Homotopy equivalence of normalized and unnormalized complexes, revisited
We consider the unnormalized and normalized complexes of a simplicial or a cosimplicial object coming from the DoldśKan correspondence for an idempotent complete additive category (kernels and cokernels are not required). The normalized complex is defined as the image of certain idempotent in the un...
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| Цитувати: | Homotopy equivalence of normalized and unnormalized complexes, revisited / V. Lyubashenko, A. Matsui // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 2. — С. 253-266. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1887522023-03-15T01:27:23Z Homotopy equivalence of normalized and unnormalized complexes, revisited Lyubashenko, V. Matsui, A. We consider the unnormalized and normalized complexes of a simplicial or a cosimplicial object coming from the DoldśKan correspondence for an idempotent complete additive category (kernels and cokernels are not required). The normalized complex is defined as the image of certain idempotent in the unnormalized complex. We prove that this idempotent is homotopic to identity via homotopy which is expressed via faces and degeneracies. Hence, the normalized and unnormalized complex are homotopy isomorphic to each other. We provide explicit formulae for the homotopy. 2021 Article Homotopy equivalence of normalized and unnormalized complexes, revisited / V. Lyubashenko, A. Matsui // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 2. — С. 253-266. — Бібліогр.: 7 назв. — англ. 1726-3255 DOI:10.12958/adm1879 2020 MSC: 18G31, 18N50 http://dspace.nbuv.gov.ua/handle/123456789/188752 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We consider the unnormalized and normalized complexes of a simplicial or a cosimplicial object coming from the DoldśKan correspondence for an idempotent complete additive category (kernels and cokernels are not required). The normalized complex is defined as the image of certain idempotent in the unnormalized complex. We prove that this idempotent is homotopic to identity via homotopy which is expressed via faces and degeneracies. Hence, the normalized and unnormalized complex are homotopy isomorphic to each other. We provide explicit formulae for the homotopy. |
| format |
Article |
| author |
Lyubashenko, V. Matsui, A. |
| spellingShingle |
Lyubashenko, V. Matsui, A. Homotopy equivalence of normalized and unnormalized complexes, revisited Algebra and Discrete Mathematics |
| author_facet |
Lyubashenko, V. Matsui, A. |
| author_sort |
Lyubashenko, V. |
| title |
Homotopy equivalence of normalized and unnormalized complexes, revisited |
| title_short |
Homotopy equivalence of normalized and unnormalized complexes, revisited |
| title_full |
Homotopy equivalence of normalized and unnormalized complexes, revisited |
| title_fullStr |
Homotopy equivalence of normalized and unnormalized complexes, revisited |
| title_full_unstemmed |
Homotopy equivalence of normalized and unnormalized complexes, revisited |
| title_sort |
homotopy equivalence of normalized and unnormalized complexes, revisited |
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Інститут прикладної математики і механіки НАН України |
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2021 |
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http://dspace.nbuv.gov.ua/handle/123456789/188752 |
| citation_txt |
Homotopy equivalence of normalized and unnormalized complexes, revisited / V. Lyubashenko, A. Matsui // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 2. — С. 253-266. — Бібліогр.: 7 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT lyubashenkov homotopyequivalenceofnormalizedandunnormalizedcomplexesrevisited AT matsuia homotopyequivalenceofnormalizedandunnormalizedcomplexesrevisited |
| first_indexed |
2025-07-16T10:57:22Z |
| last_indexed |
2025-07-16T10:57:22Z |
| _version_ |
1837800822487908352 |
| fulltext |
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 32 (2021). Number 2, pp. 253ś266
DOI:10.12958/adm1879
Homotopy equivalence of normalized and
unnormalized complexes, revisited
V. Lyubashenko and A. Matsui
Abstract. We consider the unnormalized and normalized
complexes of a simplicial or a cosimplicial object coming from the
DoldśKan correspondence for an idempotent complete additive
category (kernels and cokernels are not required). The normalized
complex is deőned as the image of certain idempotent in the unnor-
malized complex. We prove that this idempotent is homotopic to
identity via homotopy which is expressed via faces and degeneracies.
