ∼ ∼ = Ω Q ∆ ( S ) . The dga Ω Q ′ ∆ ( S ) has as underlying complex the tensor algebra T sQ ′ ∆ ( S ) togetherwith differential D ′ Ω = ∂ ′ + ∆ ′ extended as a derivation to all of T sQ ′ ∆ ( S ). Wedenote a monomial sσ ⊗ ... ⊗ sσ k ∈ T sQ ′ ∆ ( S ) by [ σ | ... | σ k ]. Let s ( x ) ∈ Q ′ ∆ ( S ) bethe generator corresponding to the degenerate 1-simplex at x . We take a quotient of T sQ ′ ∆ ( S ) by the equivalence relation generated by[ σ | ... | σ k ] ∼ [ σ | ... | σ i − | σ i +1 | ... | σ k ]if for some 1 ≤ i ≤ k we have σ i = s ( x ) (in particular [ σ ] ∼ k if σ = s ( x )); and[ σ | ... | σ k ] ∼ σ i ∈ Q ′ ∆ ( S ) n i is a degenerate simplex with n i > ≤ i ≤ k . The firstrelation corresponds to the identification in the colimit defining C (cid:3) c ( S )( x, x ) arisingfrom Remark 4.6; the second relation corresponds to modding out by degeneratechains in the definition of the normalized chain complex Q (cid:3) c ( C (cid:3) c ( S )( x, x )). Boththe differential D ′ Ω and the algebra structure of T sQ ′ ∆ ( S ) pass to the quotient T sQ ′ ∆ ( S ) / ∼ . It is clear that we have an isomorphism of dga’s Q (cid:3) c ( C (cid:3) c ( S )( x, x )) ∼ = Ω Q ′ ∆ ( S ) / ∼ since necklaces in S correspond to monomials of generators in Q ′ ∆ ( S ).We define an isomorphism of dga’s˜ ϕ : Ω Q ′ ∆ ( S ) / ∼ → Ω Q ∆ ( S ) . Given σ ∈ Q ′ ∆ ( S ) denote by σ the equivalence class of σ in Q ∆ ( S ). First define ϕ [ σ ] = [ σ ] if deg σ > ϕ [ σ ] = σ + 1 k if deg σ = 1, and ϕ (1 k ) = 1 k . Extend ϕ as an algebra map to obtain a map ϕ : Ω Q ′ ∆ ( S ) → Ω Q ∆ ( S ). It follows by a shortcomputation that the map ϕ is a chain map. Moreover, ϕ induces a map of dga’s˜ ϕ : Ω Q ′ ∆ ( S ) / ∼ → Ω Q ∆ ( S ). The map ˜ ϕ is an isomorphism of dga’s, in fact,the inverse map ψ : Ω Q ∆ ( S ) → Ω Q ′ ∆ ( S ) / ∼ is given by defining ψ [ σ ] = (cid:2) [ σ ] (cid:3) ifdeg σ > ψ [ σ ] = (cid:2) [ σ ] (cid:3) − (cid:2) k (cid:3) if deg σ = 1, and ψ (1 k ) = (cid:2) k (cid:3) and then extending ψ UBICAL RIGIDIFICATION, COBAR CONSTRUCTION, BASED LOOP SPACE 17 as an algebra map, where (cid:2) [ σ ] (cid:3) denotes the equivalence class of [ σ ] ∈ Ω Q ′ ∆ in thequotient Ω Q ′ ∆ ( S ) / ∼ . (cid:3) We now relate the dga’s Ω Q ∆ ( S ) and Γ( S )( x, x ). We will use the following lemmawhich follows from an acyclic models argument. Lemma 7.2.
