Magnitude homology and homotopy type of metric fibrations
Yasuhiko Asao Yu Tajima Masahiko Yoshinaga


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    Yassir LAIRGI1,20000-0002-7284-5489 Ludovic MONCLA10000-0002-1590-9546 Rémy CAZABET10000-0002-9429-3865 Khalid BENABDESLEM1 Pierre CLÉAU2
    September 9, 2024
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§ ABSTRACT 
    In this article, we show that each two metric fibrations with a common base and a common fiber have isomorphic magnitude homology, and even more, the same magnitude homotopy type. That can be considered as a generalization of a fact proved by T. Leinster that  the magnitude of a metric fibration with finitely many points is a product of those of the base and the fiber. We also show that the definition of  the magnitude homotopy type due to the second and the third authors is equivalent to the geometric realization of Hepworth and Willerton's pointed simplicial set.



§ INTRODUCTION

The notion of a metric fibration was defined by T. Leinster in his study of magnitude (<cit.>). It is a “fibration in the category of metric spaces”,  defined analogously to the Grothendieck fibrations of small categories, where one sees a metric space as an category enriched over ([0, ∞), ≥, +). Based on the fact that a Grothendieck fibration can also be considered as a lax functor, the first author later provided an analogous description for the metric fibration (<cit.>). A remarkable property of the metric fibration is that the magnitude of the total space of a metric fibration is a product of those of the base and the fiber if they are finite metric spaces (<cit.> Theorem 2.3.11).  In this article, we show that the same is true for the magnitude homology and the magnitude homotopy type of a metric fibration possibly with infinitely many points. Namely we have the following.

    Let π : E  B a metric fibration, and let F be its fiber. For ℓ>0, we have a homotopy equivalence 
    
    ^ℓ_∗(E) ≃⊕_ℓ_ + ℓ_ = ℓ^ℓ__∗(F)⊗^ℓ__∗(B),

    where  denotes the magnitude chain complex.


    Let π : E  B be a metric fibration and let F be its fiber. Then we have a homotopy equivalence 
    
    | M^ℓ_∙(E)| ≃⋁_ℓ_ + ℓ_ = ℓ| M^ℓ__∙(F)| ∧ | M^ℓ__∙(B)|,

     where | M^ℓ_∙(-)| is the geometric realization of the Hepworth and Willerton's pointed simplicial set (<cit.>).

In particular, we give an another proof for  the Künneth theorem for magnitude homology proved by Hepworth and Willerton (<cit.> Proposition 8.4). 

We use the terminology magnitude homotopy type as a CW complex whose singular homology is isomorphic to the magnitude homology of some metric sapce. Such a topological space first appeared in Hepworth and Willerton's paper (<cit.> Definition 8.1), and later the second and the third author gave another definition (<cit.>) by generalizing the construction for graphs due to the first author and Izumihara (<cit.>). In their paper, the second and the third author stated that the both definitions of the magnitude homotopy type, theirs and Hepworth-Willeton's, are equivalent without a proof. We gave a proof for it in the appendix (Proposition <ref>).

The main idea of the proof of our main results is to construct a contractible subcomplex D^ℓ_∗(E) of the magnitude chain complex ^ℓ_∗(E) for a metric fibration π : E  B. We have the following isomorphism (Proposition <ref>)


    ^ℓ_∗(E)/D_∗^ℓ(E)≅⊕_ℓ_ + ℓ_ = ℓ^ℓ__∗(F)⊗^ℓ__∗(B),

where F is the fiber of π. To find such a subcomplex D^ℓ_∗(E), we use the  classification horizontal, vertical, tilted, of pairs of points of E as in Figure <ref>. We define (Definition <ref>) a submodule D^ℓ_n(E) of ^ℓ_n(E) ⊂ E^n+1 as one generated by tuples (x_0, …, x_n) that contains tilted pair (x_s, x_s+1) earlier than horizontal-vertical triple (x_t, x_t+1, x_t+2) (namely s+1 ≤ t), or contains  horizontal-vertical triple (x_t, x_t+1, x_t+2) earlier than tilted pair (x_s, x_s+1) (namely t+2 ≤ s). We show that D^ℓ_∗(E) is a subcomplex of ^ℓ_∗(E) (Lemma <ref>), and that it is contractible (Proposition <ref>) by using the algebraic Morse theory. For the magnitude homotopy type, we basically follow the same argument using Δ-sets instead of chain complexes (Section <ref>).



In the remained part of this article, we show the isomorphism of magnitude homology in Section <ref>, and show the equvalence of magnitude homotopy type in Section <ref>. The Section <ref> is an appendix section in which we show the equivalence of definions of the magnitude homotopy type.