Hence, the normalized and unnormalized complex are homotopy
isomorphic to each other. We provide explicit formulae for the
homotopy.
Introduction
The study of simplicial modules began with pioneer works of Eilenberg
and Mac Lane [EM53], Dold [Dol58] and Kan [Kan58]. The equivalence
between the category of simplicial modules and the category of non-
negatively graded complexes of modules is afterwards called the DoldśKan
correspondence. The unnormalized complex corresponds to a simplicial
module in an obvious way, the normalized complex was introduced by
Eilenberg and Mac Lane [EM53, Section 4] as a quotient of unnormal-
ized complex. They prove in [EM53, Theorem 4.1] (see also Mac Lane’s
book [Mac63, Theorem VIII.6.1]) that the canonical projection of the
unnormalized complex onto the normalized one is a homotopy equivalence.
2020 MSC: 18G31, 18N50.
Key words and phrases: idempotent, simplicial object; homotopy in chain
complexes, DoldśKan correspondence.
https://doi.org/10.12958/adm1879
254 Homotopy equivalence of some complexes, revisited
As noticed in [Lyu21] the normalized complex is not only the quotient
but even a direct summand of the unnormalized one. The relevant idem-
potent has a simple expression via faces and degeneracies. This allows to
work in an additive category with split idempotents in place of category
of modules. A question arose whether there is a homotopy between this
idempotent and the identity map of the unnormalized complex which also
has a simple expression via faces and degeneracies. The present article
answers this question affirmatively.
First we prove that a homotopy of the sought form exists using a spec-
tral sequence (Section 3) associated with a double complex (Section 2).
Secondly, we provide explicit formulae for this homotopy (Section 4). All
that we do for simplicial and cosimplicial objects in an additive category
with split idempotents.
Notation
We borrow some notation from the study [Lyu21]. The simplex category
∆ has objects [n] = {0 < 1 < · · · < n}, n ∈ N = Z⩾0, morphisms are non-
decreasing maps. The cofaces are denoted as ∂i = ∂in : [n− 1] →֒ [n] ∈ ∆,
0 ⩽ i ⩽ n, i /∈ Im ∂i. The codegeneracies are denoted as σj = σjn :
[n+ 1] ↠ [n] ∈ ∆, 0 ⩽ j ⩽ n, σj(j) = σj(j + 1) = j. For 0 ⩽ j < n
P j =
(
[n]
σj
→ [n− 1]
∂j
→ [n]
)
is an idempotent, split in ∆ and, a forteriori, in Z∆. Deőne a morphism
πk ∈ Z∆([n], [n]), −1 ⩽ k < n, via formula
πk
def
= (1− P k) · · · · · (1− P 1) · (1− P 0) : [n] → [n]
with the convention π−2 = π−1 = 1[n]. It is an idempotent in Z∆ by
[Lyu21, Exercise 1.8(e)].
Denote by ∂ the sum
∑n
i=0(−1)n−i∂i ∈ Z∆([n − 1], [n]), n ⩾ 1. We
have ∂ · ∂ = 0, so there is a complex . . .
∂
−→ [n− 1]
∂
−→ [n]
∂
−→ . . . in Z∆.
Let A be an additive category. We use the unnormalized cochain
complex functor from cosimplicial objects in A to non-negatively graded
cochain complexes in A
◦C : cosA → Ch⩾0(A), ◦CB =
(
(Bn), d = B(∂) : Bn−1 → Bn
)
and the unnormalized chain complex functor from simplicial objects in A
to non-negatively graded chain complexes in A
C : sA → Ch⩾0(A), CA =
(
(An), d = A(∂op) : An → An−1
)
.
In the following A denotes an additive category with split idempotents.