For any cubical set with connections K the chain complex Q ∆ ( | K | ) isnaturally weakly equivalent to Q (cid:3) c ( K ) , where |·| : Set (cid:3) c → Set ∆ is the triangulationfunctor.Proof. This proposition follows from the Acyclic Models Theorem applied to the twofunctors Q ∆ ◦ | · | , Q (cid:3) c : Set (cid:3) c → Ch k . Define the collection of models in
Set (cid:3) c to be M = { (cid:3) c , (cid:3) c , ... } , where (cid:3) jc is thestandard j -cube with connections. It is clear that both Q ∆ ◦ | · | and Q (cid:3) c are acyclicon these models. Recall a functor F : C → Ch k is free on M if there exist a collection { M j } j ∈ J where each M j is an object in M (possibly with repetitions, possibly notincluding all of the objects in M ) together with elements m j ∈ F ( M j ) such that forany object X of C we have that { F ( f )( m j ) ∈ F ( X ) | j ∈ J , ( f : M j → X ) ∈ C ( M i , X ) } forms a basis for F ( X ). Clearly Q (cid:3) c is free on M since we can take M j = (cid:3) jc , J = { , , , ..., } , and define m j ∈ Q (cid:3) c ( M j ) = Q (cid:3) c ( (cid:3) jc ) to be the generator correspond-ing to the unique non-degenerate element in ( (cid:3) jc ) j (i.e. m j is the top non-degeneratecell of (cid:3) jc ). Note that the simplicial set | (cid:3) jc | ∼ = (∆ ) × j has j ! non-degenerate j -simplices σ j , ..., σ jj ! ∈ | (cid:3) jc | j . Hence, Q ∆ ◦ | · | is also free on M since we can take { M , M , M , M , ..., M j , ..., M jj ! , M j +11 , ... } j ∈ J where M jk = (cid:3) jc , J = { , , , ... } ,and m jk ∈ Q ∆ ( | M jk | ) the generator corresponding to the j -simplex σ jk ∈ | (cid:3) jc | j .We have a natural isomorphism of functors H ( Q ∆ ◦ | · | ) ∼ = H ( Q (cid:3) c ), in fact,for any K ∈ Set (cid:3) c there is a natural bijection between | K | and K and any twovertices x and y are connected by a sequence of 1-simplices in | K | if and only ifthey are connected by a sequence of 1-cubes in K . By the Acyclic Models Theoremthere exist natural transformations φ : Q ∆ ◦ | · | → Q (cid:3) c and ψ : Q (cid:3) c → Q ∆ ◦ | · | suchthat each composition φ ◦ ψ and ψ ◦ φ is chain homotopic to the identity map. (cid:3) We use the above lemma to relate Ω Q ∆ ( S ) and Γ( S )( x, x ). Proposition 7.3.
Let S be a -reduced simplicial set with S = { x } . The differentialgraded associative algebras Ω Q ∆ ( S ) and Γ( S )( x, x ) are naturally weakly equivalent.Proof. By Theorem 7.1 we have an isomorphismΩ Q ∆ ( S ) ∼ = Λ( S )( x, x ) = Q (cid:3) c ( C (cid:3) c ( S )( x, x )) . By Lemma 7.2 and the fact that the triangulation functor and chains functor preservethe monoidal structures, it follows that the dga’s Q (cid:3) c ( C (cid:3) c ( S )( x, x )) and Q ∆ | C (cid:3) c ( S )( x, x ) | are naturally weakly equivalent. Finally, note that we have isomorphisms Q ∆ | C (cid:3) c ( S )( x, x ) | = Q ∆ (( T ◦ C (cid:3) c )( S )( x, x )) ∼ = Q ∆ ( C ( S )( x, x )) = Γ( S )( x, x ) . (cid:3) Properties of C : Set ∆ → Cat ∆ We recall several homotopy theoretic properties of the rigidification functor C : Set ∆ → Cat ∆ , in particular, its behavior with respect to Kan weak equivalences andits relationship with path spaces. These will be used in the final section of the article.A map of simplicial sets f : S → S ′ is called a Kan weak equivalence if it is a weakequivalence in the Quillen model structure, namely, if f induces a weak homotopyequivalence of spaces | f | : | S | → | S ′ | . A map of simplicial sets f : S → S ′ iscalled a categorical equivalence if f induces a weak equivalence C ( f ) : C ( S ) → C ( S ′ )of simplicial categories in the Bergner model structure. Recall that a functor ofsimplicial categories F : C → C ′ is called a weak equivalence of simplicial categories if • F induces an essentially surjective functor at the level of homotopy cate-gories, and • for all x, y ∈ C , F : C ( S )( x, y ) → C ′ ( F ( x ) , F ( y )) is a Kan weak equivalenceof simplicial sets.The Quillen model structure on Set ∆ has Kan equivalences as weak equivalencesand Kan complexes as fibrant objects. There is a different model structure on Set ∆ ,the Joyal model structure, which has categorical equivalences as weak equivalencesand quasi-categories as fibrant objects. Moreover, the Quillen model structure is aleft Bousfield localization of the Joyal model structure. In particular, a categoricalequivalence is always a Kan weak equivalence. The converse is not true in general, buta Kan weak equivalence between Kan complexes is always a categorical equivalence.This is Proposition 17.2.8 in [Rie14], which we record below. Proposition 8.1. If f : S → S ′ is a Kan weak equivalence between Kan complexes S and S ′ then C ( f ) : C ( S ) → C ( S ′ ) is a weak equivalence of simplicial categories. A map f : C → C ′ of connected dg coalgebras is called a quasi-isomorphism if f induces an isomorphism of coalgebras after passing to homology. On the other hand,a map f : C → C ′ of connected dg coalgebras is called an Ω -quasi-isomorphism if f induces a quasi-isomorphism of dga’s Ω f : Ω C → Ω C ′ . An Ω-quasi-isomorphismbetween connected dg coalgebras is always a quasi-isomorphism. The converse is nottrue in general, namely, a quasi-isomorphism between connected dg coalgebras mightnot be an Ω-quasi-isomorphism. However, if C and C ′ are connected dg coalgebraswhich are simply connected (i.e. C = 0 = C ′ ) then a quasi-isomorphism f : C → C ′ is an Ω-quasi-isomorphism. This follows by comparing Eilenberg-Moore spectral se-quences. There are model structures of the category of connected dg coalgebrashaving each of these two notions as the weak equivalences, but we do not need thesefor the purposes of this paper.Let Set be the full subcategory of the category Set ∆ of simplicial sets whoseobjects are 0-reduced simplicial sets. Let dgCoalg k be the full subcategory of thecategory dgCoalg k of dg coalgebras whose objects are connected dg coalgebras. Thenormalized chains functor restricts to a functor Q ∆ : Set → dgCoalg k . Proposition 8.2.
The functor Q ∆ : Set → dgCoalg k sends Kan weak equivalencesto quasi-isomorphisms and categorical equivalences to Ω -quasi-isomorphisms. UBICAL RIGIDIFICATION, COBAR CONSTRUCTION, BASED LOOP SPACE 19
Proof.
The proof of the first part of the proposition is well known. For the second sup-pose f : S → S ′ is a categorical equivalence and S = { x } , S ′ = { x ′ } . Then we havean induced Kan weak equivalence of simplicial sets C ( f ) : C ( S )( x, x ) → C ( S )( x ′ , x ′ ).This induces a dga quasi-isomorphism Q ∆ C ( f ) : Q ∆ ( C ( S )( x, x )) → Q ∆ ( C ( S )( x ′ , x ′ )).The result follows since the dga’s Q ∆ ( C ( S )( x, x )) and Q ∆ ( C ( S )( x ′ , x ′ )) are naturallyweakly equivalent to the dga’s Ω Q ∆ ( S ) and Ω Q ∆ ( S ′ ), respectively, by Proposition7.3. (cid:3) For any pointed topological space (
X, b ) denote by Sing(
X, b ) the sub-simplicialset of Sing( X ) whose n -simplices are the continuous maps | ∆ n | → X that take allvertices of | ∆ n | to b . Define a new functor Q K ∆ : Set → dgCoalg k by Q K ∆ ( S ) := Q ∆ (Sing( | S | , x )), where S = { x } and Sing( | S | , x ) is the Kan complex of singularsimplices | ∆ n | → | S | sending all vertices of | ∆ n | to x ∈ | S | . In general, the functor Q ∆ does not send Kan weak equivalences of simplicial sets to Ω-quasi-isomorphisms,but Q K ∆ does. Proposition 8.3.