  §.§.§ Acknowledgements

Y. A. was supported by JSPS KAKENHI 24K16927. Y. T. was supported by JST SPRING JPMJSP2119.
 M. Y. was partially supported by JSPS KAKENHI JP22K18668 and JP23H00081.


§ ISOMORPHISM AT HOMOLOGY LEVEL



 §.§ magnitude homology


    Let (X, d) be a metric space.
    
        
  * For ℓ∈_≥ 0 and n ∈_≥ 0, we define 
        
    P_n^ℓ(X) := {(x_0, …, x_n) ∈ X^n+1| x_i≠ x_i+1, ∑_i=0^n-1d(x_i, x_i+1) = ℓ},

        and P_n(X) := ∪_ℓ P_n^ℓ(X).
        
  * For x, y, z ∈ X, we write x ≺ y ≺ z if d(x, z) = d(x, y) + d(y, z).
        
  * The magnitude chain complex (^ℓ_∗(X), ^ℓ_∗) is defined by  ^ℓ_n(X) =  P^ℓ_n(X) and
    
    _n (x_0, …, x_n) := ∑_x_i-1≺ x_i ≺ x_i+1(-1)^i(x_0, …, x̂_i, …, x_n).

Its homology ^ℓ_∗(X) is called the magnitude homology of  X.
        
    




 §.§ metric fibration



    A Lipschitz map π : E  B is a metric fibration if it satisfies the following : for all x ∈ E and b ∈ B, there uniquely exists x^b ∈π^-1b satisfying 
    
        
  * d(x, x^b) = d(π x, b),
        
  * d(x, y) = d(x, x^b) + d(x^b, y) for all y ∈π^-1b.
    



    Let π : E  B be a metric fibration. For b, b' ∈ B, a map π^-1b π^-1b' ; x ↦ x^b' is an isomorphism of metric spaces.


    <cit.> Lemma 2.3.10, <cit.> Lemma 3.4.




    
  * Let  be a monoid freely generated by words h, v, t. We denote the subset of  that consists of n words by _n.
    
  * For a metric fibration π : E  B, we define a map T : P_1(E) _1 by 
    
    T(x, x') =     if d(x, x') = d(π x, π x'), 
        if d(π x, π x') = 0, 
        if 0 < d(π x, π x')  < d(x, x').

    We extend this map to a map T : P_n(E) by T(x_0, …, x_n) = T(x_0, x_1)… T(x_n-1, x_n).
    
  * For xy∈_2 and z∈_1, we write xy =  z if there is a metric fibration π : E  B and (x, y, z) ∈ P_2(E) satisfying that x≺ y ≺ z, T(x, y, z) =  xy and T(x, z) =  z. We also define { xy} = { z∈_1 | xy =  z}.
    
    



    The words , , are abbreviations of horizontal, vertical and tilted respectively.



In the following figures, the graph on the left is (I_2 × I_2) × I_3, where I_n is the graph with vertices {1, …, n} and edges {{i, i+1}| 1 ≤ i ≤ n-1}, and the graph on the right is a non-trivial metric fibration over the complete graph K_3 with the fiber I_2. We have the following :


  * 1 ≺ 2 ≺ 6, T(1, 2, 6) = , T(1, 6) = , 
 
1 ≺ 5 ≺ 6, T(1, 5, 6) = , T(1, 6) = ,
        
  * 1 ≺ 2 ≺ 7, T(1, 2, 7) = , T(1, 7) = , 

 1 ≺ 6 ≺ 7, T(1, 6, 7) = , T(1, 7) = ,
        
  * 1 ≺ 5 ≺ 10, T(1, 5, 10) = , T(1, 10) = , 

1 ≺ 6 ≺ 10, T(1, 6, 10) = , T(1, 10) = ,
        
  * 1 ≺ 6 ≺ 11, T(1, 6, 11) = , T(1, 11) =,
        
  * 1 ≺ 2 ≺ 3, T(1, 2, 3) = , T(1, 3) =,
        
  * a ≺ e ≺ f, T(a, e, f) = , T(a, f) =.

    



For each x,  y∈_1, we have the following.

    
  * { xy} = {}⇔ xy = ⇔ xy =.
    
  * {}= {}=  {}, and { x} = { x} = {} for all x∈_1.

  * {} = {, }. 

  * For (x, y, z) ∈ P_2(E) with x≺ y ≺ z and T(x, y, z) =, we have T(x, z) = if and only if π x ≺π y ≺π z.



    
            
  * Obviously we have { xy} = {}⇒ xy =. Also we have xy = ⇒{ xy} = {}. Hence it is enough to show that  xy = implies xy =. Let (x, y, z) ∈ P_2(E) with x ≺ y ≺ z. We show that T(x, z) = implies T(x, y) = T(y, z) =. If T(x, z) =, we have π x = π z, which implies 
        
    d(x, y) + d(y, z)    = d(π x, π y) + d(x, y^π x) + d(π y, π z) + d(y^π z, z)
       = d(x, y^π x) + d(y^π x, z) +  2d(π x, π y) 
       ≥ d(x, z) + 2d(π x, π y).