V. Lyubashenko, A. Matsui 255
1. Generalities
Consider the family of idempotents of Z∆
(
Πk|[n] : [n] → [n]
)
=
{
πn−1 if n ⩽ k + 1,
πk if n > k,
indexed by k ⩾ −1. In particular, Π−1|[n] = 1. Deőne Πk|[n] = (Πk|[n])
op ∈
Z∆op([n], [n]). Consider also Π∞|[n] = πn−1 : [n] → [n] and Π∞|[n] =
(Π∞|[n])
op ∈ Z∆op([n], [n]).
Proposition 1. For any k ∈ Z⩾0∪{∞} the idempotents Πk|[n] : [n] → [n]
form a chain map Πk : (([n])n, ∂) → (([n])n, ∂) in Ch⩾0(Z∆).
Proof. Let us prove an identity in Z∆
Πk|[n−1] ·
n
∑
j=0
(−1)n−j∂j =
n
∑
j=0
(−1)n−j∂j ·Πk|[n] : [n− 1] → [n]. (1.1)
For n ⩽ k + 1 the equality
πn−2 ·
n
∑
j=0
(−1)n−j∂j = πn−2 · ∂n · πn−1 = ∂n · πn−1
=
n
∑
j=0
(−1)n−j∂j · πn−1 : [n− 1] → [n]
follows from [Lyu21, Exercise 1.8(e) and (1.16)]. For n ⩾ k+2 (1.1) reads
πk ·
n
∑
j=0
(−1)n−j∂j =
n
∑
j=0
(−1)n−j∂j · πk ∈ Z∆([n− 1], [n]) (1.2)
which we are going to prove. Since πk commutes with ∂j for j ⩾ k + 2
and due to [Lyu21, Exercise 1.8(c)] the equation reduces to
πk ·
k+1
∑
j=1
(−1)k+1−j∂j = ∂k+1 · πk ∈ Z∆([n− 1], [n]).
It follows from [Lyu21, (1.16)] that the right-hand side equals
k+1
∑
j=0
(−1)k+1−jπj−2 · ∂j .
256 Homotopy equivalence of some complexes, revisited
As a corollary,
∂k+1 · πk = ∂k+1 · πk−1 · πk = πk−1 · ∂k+1 · πk
= πk−1 ·
k+1
∑
j=0
(−1)k+1−jπj−2 · ∂j = πk−1 ·
k+1
∑
j=0
(−1)k+1−j∂j ,
hence,
(1− P k) · ∂k+1 · πk = πk ·
k+1
∑
j=0
(−1)k+1−j∂j .
It remains to notice that
P k · ∂k+1 · πk = σk · ∂k · ∂k+1 · πk = σk · ∂k · ∂k+1 · (1− σk · ∂k) · πk−1
= σk · ∂k · (∂k+1 − ∂k) · πk−1 = 0.
Corollary 1. For any k ∈ Z⩾0 ∪ {∞} and any cosimplicial object B :
Z∆ → A, any simplicial object A : Z∆op → A the idempotents B(Πk) :
◦CB → ◦CB, A(Πk) : CA → CA are cochain maps and chain maps,
respectively.
Proposition 2. Let A be an additive category with split idempotents. Then
the category Ch•(A) of cochain complexes in A has split idempotents.
Proof. Let e : M → M ∈ Ch•(A) be an idempotent, (e)2 = e. Then
for each n ∈ Z, the endomorphism en : Mn → Mn is an idempotent
in A. Therefore, it admits a splitting en =
(
Mn pn
▷ Ln ⊂
in
→Mn
)
,
in · pn = 1Ln . The exterior of the following diagram commutes:
Mn−1 pn−1
▷ Ln−1 ⊂
in−1
→Mn−1
Mn
d
↓
pn
▷ Ln
d′
↓
⊂
in
→Mn
d
↓
(1.3)
There exists a unique morphism d′ : Ln−1 → Ln which makes the both
above squares commutative, namely, d′ = in−1 ·d ·pn. In a similar diagram
with rows indexed by n− 2 and n uniqueness implies that (d′)2 = 0. We
have constructed a splitting e =
(
(M,d)
p
▷ (L, d) ⊂
i
→ (M,d)
)
in
Ch•(A).