The functor Q K ∆ : Set → dgCoalg k sends Kan weak equivalencesof simplicial sets to Ω -quasi-isomorphisms of dg coalgebras.Proof. Let
S, S ′ ∈ Set with S = { x } and S ′ = { x ′ } . If f : S → S ′ is a Kanweak equivalence then | f | : ( | S | , x ) → ( | S ′ | , x ′ ) is a homotopy equivalence of pointedspaces. The functor ( X, b ) Sing(
X, b ) from the category of pointed spaces to
Set sends homotopy equivalences of pointed spaces to Kan weak equivalences of0-reduced Kan complexes. Thus Sing( | f | ) : Sing( | S | , x ) → Sing( | S ′ | , x ′ ) is a Kanweak equivalence. It follows from Propositions 8.1 and 8.2 that Q ∆ (Sing( | f | )) : Q ∆ (Sing( | S | , x )) → Q ∆ (Sing( | S ′ | , x ′ )) is an Ω-quasi-isomorphism. (cid:3) We now explain the relationship between mapping spaces of C and different kindsof spaces of paths in a path connected topological space. This relationship is deducedfrom the homotopy theoretic properties of C as studied in Section 2.2 of [Lur09] andin [DS211] using different methods.For any simplicial category C define the simplicial nerve N ∆ ( C ) to be the simplicialset whose set of n -simplices is given by( N ∆ ( C )) n = Hom Cat ∆ ( C (∆ n ) , C ) . It follows that N ∆ : Cat ∆ → Set ∆ is the right adjoint of C : Set ∆ → Cat ∆ . If C is a topological category, then the topological nerve N T op ( C ) is defined to be thesimplicial nerve of the simplicial category Sing( C ) obtained by applying Sing to eachmorphism space of C . As its well known, for any topological monoid G , | N T op ( G ) | is a model for the classifying space BG .In Section 2.2 of [Lur09], Lurie shows that the pair of adjoint functors ( C , N ∆ )defines a Quillen equivalence between model categories Set ∆ with the Joyal modelstructure and Cat ∆ with the Bergner model structure. In particular, for any fibrantsimplicial category C (a simplicial category whose mapping spaces are Kan com-plexes) the counit map C ( N ∆ ( C )) → C is a weak equivalence of simplicial categories.This also follows from Theorem 1.5 of [DS211].Let X be a path connected topological space and let x, y ∈ X . Define the spaceof Moore paths in X between x and y to be P Mx,y X = { ( γ, r ) | γ : [0 , ∞ ) → X, γ (0) = x, γ ( s ) = y for r ≤ s, r ∈ [0 , ∞ ) } topologized as a subset of Map([0 , ∞ ) , X ) × [0 , ∞ ),where Map([0 , ∞ ) , X ) is equipped with the compact-open topology. Define a functor P : T op → Cat
T op from the category of topological spaces to the category of topological categories asfollows. For any X ∈ T op the objects of P ( X ) are the points of X . For any x, y ∈ X ,define the space of morphisms P ( X )( x, y ) := P Mx,y X with composition rule inducedby concatenation of paths. We call P : T op → Cat
T op the path category functor .The functor C : Set ∆ → Cat ∆ is a simplicial model for the path category functoras shown in Proposition 8.4 below. Denote by Sing( P X ) the simplicial categoryobtained by applying Sing to the morphism spaces of the topological category P X . Proposition 8.4.