       Since we have x ≺ y ≺ z, we obtain that d(π x, π y) = d(π z, π y) = 0, namely T(x, y) = T(y, z) =.
        
  * Note that we have = and = by Example <ref> (1), and we also have ( = ) and ( = ) by the definition of the metric fibration. Hence we obtain {}= {} = {}. Suppose that T(x, y, z) =  x for (x, y, z) ∈ P_2(E),  x∈_1 and x ≺ y ≺ z. Then we have d(x, z) = d(x, y) + d(y, z) > d(π x, π y) + d(π y, π z) ≥ d(π x, π z) > 0 by T(y, z) = and (1). Hence we obtain T(x, z) =, and by Example <ref> (2), (3) and (4), we obtain { x} = {}. We can similarly show that  { x} = {}.
        
  * We have {}⊂{, } by (1), and the inverse inclusion follows from Example <ref> (5) and (6).
        
  * By T(x, y, z) = and x≺ y ≺ z, we have 
        
    d(x, z) = d(x, y) + d(y, z) = d(π x, π y) + d(π y, π z).

        Hence T(x, z) = implies that d(π x, π z) = d(x, z) = d(π x, π y) + d(π y, π z), and π x ≺π y ≺π z implies that d(x, z) = d(π x, π z).
    




 §.§ a subcomplex D_∗^ℓ(E)⊂^ℓ_∗(E)

In the following, we construct a chain subcomplex D^ℓ_∗(E) ⊂^ℓ_∗(E) that consists of  tuples of special types P_n^ℓ, (E) and P_n^ℓ, (E).  We define the set P_n^ℓ, (E) ⊂ P^ℓ_n(E) as tuples containing tilted pair (x_s, x_s+1) earlier than horizontal-vertical triple (x_t, x_t+1, x_t+2) (namely s+1 ≤ t). Dually, we define the set P_n^ℓ, (E) ⊂ P^ℓ_n(E)  as tuples containing  horizontal-vertical triple (x_t, x_t+1, x_t+2) earlier than tilted pair (x_s, x_s+1) (namely t+2 ≤ s). Formally we define them as follows.

    For a metric fibration π : E  B, we define subsets P_n^ℓ, (E), P_n^ℓ, (E) ⊂ P^ℓ_n(E) by 
    
    P_n^ℓ, (E)    := {x ∈ P^ℓ_n(E) | Tx ∈^m^m' for  m, m'≥ 0}, 
    
            P_n^ℓ, (E)    := {x ∈ P^ℓ_n(E) | Tx ∈^m^m'+1 for  m, m'≥ 0}.

    We also define a submodule  D^ℓ_n(E) :=  P_n^ℓ, , (E) ⊂^ℓ_n(E), where P_n^ℓ, , (E) = P_n^ℓ, (E)∪ P_n^ℓ, (E).











    We have _n x ∈ D^ℓ_n-1(E) for x ∈ P_n^ℓ, , (E). Namely, D^ℓ_∗(E) ⊂^ℓ_∗(E) is a chain subcomplex.


     It follows from Lemma <ref>.




    Let π : E  B be a metric fibration.  We fix b ∈ B and F := π^-1b. Then we have an isomorphism of chain complexes 
    
    ^ℓ_∗(E)/D_∗^ℓ(E)≅⊕_ℓ_ + ℓ_ = ℓ^ℓ__∗(F)⊗^ℓ__∗(B).

    



    Note that the module ^ℓ_n(E)/D_n^ℓ(E) is freely generated by tuples x ∈ P^ℓ_n(E) with Tx = ^m^n-m for some 0 ≤ m ≤ n. For each n≥ 0,  we define a homomorphism φ_n :  ^ℓ_n(E)/D_n^ℓ(E) ⊕_ℓ_ + ℓ_ = ℓ
 m≥ 0^ℓ__m(F)⊗^ℓ__n-m(B) by 
    
    φ_n(x_0, …, x_n) = (x_0^b, …, x_m^b)⊗ (π x_m, …, π x_n),

    where we suppose that T(x_0, …, x_n) = ^m^n-m. This homomorphism has an inverse ψ_n defined by 
    
    ψ_n((f_0, …, f_m)⊗ (b_0, …, b_n-m)) = (f_0^b_0, …, f_m^b_0, f_m^b_0b_1, f_m^b_0b_1b_2, …, f_m^b_0… b_n-m),

    where we denote a point (f_m^b_0)^b_1 by f_m^b_0b_1 and similarly for further iterations.
    Hence it reduces to show that φ_∗ is a chain map. We denote the boundary operator on ^ℓ_∗(E)/D_∗^ℓ(E) induced from ^ℓ_∗ by [^ℓ]_∗ in the following. For (x_0, …, x_n) ∈^ℓ_n(E)/D_n^ℓ(E) with T(x_0, …, x_n) = ^m^n-m, we have 
    
