Multisimplicial chains and configuration spaces

This paper presents a generalization to multisimplicial sets of previously defined $E_\infty$-coalgebra structures on the chains of simplicial and cubical sets. We focus on the surjection chain complexes of McClure--Smith as a main example and construct a zig-zag of complexity preserving quasi-isomorphisms of $E_\infty$-coalgebras relating these to both the singular chains on configuration spaces and the Barratt--Eccles chain complexes.


Introduction
The cochain complex of a simplicial set is equipped with the classical Alexander-Whitney product defining the ring structure in cohomology.This cochain level structure has several explicit extensions to an E ∞ -algebra [MS03; BF04; Med20a] encoding commutativity and associativity up to coherent homotopies.The importance of E ∞ -algebras in homotopy theory is well known.For example, Mandell showed that finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E ∞ -algebras [Man06].Our first objective is to define a natural product together with an E ∞ -algebra extension on the cochains of multisimplicial sets [Gug57].These are generalizations of both simplicial and cubical sets which are useful for concrete computations since they can model homotopy types using fewer cells.For example, the proof of the non-formality of the cochain algebra of planar configuration spaces [Sal20] used a simplicial model and the Alexander-Whitney product on its cochains.By using a multisimplicial model and the product defined here, these computations become simpler and faster, paving the way for extending this result to higher dimensions.
Multisimplicial sets are contravariant functors from products of the simplex category △ to Set.Explicitly, for any positive integer k the category mSet (k) of k-fold multisimplicial sets is the presheaf category Fun((△ op ) ×k , Set).There is a notion of geometric realization for multisimplicial sets, which results in a CW complex having, for each non-degenerate multisimplex, a cell modeled on a product of geometric simplices ∆ n1 × • • • × ∆ n k .We are interested in modeling homotopy types algebraically, for which we consider the composition of the geometric realization and the functor of cellular chains C.This composition defines N : mSet (k) → Ch, the functor of (normalized) chains.In §2.5 we define a lift of N to the category of E ∞ -coalgebras, and, consequently, a lift of the functor of cochains to the category of E ∞ -algebras.We do so using the finitely presented E ∞ -prop introduced in [Med20a] and its monoidal properties.Specifically, using the isomorphism we extend the image of the prop generators constructed in [Med20a] from the chains of standard simplices to those of standard multisimplices.These generators are the Alexander-Whitney coproduct, the augmentation map, and an algebraic version of the join product.The resulting E ∞ -coalgebra structure generalizes those defined in [MS03; BF04;Med20a] for simplicial chains and in [KM22] for cubical chains.
As an application, we study the Steenrod construction for multisimplicial chains in §2.7 emphasizing the explicit nature of our construction.
Let us now focus on the relationship between multisimplicial and simplicial theories.The restriction to the image of the diagonal inclusion △ op → (△ op ) ×k of any k-fold multisimplicial set X defines its associated diagonal simplicial set X D .There is a natural homeomorphism of realizations |X| ∼ = |X D | [Qui73].Under this homeomorphism the cells of |X D | arise from those of |X| through subdivision, a procedure described algebraically by the Eilenberg-Zilber quasi-isomorphism The functor induced by the diagonal restriction has a right adjoint N (k) , the multisimplicial nerve of a simplicial set, making the categories of k-fold multisimplicial and simplicial sets equivalent in Quillen's sense.Furthermore, there is a natural inclusion I : N(Y ) → N(N (k) Y ) which is also a quasi-isomorphism.On one hand, the EZ map preserves the counital coalgebra structure, but it does not respect the higher E ∞ -structure.On the other, the map I is an E ∞ -coalgebra quasi-isomorphism as proven in §3.3.We use this fact to prove in §3.4 that, for any topological space X, the linear map from its singular simplicial chains to its singular k-fold multisimplicial chains, given by precomposing a continuous map (∆ n → X) with the projection ( In the second part of the paper, we use these constructions to study a multisimplicial model of the canonical filtration and showed that X (r) is connected to the singular chains of Conf r (R ∞ ) via a zigzag of filtration preserving S r -equivariant quasi-isomorphisms.Presumably it was observed by both McClure-Smith and Berger-Fresse that X (r) can be interpreted as the chains of an r-fold multisimplicial set Sur(r), which we introduce in §4.2 with a filtration There is an operad structure on {X d (r)} r≥1 for each d ≥ 1, but we do not focus on it since it is not induced from one at the multisimplicial level.By the constructions in Section 2 the complex N Sur(r) is equipped with an E ∞ -coalgebra structure, which we connect to the singular chains of Conf r (R ∞ ) via an explicit zig-zag of filtration preserving S r -equivariant quasi-isomorphisms of E ∞ -coalgebras.