Corollary 2. Let A be an additive category with split idempotents. Then
the categories Ch•(A) of chain complexes in A, Ch⩾0(A), Ch⩾0(A) have
split idempotents.
V. Lyubashenko, A. Matsui 257
This corollary and a version of diagram (1.3) show that for any k ∈
Z⩾0 ∪ {∞} the natural transformations -(Πk) : ◦C → ◦C, -(Πk) : C → C
split as follows:
-(Πk) =
(
◦C
◦p
▷ Im(-(Πk)) ⊂
◦i
→ ◦C
)
, ◦i · ◦p = 1,
-(Πk) =
(
C
p
▷ Im(-(Πk)) ⊂
i
→ C
)
, i · p = 1,
where Im(-(Πk)) : cosA → Ch⩾0(A), Im(-(Πk)) : sA → Ch⩾0(A) are
functors and ◦p, ◦i, p, i are natural transformations. In particular, Im(-(Πk)),
Im(-(Πk)) are the normalised cochain complex functor ◦N and the nor-
malised chain complex functor N , respectively (in the form of [Lyu21,
Corollary 1.12]).
2. Double complex
Let us add to ∆ initial object [−1] = ∅. The obtained category is
denoted ∆+. This is the category of all őnite ordinals and non-decreasing
maps. We view ∆ as a full subcategory of ∆+. Denote by ∂ the sum
∑n
i=0(−1)n−i∂i ∈ Z∆+([n − 1], [n]), n ⩾ 0. In particular, the only map
∂0 : [−1] → [0] is also denoted by ∂. We have ∂ ·∂ = 0, so ∂ can be viewed
as a differential.
We have a double complex D+ (whose all squares anticommute)
↑ ↑ ↑
Z∆(∂,1)
→ Z∆([1], [3])
−Z∆(∂,1)
→ Z∆([0], [3])
Z∆+(∂,1)
→ Z∆+([−1], [3])
−Z∆(∂,1)
→ Z∆([1], [2])
Z∆(1,∂)↑
Z∆(∂,1)
→ Z∆([0], [2])
Z∆(1,∂)↑
−Z∆+(∂,1)
→ Z∆+([−1], [2])
Z∆+(1,∂)↑
Z∆(∂,1)
→ Z∆([1], [1])
Z∆(1,∂)↑
−Z∆(∂,1)
→ Z∆([0], [1])
Z∆(1,∂)↑
Z∆+(∂,1)
→ Z∆+([−1], [1])
Z∆+(1,∂)↑
−Z∆(∂,1)
→ Z∆([1], [0])
Z∆(1,∂)↑
Z∆(∂,1)
→ Z∆([0], [0])
Z∆(1,∂)↑
−Z∆+(∂,1)
→ Z∆+([−1], [0])
Z∆+(1,∂)↑
(2.1)
The őrst horizontal (homological) degree of Z∆+([m], [n]) is m, the
second vertical (cohomological) degree is n. So we view this double complex
as chain in horizontal direction and cochain in vertical direction. Otherwise,
we use the conventions and notations of [Wei94]. The total chain complex
Tot
∏
D+ associated with double complex D+ is
258 Homotopy equivalence of some complexes, revisited
· · ·
[-,∂]
→
∞
∏
m=0
Z∆([m+ 2], [m])
2
[-,∂]
→
∞
∏
m=0
Z∆([m+ 1], [m])
1
[-,∂]
→
∞
∏
m=0
Z∆([m], [m])
0
[-,∂]
→
∞
∏
m=−1
Z∆+([m], [m+ 1])
−1
[-,∂]
→ · · · .