Let X be a path connected topological space. The simplicial cate-gories C ( Sing ( X )) and Sing ( P X ) are weakly equivalent.Proof. Choose b ∈ X . The topological category P X is weakly equivalent to Ω X , thetopological category with a single object b and as morphism space Ω X ( b, b ) = Ω Mb X the space of based Moore loops at b with composition law given by concatenationof loops. A weak equivalence P X → Ω X of topological categories is given by fixinga collection of paths O = { γ x } x ∈ X where γ x is a path from b to x . More precisely,we have a functor F O : P X → Ω X given on objects by sending all objects of P X to the single object of Ω X and on morphisms F O : P X ( x, y ) → Ω X ( b, b ) is thecontinuous map F O ( γ ) = γ − y ∗ γ ∗ γ x , where ∗ denotes concatenation. The functor F O is clearly a weak equivalence of topological categories. The topological nerve N T op = N ∆ ◦ Sing :
Cat
T op → Set ∆ sends weak homotopy equivalences of topo-logical categories to Kan weak equivalence of simplicial sets. Thus, the simplicialsets N T op ( P X ) and N T op (Ω X ) are Kan weakly equivalent. Moreover, the geometricrealization | N T op (Ω X ) | is a model for B (Ω X ), the classifying space of the topologicalmonoid of based loops. It follows from B (Ω X ) ≃ X that the simplicial sets N T op ( P X )and Sing( X ) are Kan weakly equivalent. On the other hand, since the homotopy cat-egory of N T op ( P X ) is a groupoid it follows that N T op ( P X ) is a Kan complex [Joy02].By Proposition 8.1 we have that C ( N T op ( P X )) and C (Sing( X )) are weakly equivalentas simplicial categories. Since C ◦ N ∆ ( C ) ≃ C for any C ∈ S ∆ whose mapping spacesare Kan complexes, it follows that C ( N T op ( P X )) = C ( N ∆ (Sing( P X ))) ≃ Sing( P X ).Hence, the simplicial categories C (Sing( X )) and Sing( P X ) are weakly equivalent. (cid:3) We have the following corollary.
Corollary 8.5.
Let X be a path connected topological space and b ∈ X . The simpli-cial categories with one object C ( Sing ( X, b )) and Sing (Ω X ) are weakly equivalent.Proof. For path connected X the inclusion Sing( X, b ) ֒ → Sing( X ) is a Kan weakequivalence of Kan complexes, so C (Sing( X ))( b, b ) ≃ C (Sing( X, b ))( b, b ). Hence, byProposition 8.4, C (Sing( X, b )) ≃ Sing(Ω X ). (cid:3) We finish this section by describing more explicitly the weak equivalence of sim-plicial sets between C (Sing( X ))( x, y ) and Sing( P X )( x, y ) given by Proposition 8.4.We review this for completeness but it is not strictly necessary to follow Section 9.We follow Chapter 2 of [Lur09]. UBICAL RIGIDIFICATION, COBAR CONSTRUCTION, BASED LOOP SPACE 21
Define a cosimplicial object J • : ∆ → ( ∂ ∆ ↓ Set ∆ ), by letting J n to be thequotient of the standard simplex ∆ n +1 by collapsing the last face (i.e. the facespanned by vertices [0 , ..., n ]) to a vertex. The quotient simplicial set J n has exactlytwo vertices which we denote by the integers 0 and n + 1. For any S ∈ Set ∆ and x, y ∈ S , there is a simplicial set Hom RS ( x, y ) called the right mapping space definedby letting Hom RS ( x, y ) n be the set of all morphisms of simplicial sets ϕ : J n → S such that ϕ (0) = x and ϕ ( n + 1) = y , together with structure face and degeneracymaps defined to coincide with the corresponding structure maps of on S n +1 . Definea cosimplicial simplicial set Q • by letting Q n := C ( J n )(0 , n + 1) and denote by | − | Q • : Set ∆ → Set ∆ the realization functor associated to Q • . Recall Proposition2.2.4.1 of [Lur09]: Proposition 8.6.