In a similar way, Berger and Fresse [BF04] studied a chain complex E(r) of Z[S r ]-modules with a filtration This complex comes from the chains on a simplicial set introduced by Barratt and Eccles [BE74] equipped with a filtration [Smi89].(As before we disregard the operadic structure.)Since E(r) is induced from a simplicial set, it is endowed with an E ∞ -coalgebra structure, and it is not hard to see that the zig-zag of filtration preserving S r -equivariant quasiisomorphisms used to compare it to the singular chains of Conf r (R ∞ ) respects this higher structure.Consequently, X (r) and E(r) can be related by an explicit zig-zag of such maps.
It is desirable to have a direct map between the multisimplicial and simplicial models.Berger-Fresse constructed two such filtrations preserving S r -equivariant quasi-isomorphisms TR : N E(r) → N Sur(r) and TC : N Sur(r) → N E(r).
The first one, introduced in [BF04, 1•3], is unfortunately not a coalgebra map.Therefore we will focus on the second one, which was introduced in [BF02].Our contribution, presented in §4.4, is the construction of a factorization TC : N Sur(r) where the second map is induced from a filtration preserving S r -equivariant weakequivalence of simplicial sets.Therefore, we prove that TC is a coalgebra map since EZ is one.

Multisimplicial algebraic topology
2.1.Multisimplicial sets.Let us consider an arbitrary positive integer k.The k-fold multisimplex category △ ×k is the k-fold Cartesian product of the simplex category △.The category is referred to as the category of k-fold multisimplicial sets.We remark that mSet (1)  and mSet (2) are naturally equivalent to the categories of simplicial and bisimplicial sets respectively.A representable k-fold multisimplicial sets is denoted by △ n1,...,n k .Explicitly, a k-fold multisimplicial set X consists of a collection of sets indexed by k-tuples of non-negative integers (m 1 , . . ., m k ) together with face maps d j i : X m1,...,mj ,...,m k → X m1,...,mj−1,...,m k and degeneracy map s j i : X m1,...,mj,...,m k → X m1,...,mj +1,...,m k for 1 ≤ j ≤ k and 0 ≤ i ≤ m j such that, referring to j as the direction of these maps, two of them satisfy the simplicial identities when they have the same direction and commute when they do not.An element of X m1,...,m k is called an (m 1 , . . ., m k )multisimplex and it is said to be degenerate if it is in the image of a degeneracy map.
The geometric realization functor has a right adjoint Sing (k) : Top → mSet (k)   defined on a topological space X, as usual, by the expression

Algebraic realization. The functor of chains
is the Yoneda extension of the functor defined on representable objects by It is naturally isomorphic to the composition of the geometric realization functor and the functor of cellular chains with respect to the canonical cellular structure.
Explicitly, for a k-fold multisimplicial set X the k-module N(X) n is freely generated by the non-degenerate (n 1 , . . ., n k )-multisimplices with For any topological space X the chain complex N Sing (k) (X) is denoted S (k) (X) and referred to as the k-fold singular chains of X.

Coalgebra structure.
A counital coalgebra structure on a chain complex C is a pair of chain maps ∆ : The tensor product of two counital coalgebras C and C ′ is itself a counital coalgebra with structure maps given by where τ transposes the second and third factors.