    [^ℓ]_n(x_0, …, x_n)    = ∑_x_i-1≺ x_i ≺ x_i+1
     T(x_i-1, x_i+1) ≠(-1)^i(x_0, …, x̂_i, …, x_n) 
       = ∑_x_i-1≺ x_i ≺ x_i+1
     1 ≤ i ≤ m-1(-1)^i(x_0, …, x̂_i, …, x_m, …, x_n) 
       + ∑_π x_i-1≺π x_i ≺π x_i+1
     m+1 ≤ i ≤ n-1(-1)^i(x_0, …, x_m, …,  x̂_i, …,  x_n),

    by Lemma <ref> (2) and (4). Hence we obtain that 
    
    φ_n-1[^ℓ]_n(x_0, …, x_n)    = ∑_x^b_i-1≺ x^b_i ≺ x^b_i+1
     1 ≤ i ≤ m-1(-1)^i(x^b_0, …, x̂_i^b, …, x^b_m) ⊗ (π x_m, …, π x_n) 
       + ∑_π x_i-1≺π x_i ≺π x_i+1
     m+1≤ i ≤ n-1(-1)^i(x^b_0, …, x^b_m) ⊗ (π x_m, …,  π x_i, …, π x_n).

     On the other hand, for φ_n(x_0, …, x_n) = (x^b_0, …, x^b_m)⊗ (π x_m, …, π x_n) ∈^ℓ__m(F)⊗^ℓ__n-m(B), we have
    
    (_m^ℓ_⊗_n-m^ℓ_)φ_n(x_0, …, x_n)     = ∑_x^b_i-1≺ x^b_i ≺ x^b_i+1
     1 ≤ i ≤ m-1(-1)^i(x^b_0, …, x̂_i^b, …, x^b_m)⊗ (π x_m, …, π x_n) 
       +  ∑_π x_i-1≺π x_i ≺π x_i+1
     m+1≤ i ≤ n-1(-1)^i(x^b_0, …, x^b_m)⊗ (π x_m, …, π x_i, …,   π x_n).

    Thus we obtain that φ_n-1[^ℓ]_n =  (_m^ℓ_⊗_n-m^ℓ_)φ_n.




 §.§ Algebraic Morse Theory

We recall the algebraic Morse theory studied in <cit.>. Let C_∗ = (C_∗, ∂_∗) be a chain complex with a decomposition C_k = ⊕_a ∈ I_n C_n, a and C_n, a≅ for each k. For a ∈ I_n+1 and b ∈ I_n, let f_ab C_n+1, a  C_n, b be the composition
C_n+1, a↪ C_n+1  C_n↠ C_n, b. We define a directed graph Γ_C_∗ with vertices ∐_n I_n and directed edges {a → b | f_ab≠ 0}. We recall terminologies on the matching.  



  * A matching M of a directed graph Γ is a subset of directed edges M ⊂ E(Γ) such that each two distinct edges in M have no common vertices. 

  * For a matching M of a directed graph, vertices that are not the endpoints of any edges in M are called critical. 

  * For a matching M of a directed graph Γ, we define a new directed graph Γ^M by inverting the direction of all edges in M.



A matching M on Γ_C_∗  is called a Morse matching if it satisfies the following.


  * f_ab is an isomorphism if a → b ∈ M. 

  * Γ_C_∗^M is acyclic, that is, there are no closed paths in Γ_C_∗^M of the form
a_1→ b_1→…→ b_p-1→ a_p=a_1
with a_i∈ I_n+1 and b_i∈ I_n for some p.


 

 For a matching M on Γ_C_∗, we denote the subset of I_n that consists of critical vertices by I_n. 

 
 For a Morse matching M on Γ_C_∗, we have a chain complex (C_n = ⊕_a ∈I_n C_n, a, _∗) that is  homotopy equivalent to (C_∗, _∗).
 

 

 §.§ matching on D_∗^ℓ(E)

We apply algebraic Morse theory to the chain complex (D^ℓ_∗(E), ^ℓ_∗) with the decomposition D^ℓ_n(E) = ⊕_a ∈ P^ℓ, , _n(E) D_n, a and D_n, a≅. For a = (x_0, …, x_n+1) ∈ P^ℓ, , _n+1(E) and b ∈ P^ℓ, , _n(E), we write b = ^ℓ_n+1, ia if b = (x_0, …, x̂_i, …, x_n+1). It is immediately verified that f_ab is an isomorphism for a ∈ P^ℓ, , _n+1(E) and b ∈ P^ℓ, , _n(E) if and only if b = ^ℓ_n+1, ia for some i.