Let S be an quasi-category containing a pair of objects x and y .There is a natural Kan weak equivalence of simplicial sets f : | Hom RS ( x, y ) | Q • → C ( S )( x, y ) . In Proposition 2.2.2.7 of [Lur09], Lurie shows there is a Kan weak equivalence ofsimplicial sets g : | S | Q • ∼ = colim ∆ n → S C ( J n )(0 , n + 1) → colim ∆ n → S ∆ n ∼ = S for any simplicial set S . Hence, for a quasi-category S and x, y ∈ S we have a zigzag of Kan weak equivalencesHom RS ( x, y ) g ←− | Hom RS ( x, y ) | Q • f −→ C ( S )( x, y ) . Now consider the above zig zag of Kan weak equivalences in the case S = Sing( X )for a topological space X . There is a Kan weak equivalence of simplicial sets θ : Hom R Sing( X ) ( x, y ) → Sing( P Mx,y X )given as follows. A simplex ϕ : J n → Sing( X ) ∈ Hom R Sing( X ) ( x, y ) corresponds to acontinuous map σ ϕ : | ∆ n +1 | → X which collapses the last face of | ∆ n +1 | to x andsends the last vertex of | ∆ n +1 | to y . For each point p in the last face of | ∆ n +1 | thereis a straight line segment from p to the last vertex of | ∆ n +1 | . These straight linesegments give a family of disjoint paths inside | ∆ n +1 | which start in the last face andend in the last vertex and such a family is parametrized by | ∆ n | . The continuousmap σ ϕ induces a continuous map | ∆ n | → P Mx,y X which corresponds to a simplex θ ( ϕ ) : ∆ n → Sing( P Mx,y X ). The map θ is clearly a Kan weak equivalence of simplicialsets. It follows from the above zig zag formed by Kan weak equivalences f and g that C (Sing( X ))( x, y ) ≃ Sing( P Mx,y X ).9. Algebraic models for loop spaces
In this section we deduce an extension of a classical theorem of Adams from ourprevious results and discuss a few consequences. We start by showing that for apath connected pointed space (
X, b ), Λ(Sing(
X, b ))( b, b ) and S ∗ (Ω Mb X ; k ) are weaklyequivalent as dga’s. Proposition 9.1.
Let ( X, b ) be a pointed path connected topological space. Thedifferential graded associative algebras Λ( Sing ( X, b ))( b, b ) and S ∗ (Ω Mb X ; k ) are weaklyequivalent. Proof.
By definition Λ(Sing(
X, b ))( b, b ) = Q (cid:3) c ( C (cid:3) c (Sing( X, b ))( b, b )). By Lemma7.2 we have a quasi-isomorphism of chain complexes Q (cid:3) c ( C (cid:3) c (Sing( X, b ))( b, b )) ≃ Q ∆ ( | C (cid:3) c (Sing( X, b ))( b, b ) | ) . Moreover, this is quasi-isomorphism is a weak equivalence of dga’s since the monoidalstructures are preserved under the triangulation functor. By Proposition 5.3, we havean isomorphism Q ∆ ( | C (cid:3) c (Sing( X, b ))( b, b ) | ) ∼ = Q ∆ ( C (Sing( X, b ))( b, b )) . Finally, by Corollary 8.5, we have Q ∆ ( C (Sing( X, b ))( b, b )) ≃ S ∗ (Ω Mb X ; k )as dga’s. (cid:3) In [Ada52], Adams introduced the cobar construction and constructed a chainmap of dga’s ϕ : Ω Q ∆ (Sing( X, b )) → C (cid:3) ∗ (Ω Mb X ; k ), where C (cid:3) ∗ (Ω Mb X ; k ) denotes thenormalized singular cubical chains on Ω Mb X . Moreover, Adams showed that if X is simply connected then ϕ is a quasi-isomorphism. The proof of this fact reliedon associating a spectral sequence to Ω Q ∆ (Sing( X, b )) and then comparing it to theSerre spectral sequence for the fibration Ω Mb X → P X → X . The simple connectivityassumption was used in order for the hypotheses of the Zeeman comparison theoremfor spectral sequences to be satisfied.We now deduce an extension of Adams’ classical theorem (Corollary 9.2 below)to the case when X is a path connected space with possibly non-trivial fundamentalgroup. Note that we have not relied on spectral sequence arguments but rather oncategorical and space level arguments as discussed in the previous section. Corollary 9.2.
For any pointed path connected space ( X, b ) , the differential gradedalgebras Ω( Q ∆ ( Sing ( X, b ))) and S ∗ (Ω Mb X ; k ) are weakly equivalent.Proof. This follows directly from Theorem 7.1 and Proposition 9.1. (cid:3)
We conclude with two remarks and an application to model the free loop space.
Remark 9.3.