For each n ∈ N, the complex N(△ n ) is naturally equipped with a counital coalgebra structure defined by: We will refer to it as the Alexander-Whitney structure.
Using the tensor product structure, we deduce a natural counital coalgebra structure on the chains of representable multisimplicial sets and, via a Yoneda extension, one on the chains of general multisimplicial sets.
Explicitly, for a k-fold multisimplicial set X and (m 1 , . . ., m k )-multisimplex where the front (i 1 , . . ., i k )-face of x is the multisimplex [Med20a], the collection of all maps {C → C ⊗r } r∈N generated by ∆, ǫ and * make C into an E ∞ -coalgebra, that is to say, a coalgebra over certain operad UM that is a cofibrant resolution of the terminal operad.
As proven in [MR21], the counital coalgebra structure on the tensor product of two M-bialgebras C and C ′ can be naturally extended to an M-bialgebra structure using For any integer n, the join product where π is the permutation that orders the vertices.It is proven in [Med20a] that on the chains of representable simplicial sets the Alexander-Whitney structure together with the join product make N(△ n ) into a natural M-bialgebra and, consequently, a natural E ∞ -coalgebra.We mention that this structure is induced by one preset at the level of geometric realizations [Med21b].
Using the tensor product structure, we deduce a natural M-bialgebra structure on the chains of representable multisimplicial sets and consequently a natural E ∞ -coalgebra structure, which extends along the Yoneda inclusion to the chains on any multisimplicial set X.
Explicitly, for two basis elements of N △ n1,...,n k we have where, with the convention x <1 = x >n = 1 ∈ k, We remark that since the category of M-bialgebras is not cocomplete, we do not necessarily have an M-bialgebra structure on N(X) for a general multisimplicial set X.An example for which such structure does not exist is given by one such X whose geometric realization consists of just two points.
2.6.Cubical theory.Since the complex of chains of the k-fold multisimplicial set △ 1 × • • • × △ 1 is isomorphic to the chains on the standard cubical set k , it is natural to compare the E ∞ -coalgebra structure defined here with that presented in [KM22] for cubical sets.As counital coalgebras N(△ 1 × • • • × △ 1 ) and N( k ) are isomorphic, and, denoting the product of the M-bialgebra defined there by * , we have x * y = (−1) |x| x * y under this chain isomorphism.The sign convention used here is more natural, used for example to endow Adams' cobar construction with the structure of a monoidal E ∞ -coalgebra [MR21].
2.7.Steenrod construction.In [Ste47], Steenrod introduced natural operations on the mod 2 cohomology of spaces, the celebrated Steenrod squares via an explicit construction of natural linear maps ∆ i : N(X) → N(X) ⊗ N(X) for any simplicial set X, satisfying up to signs the following homological relations with the convention ∆ −1 = 0.These so-called cup-i coproducts appear to be fundamental, as they are axiomatically characterized [Med22] and induce the nerve of strict infinity categories [Med20b].A description of cup-i coproducts for multisimplicial sets can be deduced from our E ∞ -coalgebra structure.It is given recursively by Steenrod also introduced operations on the mod p cohomology of spaces when p is an odd prime [Ste52;Ste53].To define these effectively, generalization of the cupi coproducts were introduced in [KM21].After the present work, these so-called cup-(p, i) coproducts are defined on multisimplicial chains, and their formulas are explicit enough to be implemented in the computer algebra system ComCH [Med21a], where constructions of Cartan and Adem coboundaries [Med20c; BMM21] for multisimplicial sets are also be found.

Comparison with the simplicial theory
We will use sSet to denote the category of 1-fold multisimplicial sets mSet (1)  referring to its objects and morphisms as simplicial sets and simplicial morphisms as usual.
3.1.Diagonal simplicial set.For any k ∈ N, the diagonal The functor (−) D : mSet (k) → sSet admits a right adjoint N (k) : sSet → mSet (k) , defined, as usual, by the expression These functors define a Quillen equivalence.A proof of this fact can be given using [Mal05, Proposition 1.6.8]or adapting that in [Moe89, Proposition 1.2].