 
 
 

  * For a = (x_0, …, x_n) ∈ P^ℓ, _n(E) with Ta ∈^m^m', we define 

    a^ := (x_0, …, x_m+m', x_m+m'^π x_m+m'+1,  x_m+m'+1, …, x_n).


  * For (x_0, …, x_n) ∈ P^ℓ_n(E), we define 

    |(x_0, …, x_n)| := ∑_T(x_i, x_i+1)=  i.



Namely, we obtain a tuple a^ by filling the gap of the first tilted part of a. The filled part becomes horizontal-vertical triple.

    Let a_1 ≠ a_2 ∈ P^ℓ, _n(E). If a_2 = ^ℓ_n+1, ia_1^ for some i,  then we have |a_1^| < |a_2^|.


    Suppose that Ta_1 = ^m^m' xw for some x∈_1 and w∈. Then we have Ta_1^ =  ^m^m'+1 xw. If we have ^ℓ_n+1, ia_1^ = a_2 ∈ P^ℓ, _n(E), then we should have  
    
    Ta_2 ∈{^m-1^m' xw,  ^m^m”^m'-m”-2 xw,  ^m^m'+1 w},

    by Lemma <ref>. In each case, we have 
    
    Ta_2^∈{^m-1^m' xw,  ^m^m”+1^m'-m”-2 xw,  ^m^m'+2 w}

    respectively. In all cases, we have |a_1^| < |a_2^|. 

We define a matching M on D_∗^ℓ(E) by 

    M = {f_a^a : a^→ a | a ∈ P^ℓ, _n(E) }.

This is apparently a matching, and is also acyclic by Lemma <ref>. Further, there is no critical vertex in Γ_D_∗^ℓ(E). Thus we obtain the following by Proposition <ref>.


    The chain complex D_∗^ℓ(E) is contractible.



    Let π : E  B a metric fibration, and let F be its fiber. For ℓ>0, we have a homotopy equivalence and an isomorphism
    
    ^ℓ_∗(E) ≃^ℓ_∗(E)/D^ℓ_∗(E) ≅⊕_ℓ_ + ℓ_ = ℓ^ℓ__∗(F)⊗^ℓ__∗(B).



    It follows from Propositions <ref>, <ref> and the fact that each quasi-isomorphism between levelwise free chain complexes is induced from a homotopy equivalence.


    Note that, by Corollary <ref>, we reprove  the Künneth theorem in <cit.> Proposition 8.4, namely ^ℓ_∗(F× B) ≅ H_∗(⊕_ℓ_ + ℓ_ = ℓ^ℓ__∗(F)⊗^ℓ__∗(B)).




§ EQUIVALENCE OF MAGNITUDE HOMOTOPY TYPE




 §.§ Δ-set


We denote the category of finite ordinals {0 < 1< … < n} = :[n]'s and order preserving maps between them by Δ. We define maps δ_n,i : [n-1]  [n] and σ_n,i : [n+1]  [n] for 0 ≤ i ≤ n by δ_n, i j = 
        j     j < i, 

        j+1     j ≥ i, and σ_n, i j = 
        j     j ≤ i, 

        j-1     j > i. We abbreviate them to δ_i and σ_i. Note that all order preserving map f : [m]  [n] can be uniquely decomposed as a composition of order preserving maps f = ϕ_1(f)ϕ_2(f) such that ϕ_1(f) is injective and ϕ_2(f) is surjective. Also, we can decompose ϕ_1(f) and ϕ_2(f) into compositions of δ_i's and σ_i's respectively.

    A family of sets X_∙ = {X_n}_n≥ 0 equipped with maps d_i : X_n  X_n-1 (0 ≤ i ≤ n) is called a Δ-set if it satisfies d_id_j = d_j-1d_i for i<j. Equivalently, a Δ-set is a functor Δ_ inj^ op, where Δ_ inj is the category of finite ordinals and order preserving injections that are generated from δ_i's. We define the category of Δ-sets by Δ := ^Δ_ inj^ op.




 Note that the inclusion j : Δ_ injΔ induces a functor j^∗ : Δ. Namely, for a simplicial set S_∙, we can obtain a Δ-set j^∗ S_∙ by forgetting the degeneracy maps. The functor j^∗ has the left adjoint (<cit.> Theorem 1.7)  j_! : Δ defined by 
 
    (j_!X_∙)_n = {(p, f) | p ∈ X_n-k, f : [n] ↠ [n-k] ∈Δ , 0≤ k ≤ n}.