So the degree of
∏
Z∆+([m+ k], [m]) is k. The differential dh+ dv is [-, ∂]
since we use the right operators. Thus, [f, ∂] = f ·∂− (−1)deg f∂ ·f , where
deg f = m− n for f ∈ Z∆+([m], [n]).
Removing from (2.1) the rightmost column (indexed by −1) and the
map from 0th column to −1st column, we get a doulde complex D. The
associated total complex Tot
∏
D is
· · ·
[-,∂]
→
∞
∏
m=0
Z∆([m+ 2], [m])
2
[-,∂]
→
∞
∏
m=0
Z∆([m+ 1], [m])
1
[-,∂]
→
∞
∏
m=0
Z∆([m], [m])
0
[-,∂]
→
∞
∏
m=0
Z∆+([m], [m+ 1])
−1
[-,∂]
→ · · · .
3. Spectral sequence
Besides the double complex D let us consider double complex D̄ of the
same shape as D which is Z concentrated in bidegree (0, 0). Denote by p00 :
D → D̄, ψ0 7→ 1, the projection chain map. Associate with D a differential
abelian group G =
⊕
t∈Z
∏
n⩾max{0,−t}D
n
t+n, Dn
m = Z∆([m], [n]), graded
by summands Gt =
∏
n⩾max{0,−t}D
n
t+n = Tot
∏
t D. Notice that d(Gt) ⊂
Gt−1. Thus, G = ⊕t∈ZTot
∏
t D. Consider the decreasing őltration of G
F pG =
⊕
t∈Z
∏
n⩾max{p,−t}
Dn
t+n, p ∈ N,
formed by pth and above rows of D. Since G = F 0G, this őltration
is exhaustive. Since ∩p∈NF
pG = 0, this őltration is weakly convergent
[McC01, §3.1, p. 62].
Let us compute few initial terms of the associated spectral sequence:
Ep
0;qG = F pGq−p/F
p+1Gq−p = Dp
q , d0 = dh;
Ep
1;qG = Hq−p(F
pG/F p+1G) = Hq(D
p
∗, dh), d1 = dv.
V. Lyubashenko, A. Matsui 259
Since the topological simplex ∆p
top is contractible, the cell complex
C•(∆
p
top) = N(Z∆p) is homotopy isomorphic to Z, concentrated in degree
0. By Dold-Kan correspondence the simplicial abelian group Z∆p is
homotopy isomorphic to the constant presheaf Z, therefore, the complex
C(Z∆p) is homotopy isomorphic to Z, and the former is isomorphic to
the pth row of D. We conclude that
Ep
1;qG =
{
Z if q = 0,
0 if q > 0.
More precisely, Ep
1;0G→ Z∆+([−1], [p]) is an isomorphism and the differ-
ential dv induces
(
Z∆+([−1], ∂) : Z∆+([−1], [p]) → Z∆+([−1], [p+1])
)
=
{
0 for p even,
1Z for p odd.
The complex E•
1;0 is homotopy isomorphic to Z, hence,
Ep
2;qG =
{
Z for p = q = 0,
0 otherwise,
and d2 = 0.
Thus, Ep
2;qϕ : Ep
2;qG→ Ep
2;qḠ is an isomorphism for all p, q ∈ N.
The induced őltration of H(G, d) is deőned by
F pH(G, d) = Im
(
H(→֒) : H(F pG, d) → H(G, d)
)
.
It follows from Acyclic Assembly Lemma 2.7.3.1 [Wei94] that Tot
∏
D+
is acyclic. In fact, pth row of D+ is isomorphic to ∧[•]
Z
[p] = ∧1+•
Z
1+p,
• ⩾ −1, with the differential d, given by right derivation, whose restriction
to Z
[p] = Z
1+p is determined by (ei)d = 1, 0 ⩽ i ⩽ p. A contracting
homotopy for ∧[•]
Z
[p] is given e.g. by - ∧ ep, therefore, pth row of D+ is
acyclic. Conventions for double complexes in [Wei94] differ from ours
by reŕection with respect to line x + y = 0. In particular, the fourth
quadrant is the fourth quadrant in both conventions, however, horizontal
and vertical directions exchange roles. Therefore, Acyclic Assembly Lemma
2.7.3.1 [Wei94] is applicable and we conclude that Tot
∏
D+ is acyclic.