3.2.Eilenberg-Zilber map.Recall that an (n 1 , . . ., n k )-shuffle σ is a permutation in S n satisfying We denote the set of such permutations by Sh(n 1 , . . ., n k ).For any σ ∈ Sh(n 1 , . . ., n k ) the inclusion If e is the identity permutation, we denote i e simply as i.The set {i σ | σ ∈ Sh(n 1 , . . ., n k )} defines a triangulation of ∆ n1 × • • • × ∆ n k making it isomorphic, in the category of cellular spaces, to the geometric realization of the simplicial set △ n1 × • • • × △ n k .Using this identification, the identity map induces a cellular map agrees, under the natural identifications, with the traditional Eilenberg-Zilber map.
Since the traditional Eilenberg-Zilber map preserves counital coalgebra structures we have the following.Theorem 3.2.1.For every multisimplicial set X the map EZ : N(X) → N(X D ) is a quasi-isomorphism of counital coalgebras.
We remark that the Eilenberg-Zilber map is not a morphism of E ∞ -coalgebras.For example, as shown in [KM22, §5.4], we have

Canonical inclusion.
Let Y be a simplicial set and n an integer.Consider the function Y n → N (k) Y n,0,...,0 sending a simplex with characteristic map ζ : △ n → Y to the composition These functions induce a chain map and we have the following.
Proof.The structure-preserving properties of this map are immediate.It to be shown that it induces a homology isomorphism.Consider the composition of quasi-isomorphisms where the second map is induced by the counit of the adjunction.We will now verify that it is left inverse to I. Consider a simplex y with characteristic map ζ : △ n → Y .The multisimplex I(y) is given by the simplicial map Since the only (n, 0, . . ., 0)-shuffle is the identity, the simplex EZ • I(y) is the simplicial map Finally, the image of this simplex under the counit is the evaluation of 3.4.Singular chains.
Theorem 3.4.1.Let X be a topological space.The chain map defined by precomposing a continuous map (∆ n → X) with the projection Proof.This map factors as the composition of two quasi-isomorphisms of E ∞coalgebras.The first is I : S(X) → N(N (k) Sing(X)), which was studied in § 3.3.The second is induced by a multisimplicial isomorphism defined as follows.Using the adjunction of §2.2, any simplicial map which precomposing with ez gives a continuous map It is not hard to see that every such map arises this way since ez is a homeomorphism.

Models of configuration spaces
We are interested in modeling algebraically the S r -equivariant homotopy type of the space of configurations of r labeled and distinct points in Euclidean ddimensional space.Multisimplicial sets can be used to provide an explicit chain complex model with a small number of generators, which, using the E ∞ -structure defined in this paper, retains all homotopical information by Mandell's theorem [Man06].
In the first subsection, we recall a method due to Berger detecting spaces homotopy equivalent to euclidean configuration spaces by means of a filtration indexed by a complete graph poset.In the second subsection, we construct the multisimplicial model and show that is equipped with such a filtration.In the third subsection, we recall the construction of the simplicial Barratt-Eccles model and show that is equipped with a similar filtration.In the fourth subsection, we relate the multisimplicial and simplicial chain models by an explicit map.In the last subsection, we give some examples of the sizes of the two models, showing that the multisimplicial is smaller.4.1.Recognition of configuration spaces.Let Conf r (R d ) denote the configuration space of r-tuples of pairwise disjoint vectors in R d .This space is equipped with a free action of the symmetric group S r of permutations of {1, . . ., r} swapping elements of a r-tuple.Definition 4.1.1.A complete graph on r vertices is a pair (µ, σ) with µ a collection of non-negative integers µ ij for all 1 ≤ i < j ≤ r, and σ is an ordering of {1, . . ., r}.We write σ ij for the restriction of the ordering σ to the set {i, j}.Graphically (µ, σ) is a simple directed graph in the edge corresponding to i < j directed according to σ ij and labeled by µ ij .Please consult Figure 1 for an example.Let us denote the set of complete graphs with r vertices by K(r) equipped with the poset structure for each pair i < j.It is equipped with an exhaustive filtration by subposets where K d (r) consists of those graphs with max(µ ij ) < d.