The structure maps d_i : (j_!X_∙)_n  (j_!X_∙)_n-1, s_i : (j_!X_∙)_n  (j_!X_∙)_n+1 for 0≤ i ≤ n are defined by 

 
    d_i(p, f)    = ((ϕ_1(fδ_i))^∗ p, ϕ_2(fδ_i)),
    
         s_i(p, f)    = (p, fσ_i),

where we use the following composition and factorization of maps: 

    [n-1] [r]^-δ_i[dr]_-ϕ_2(fδ_i)    [n] [r]^-f    [n-k] 
        [m]  [ur]_-ϕ_1(fδ_i)   .


    
        
  * For a metric space X, ℓ∈_≥ 0 and n ∈_≥ 0, we define m^ℓ_n(X) := P^ℓ_n(X)∪{∗}. We also define maps d_i :  m^ℓ_n(X)  m^ℓ_n-1(X) for 0 ≤ i ≤ n by
        
    d_i(∗)    = ∗, 
    
                d_i(x_0, …, x_n)    = 
                    (x_0, …, x̂_i, …, x_n)    if  x_i-1≺ x_i ≺ x_i+1, 1≤ i ≤ n-1, 
    ∗   otherwise.

Then it is immediate to verify that m^ℓ_∙(X) is a Δ-set.


  * For a metric space X, we denote Hepworth and Willerton's simplicial set (<cit.> Definition 8.1) by M^ℓ_∙(X). That is defined by 

    M^ℓ_n(X) = {(x_0, …, x_n) ∈ X^n+1|∑_i=0^n-1d(x_i, x_i+1) = ℓ}∪{∗},

for ℓ∈_≥ 0 and n ∈_≥ 0. The maps d_i's are defined by the same formula as those of m^ℓ_∙, and s_i's are defined by s_i(x_0, …, x_n) = (x_0, …, x_i, x_i, …, x_n) and s_i(∗) = ∗.


  * For a point ∗∈, defined by ∗_n = {∗}, we have

    (j_!j^∗∗)_n ≅{f : [n] ↠ [n-k] | 0 ≤ k ≤ n},

and d_if = ϕ_2(fδ_i), s_i f = fσ_i for f : [n] ↠ [n-k]. Note that the non-degenerate simplices of (j_!j^∗∗)_∙ are only identities id_[n], and its geometric realaization |(j_!j^∗∗)_∙| is S^∞.


  * For a metric space X and ℓ∈_≥ 0, we define a simplicial set M^ℓ_∙(X) by 
    M^ℓ_n(X) = {(x_0, …, x_n) ∈ X^n+1|∑_i=0^n-1d(x_i, x_i+1) = ℓ}∪{f : [n] ↠ [n-k] | 0 ≤ k ≤ n}.

We define 

    d_i(f)    = ϕ_2(fδ_i), 
    
                d_i(x_0, …, x_n)    = 
                    (x_0, …, x̂_i, …, x_n)    if  x_i-1≺ x_i ≺ x_i+1, 1≤ i ≤ n-1, 
     id_[n-1]   otherwise.


and 

    s_i(f)    = fσ_i, 
    
                s_i(x_0, …, x_n)    = (x_0, …, x_i, x_i, …, x_n).

    




    We have j_! m^ℓ_∙(X) ≅ M^ℓ_∙ (X).



In the following, we denote the maps j_! m^ℓ_n(X)  j_! m^ℓ_m(X) and  M^ℓ_n(X)  M^ℓ_m(X) induced from a map f : [m]  [n] by f^ m and f^ M respectively. We also denote the structure maps d_i, s_i's of j_! m^ℓ_∙(X) and M^ℓ_∙(X) by d_i^ m, s_i^ m and d_i^ M, s_i^ M's respectively. We define a map F_n : (j_! m^ℓ_∙(X))_n  M^ℓ_n (X) by

    F_n (p, f) =  f     p= ∗
    
    f^ Mp     p ≠∗,

where we identify an element p ∈ P^ℓ_n-k(X) ⊂ m^ℓ_n-k(X) with an element p ∈ M^ℓ_n-k (X). This map is obviously a bijection, hence it reduces to show that this defines a morphism of simplicial sets. Now we have 


    F_n+1 s_i^ m(p, f)    = F_n+1(p, fσ_i)= fσ_i    p= ∗
    
    s_i^ Mf^ Mp     p≠∗ =s_i^ MF_n(p,f).