It follows from the above discussion that we may recover the homol-ogy of the based loop space of | S | by taking the cobar construction on any con-nected dg coalgebra Ω-quasi-isomorphic to Q K ∆ ( S ). In general, Q ∆ ( S ) and Q K ∆ ( S )are quasi-isomorphic but not necessarily Ω-quasi-isomorphic. However, if S = { x } and S = { s ( x ) } , where s ( x ) denotes the degenerate 1-simplex at x , then Q ∆ ( S )and Q K ∆ ( S ) are simply connected dg coalgebras and the natural map of dg coalge-bras ι : Q ∆ ( S ) → Q K ∆ ( S ) is a quasi-isomorphism. Thus, by Propoisition 2.2.7 in[LoVa12], ι is an Ω-quasi-isomorphism. Consequently, Ω Q ∆ ( S ) is weakly equivalentas a dg algebra (i.e. quasi-isomorphic) to S ∗ (Ω Mx | S | ; k ). Remark 9.4.
In the case of a simplicial complex, an explicit and smaller modelfor the based loop space can be given using a Kan fibrant replacement functor. Let K be a simplicial complex with an ordering of its vertices and let v be a vertex of K . Let f K be the simplicial set obtained by defining the face maps in accordancewith the ordering of the vertices and adding degeneracies freely to K . The cobarconstruction on Q ∆ ( f K ) might not yield the homology of the based loop space of | f K | . However, we may consider the Kan fibrant replacement Ex ∞ ( f K ) of f K . UBICAL RIGIDIFICATION, COBAR CONSTRUCTION, BASED LOOP SPACE 23 Ex ∞ ( f K ) is a Kan complex weakly equivalent to f K , so it follows that the Kancomplexes Ex ∞ ( f K ) and Sing( | f K | ) are weakly equivalent. Thus C (Ex ∞ ( f K )), C (Sing( | f K | )), and Sing( P | f K | ) are weakly equivalent simplicial categories. There-fore Λ(Ex ∞ ( f K ))( v, v ) is a dga model for the based loop space of | f K | at v . Thisremark explains an example of Kontsevich outlined in [Kon09]. In [HT10], a similarconstruction was also described for any simplicial set, which was then compared toKan’s loop group construction.Finally, a chain complex model for the free loop space of a path connected topo-logical space may be obtained as follows. For any dga A denote by CH ∗ ( A ) theHochschild chain complex of A . For the definition we refer the reader to any stan-dard reference such as [Lod98]. Corollary 9.5.
For any pointed path connected space ( X, b ) , the Hochschild chaincomplex CH ∗ (Ω( Q ∆ ( Sing ( X, b )))) is quasi-isomorphic to S ∗ ( LX ; k ) , the singularchains on the free loop space of X .Proof. This is a direct consequence of the fact that the Hochschild chain complex ofthe dga S ∗ (Ω M X ; k ) is quasi-isomorphic to S ∗ ( LX ; k ) (a theorem usually attributedto Goodwillie [Goo85]), Corollary 9.2, and the invariance of Hochschild chains underweak equivalences of dga’s. (cid:3) As explained in remark 2.23 of [Hes16], for any connected dg coalgebra C there isa quasi-isomorphism of chain complexes coCH ∗ ( C ) ≃ CH ∗ (Ω C )where coCH ∗ ( C ) denotes the coHochschild chain complex of C ; we refer to [Hes16]for definitions and further details. As a consequence, we obtain a model for the freeloop space LX of a path connected space space X that does not require passing tothe based loop space, which we expect to be convenient in studying string topology. Corollary 9.6.
For any pointed path connected space ( X, b ) , the coHochschild com-plex coCH ∗ ( Q ∆ ( Sing ( X, b ))) is quasi-isomorphic to S ∗ ( LX ; k ) .Proof. This follows directly from Corollary 7.2 and the fact that coCH ∗ ( C ) ≃ CH ∗ (Ω C ) for any connected dg coalgebra C . (cid:3) . References [Ada52] J. Adams, On the cobar construction,
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Manuel Rivera, Department of Mathematics, University of Miami, 1365 MemorialDrive, Coral Gables, FL 33146
E-mail address [email protected]
Mahmoud Zeinalian, Department of Mathematics, City University of New York, LehmanCollege, 250 Bedford Park Blvd W, Bronx, NY 10468