Definition 4.1.2.For a given poset A, a cellular A-decomposition of a topological space X is a family of subspaces {X a } a∈A such that: The relevance of this notion is the well-known fact that if a topological space X admits a cellular A-decomposition, then the natural maps are cellular homotopy equivalences.Please consult [Ber97, §1.7] for a proof.Let C d (r) be the space of r little d-dimensional cubes, which is equipped with an equivariant homotopy equivalence to Conf r (R d ) picking the center of cubes.Brun and others in [BFV07] show that C d (r) has a cellular K ex d (r)-decomposition {C a }, where K ex d (r) is a poset containing the poset K d (r) and the inclusion of posets induces an equivariant homotopy equivalence on realizations.Combining these results we have is a zig-zag of equivariant homotopy equivalences.
Definition 4.1.4.Let X be a multisimplicial (or simplicial) set.A K(r)-filtration of X is a family of (multi)simplicial subsets {X a } indexed by a ∈ K(r) so that (1) a ≤ b implies X a ⊆ X b ; (2) |X a | is a cellular K(r)-decomposition of the realization |X| In particular this implies that X = colim a∈K(r) X a .Let X d = colim a∈CG d (r) X a .There is a nested sequence X 1 ⊂ X 2 ⊂ • • • For a given (multi)simplex x ∈ X we will refer to min{d | x ∈ X d } as the complexity of x.
4.2.Multisimplicial model.We define for each positive integer r a multisimplicial set Sur(r) equipped with a K(r)-filtration.The functor of chains applied to the nested sequence Sur 1 (r) ⊂ Sur 2 (r) ⊂ • • • will recover the algebraic models Spaces Y r 0 homeomorphic to |Sur(r)| were studied in the work of McClure-Smith [MS03].homeomorphism between Y r 0 and |Sur(r)| is described explicitly in the appendix of [Sal09].
Next we define a K(r)-filtration on Sur(r).For i < j, let f ij be the subsequence of f (1) • • • f (m + r) obtained by omitting all occurrences of elements different from i and j.For (µ, σ) ∈ K(r) we say that f ∈ Sur(r) (µ,σ) if for each i < j, either i and j alternate strictly less than µ ij times in the sequence f ij , or they do so exactly µ ij times and the ordering formed by the first occurrences of i and j in f ij agrees with σ ij .
The surjection f has complexity d or less if the alternation number of each f ij is less than d + 1, i.e., if the non-degenerate dimension of f ij in Sur(2) is d or less for each i < j.We notice that the action of S r on Sur(r) preserves the nested sequence Sur 1 (r) ⊂ Sur 2 (r) ⊂ . . .For the proof that |Sur(r)| has indeed an induced cellular K(r)-decomposition we refer to Lemma 14.8 in [MS04].Applying the functor of singular chains to the zig-zag of Proposition 4.1.3produces a zig-zag of equivariant quasi-isomorphisms of UM-coalgebras connecting S|Sur d (r)| and S Conf r (R d ).We can extend it using the following zig-zag of maps of the same kind As announced in the introduction, this construction relates the chains on the multisimplicial model of configuration space and its singular chains via an explicit zig-zag of equivariant quasi-isomorphisms of E ∞ -coalgebras.4.3.Simplicial model.We recall the Barratt-Eccles simplicial set E(r) defined for each r ∈ N that is equipped with a K(r)-filtration.Applying the functor of chains to the nested sequence will provide the algebraic models of configuration spaces studied by Berger and Fresse in [BF04].
The n-simplices of E(r) are tuples of n + 1 elements of the symmetric group S r .Its face and degeneracy maps are defined by removing and doubling elements respectively.There is an operad structure on these simplicial sets, but we do not consider it here.