We also have

    F_n-1 d_i^ m(p, f)    = F_n-1(ϕ_1^ m p, ϕ_2) =ϕ_2    ϕ_1^ m p= ∗
    ϕ_2^ Mϕ_1^ M p    ϕ_1^ m p≠∗,

where we abbreviate ϕ_1(fδ_i), ϕ_2(fδ_i) to  ϕ_1, ϕ_2 respectively, and we identify ϕ^ m_1p ∈ m^ℓ_∙(X) with ϕ^ M_1p ∈ M^ℓ_∙ (X). Also, we have
 
    d_i^ MF_n(p, f)    =  d_i^ Mf     p= ∗, 
    
     d_i^ Mf^ M p     p ≠∗,
       = ϕ_2     p= ∗, 
    ϕ_2^ Mϕ_1^ M p     p≠∗,
       = ϕ_2     p= ∗, 
    ϕ_2     p ≠∗, ϕ_1^ m p =∗ ,
    ϕ_2^ Mϕ_1^ M p     p≠∗, ϕ_1^ m p ≠∗, 
    
       =ϕ_2    ϕ_1^ m p= ∗, 
    ϕ_2^ Mϕ_1^ M p    ϕ_1^ m p≠∗.

Hence F_∙ is an isomorphism of simplicial sets.






    We have a homotopy equivalence | M^ℓ_∙(X)| ≃ | M^ℓ_∙(X)|.


Obviously we have an inclusion j_!j^∗∗ M^ℓ_∙(X), and its quotient map M^ℓ_∙(X) M^ℓ_∙(X). Hence it induces a sequence |j_!j^∗∗|  | M^ℓ_∙(X)| | M^ℓ_∙(X)|. Since |j_!j^∗∗| ≃ S^∞ is a subcomplex of | M^ℓ_∙(X)|, we conclude that | M^ℓ_∙(X)| ≃ | M^ℓ_∙(X)|.





























 §.§ D^ℓ_∙(E) ⊂ m^ℓ_∙(E)



    For a metric fibration π : E  B, we define a Δ-subset D^ℓ_∙(E) ⊂ m^ℓ_∙(E) by D^ℓ_n(E) = P^ℓ, , _n(E)∪{∗} for ℓ∈_≥ 0.

We can verify that D^ℓ_∙(E) is indeed a Δ-set by Lemma <ref>.


    |j_!D^ℓ_∙(E)| is contractible.


    By the same argument as the proof of Proposition <ref>, |j_!D^ℓ_∙(E)| is homotopy equivalent to the geometric realization of a simplicial subset K_∙⊂ M^ℓ_∙(E) generated from the family of sets P^ℓ, , _∙(E). Since the non-degenerate simplices of K_∙ are elements of P^ℓ, , _n(E)'s, the chain complex C_∗ K is homotopy equivalent to the chain complex D^ℓ_∗(E) of Definition <ref>, which is contractible. Therefore it reduces to show that |K_∙| is simply connected. Recall that the fundamental groupoid Π_1 |K_∙| is equivalent to the fundamental groupoid Π_1 K_∙, whose objects are vertices of K_∙ and morphisms are generated by edges of K_∙ with the identification d_0σ d_2σ∼ d_1σ for σ∈ K_2. Now Π_1 K_∙ has only one object, and each morphism is a sequence of tuples (x_0, x_1) with T(x_0, x_1) =. Since we have (x_0, x_1)=d_1(x_0, x_1)^∼ d_0(x_0, x_1)^d_2(x_0, x_1)^∼∗, this groupoid is a trivial group.


    We have a homotopy equivalence |j_! m^ℓ_∙(E)| ≃ |j_! m^ℓ_∙(E)/j_!D^ℓ_∙(E)|.


   Same as Proposition <ref>.



    We have m^ℓ_∙(E)/D^ℓ_∙(E) ≅ m^ℓ_∙(F× B)/D^ℓ_∙(F× B), where F = π^-1b for a fixed b∈ B.


    We define a map φ_∙ :  m^ℓ_∙(E)/D^ℓ_∙(E)  m^ℓ_∙(F× B)/D^ℓ_∙(F× B) by φ_n(∗) = ∗ and 
     
    φ_n(x_0, …, x_n)  
       =((x_0^b, π x_0), …, (x_i^b, π x_i),  …, (x_m^b, π x_m),(x_m^b, π x_m+1), …, (x_m^b, π x_m+j) …,  (x_m^b, π x_n)),

    where we suppose that T(x_0, …, x_n) = ^m^n-m. This map has an inverse ψ_∙ defined by 
    
    ψ_n((f_0, b_0), …, (f_m, b_0), (f_m, b_1), …, (f_m, b_n-m))= (f_0^b_0, …, f_m^b_0, f_m^b_0b_1, f_m^b_0b_1b_2, …, f_m^b_0… b_n-m).

    Hence it reduces to show that φ_∙ is a morphism of Δ-sets, but it can be verified in the same manner as Propposition <ref>. 



    Let π : E  B be a metric fibration and let F be its fiber. Then we have a homotopy equivalence | M^ℓ_∙(E)| ≃ | M^ℓ_∙(F× B)|. 