In particular w has complexity d or less if for each i < j the non-degenerate dimension of w ij = ((w 0 ) ij , . . ., (w n ) ij ) in E(2) is d or less for all i < j.We notice that the action of S r on E(r) preserves the nested sequence For a proof that this is a K(r)-filtration we refer to Example 2.8 in [Ber97].Please consult [Smi89; Kas93; Ber97] for more details.
Applying the functor of singular chains to the zig-zag of Proposition 4.1.3produces a zig-zag of equivariant quasi-isomorphisms of UM-coalgebras connecting S|E d (r)| and S Conf r (R d ).Using the unit of the Quillen equivalence extends this zig-zag to one relating N E d (r) and S Conf r (R d ), which can be combined with the zig-zag constructed in the previous subsection.As announced in the introduction, this construction relates the chains on the multisimplicial model of configuration space and those the simplicial model via an explicit zig-zag of equivariant quasiisomorphisms of E ∞ -coalgebras.

Table completion.
It is desirable to have a direct S r -equivariant quasi-isomorphism between these algebraic models.Two filtration preserving quasi-isomorphisms were constructed by Berger-Fresse TR : N E(r) → N Sur(r) and TC : N Sur(r) → N E(r).
The first one, introduced in [BF04, 1•3], is not a coalgebra map, as the reader familiar with its definition can easily verify.We will focus on the second one which was introduced in [BF02] and termed table completion.We will construct a factorization TC : N Sur(r) where the second map is induced from a simplicial map defined below.This factorization proves that TC is a coalgebra map since both factors are.We warn the reader that since EZ does not respect the E ∞ -coalgebra structure, neither does TC.
The restriction to the diagonal defines a K(r)-filtration of Sur(r) D and a nested sequence Sur 1 (r) D ⊂ Sur 2 (r) D ⊂ • • • on Sur(r) D that is preserved by the action of S r on Sur(r) D .In terms of cellular K(r)-decompositions, given (µ, σ) ∈ K(r) then f ∈ Sur(r) D (µ,σ) if for each i < j, either i and j alternate strictly less than µ ij times in the sequence f ij , or they do so exactly µ ij times and the ordering formed by the first occurrences of i and j in f ij agrees with σ ij .
Since the complexity of an element is unchanged by degeneracy maps, it can easily be seen that EZ: N Sur(r) → N Sur(r) D preserves K(r)-filtrations.
Let us now define the simplicial map tc.For f as above, let tc(f ) = (σ 0 , . . ., σ m ) with σ j represented by the subsequence of f containing the (j + 1) st occurrence of each ℓ ∈ {1, . . ., r}.For example, we have tc(122333112) = (123, 231, 312).Proof.It is clear that tc is a simplicial map, and verifying its relationship with Berger-Fresse's chain map is straightforward using the prism interpretation of a surjection as in [BF02, §2.1] and explicit combinatorial formula for T C in [BF02, §3.1].
To check that tc preserves K(r)-filtrations let us first notice that tc(f ) ij = tc(f ij ), so without loss of generality we can assume r = 2.In this case it is clear that nondegenerate simplices are sent to non-degenerate simplices (of the same dimension), so for these the complexity is preserved.We conclude the same for degenerate simplices using that tc is a simplicial map and that degeneracy maps leave complexity unchanged.
S|Sur d (r)| ∼ = S|Sur d (r) D | → N(Sur d (r) D ) → N(N (r) (Sur d (r) D )) ← N Sur d (r).The first map is induced by the homeomorphism |Sur d (r)| ∼ = |Sur d (r) D |, the second by the unit of the Quillen equivalence between simplicial sets and topological spaces, the third is the comparison map of §3.3, and last one is induced by the unit of the Quillen equivalence between multisimplicial sets and simplicial sets.
Theorem 4.4.1.The simplicial map tc : Sur(r) D → E(r) satisfies TC = N(tc) • EZ and induces a weak equivalence tc d : Sur d (r) D → E d (r) for every r, d ∈ N.