    We have homotpy equivalences
    
    | M^ℓ_∙(E)| ≃ |j_!  m^ℓ_∙(E)|  ≃  | j_! m^ℓ_∙(E)/j_!D^ℓ_∙(E)| 
       ≅  |j_! m^ℓ_∙(F× B)/j_!D^ℓ_∙(F× B)|   ≃  | j_! m^ℓ_∙(F× B)|  ≃  |  M^ℓ_∙(F× B)|,

by Propositions <ref>, <ref>, <ref> and <ref>. Note that j_! commutes with quotients since it is a left adjoint.

From Tajima and Yoshinaga's Künneth theorem for magnitude homotopy type (<cit.> Theorem 4.27) together with the coincidence of two definitions of magnitude homotopy types (Proposition <ref>), we have the following.

    Let π : E  B be a metric fibration and let F be its fiber. Then we have a homotopy equivalence | M^ℓ_∙(E)| ≃⋁_ℓ_ + ℓ_ = ℓ| M^ℓ__∙(F)| ∧ | M^ℓ__∙(B)|. 




§ APPENDIX


In this appendix, we prove the following proposition which is stated in <cit.> without a proof.


   Let X be a metric space and ℓ∈_≥ 0. Tajima and Yoshinaga's magnitude homotopy type ℳ^ℓ(X) is homeomorphic to the geometric realization | M^ℓ_∙(X)| of Hepworth and Willerton's simplicial set M^ℓ_∙(X). 


Recall from <cit.> that the CW complex ℳ^ℓ(X) is defined as the quotient |Δ Cau^ℓ(X)|/|Δ' Cau^ℓ(X)| of the geometric realization of simplicial complexes Δ Cau^ℓ(X) and Δ' Cau^ℓ(X). Here, the simplicial complex Δ Cau^ℓ(X) is the order complex of the poset Cau^ℓ(X) = ∐_a, b ∈ X Cau^ℓ(X ; a, b) defined by 

    Cau^ℓ(X ; a, b) = {(x, t) ∈ X × [0, ℓ] | d(a, x) ≤ t, d(x, b) ≤ℓ - t},

where (x, t) ≤ (x', t') if and only if d(x, x') ≤ t'-t. Then the simplicial complex Δ Cau^ℓ(X) = ∐_a, b ∈ XΔ Cau^ℓ(X ; a, b) is defined by

    Δ Cau^ℓ(X ; a, b) = {{(x_0, t_0), …, (x_n, t_n)}| d(x_i, x_i+1) ≤ t_i+1-t_i  for -1 ≤ i ≤ n },

where we put x_-1 = a, x_n+1 = b, t_-1 = 0, t_n+1 = ℓ. Since we can extend each partial order to a total order, the simplicial complex Δ Cau^ℓ(X ; a, b) can be considered as an ordered simplicial complex, and each face of it can be expressed as a tuple ((x_0, t_0), …, (x_n, t_n)) which is not just a set of points {(x_0, t_0), …, (x_n, t_n)}. The simplicial subcomplex Δ' Cau^ℓ(X) = ∐_a, b ∈ XΔ Cau^ℓ(X ; a, b) is defined by

    Δ' Cau^ℓ(X ; a, b) = {((x_0, t_0), …, (x_n, t_n)) ∈Δ Cau^ℓ(X ; a, b) |∑_i=0^n-1d(x_i, x_i+1) <ℓ},

which is also ordered. Here we note that we have d(x_i, x_i+1) = t_i+1-t_i for all -1≤ i ≤ n if and only if ∑_i=0^n-1d(x_i, x_i+1) = ℓ by Proposition 4.2 of <cit.>.

    Note first that each ordered simplicial complex X can be turned into a Δ-set X in a natural manner, and we obtain a simplicial set j_!X. Obviously, the geometric realization of the ordered simplicial complex X is homeomorphic to the geometric realization |j_!X| by the definitions. Also, for a pair Y ⊂ X of ordered simplicial complexes, we have |X|/|Y| ≅ |j_!X|/|j_!Y| ≅ |j_!X/j_!Y|. Hence we have 
    
    ℳ^ℓ(X)    = |Δ Cau^ℓ(X)|/|Δ' Cau^ℓ(X)| 
       ≅⋁_a, b|Δ Cau^ℓ(X; a, b)|/|Δ' Cau^ℓ(X; a, b)| 
       ≅⋁_a, b|j_!Δ Cau^ℓ(X; a, b)/j_!Δ' Cau^ℓ(X; a, b)|
       ≅ |⋁_a, bj_!Δ Cau^ℓ(X; a, b)/j_!Δ' Cau^ℓ(X; a, b)|.

    Now, by Proposition 4.2 of <cit.>,  we have ⋁_a, bj_!Δ Cau^ℓ(X; a, b)/j_!Δ' Cau^ℓ(X; a, b) =  M^ℓ_∙(X). 

99

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