In few lines, for p ⩾ max{0,−k − 1} an element α ∈ ZkF
pTot
∏
D+ ⊂
∏∞
m=p Z∆
+([m + k], [m]) has among its coordinates a dh-cycle αp ∈
Z∆+([p+k], [p]). Necessarily αp = (βp)dh for some βp ∈ Z∆([p+k+1], [p]).
Then α− [βp, ∂] ∈ ZkF
p+1Tot
∏
D+ and we őnd βp+1, and so on. Clearly,
α = [β, ∂].
260 Homotopy equivalence of some complexes, revisited
These reasonings show also that F pHk(G, d) = 0 for p ⩾ max{0, 1−k},
hence, the őltered abelian group (H(a, d), F ) is complete, that is, the
mapping
H(G) → lim
p
H(G)/F pH(G)
is an isomorphism. Hence, by deőnition, őltered differential module
(G,F, d) is strongly convergent.
Similarity one produces a őltered differential abelian group Ḡ from D̄.
The őltration (Ḡ, F, d = 0) is exhaustive, weakly convergent and complete.
In fact, F pḠ = 0 for p > 0. The map p00 : D → D̄ induces a őltration
preserving chain map ϕ : G→ Ḡ.
Theorem 3.9 of [McC01] implies
Proposition 3. The map ϕ induces an isomorphism on homology, H(ϕ) :
H(G, d) → H(Ḡ, d) = Ḡ.
Brieŕy, cohomology of Tot
∏
D reduces to cohomology of the rightmost
column of ∆+, which is Z concentrated in degree 0.
Let A be an additive category, B ∈ cosA, A ∈ sA be cosimplicial and
simplicial objects, respectively. Consider a cycle φ ∈ Z0Tot
∏
D. Its image
B(φ) : ◦C•B → ◦C•B is a cochain endomorphism of the cochain complex
◦C•B = ((Bn), B∂) associated with B. The image A(φop) : C•A→ C•A
is a chain endomorphism of the chain complex C•A = ((An), d = A(∂op))
associated with A. Suppose now that φ ∈ B0Tot
∏
D is the boundary
φ = [η, ∂], η ∈ Tot
∏
1 D. Then B(φ) = Bη · B∂ + B∂ · Bη and A(φop) =
A(ηop)◦d+d◦A(ηop) are null-homotopic such that B(φ)0 = 0 : B0 → B0
and A(φop)0 = 0 : A0 → A0.
Summing up, Proposition 3 implies
Corollary 3. Let A be an additive category with split idempotents. Let
φ ∈ Z0Tot
∏
D, φn ∈ Z∆([n], [n]) be a cycle such that ψ0 = 0. Then
B(φ) : ◦C•B → ◦C•B and A(φop) : C•A→ C•A are null-homotopic. The
contracting homotopy is given by the sequence B(ηn), A(η
op
n ) respectively,
where φ = [η, ∂], ηn ∈ Z∆([n+ 1], [n]), n ⩾ 0.
Applying this corollary to the cycle φ = 1−Π∞ we conclude that the
morphisms in the decompositions ofB(Π∞)=
(
◦CB
◦p
▷ ◦NB ⊂
◦i
→ ◦CB
)
and A(Π∞) =
(
CA
p
▷ NA ⊂
i
→ CA
)
are homotopy inverse to each
other.
V. Lyubashenko, A. Matsui 261
4. Homotopies between projections
Projections Πk and Πk+1 are homotopic for k ⩾ −1 by Corollary 3.
Let us őnd the homotopy between them explicitly.
Proposition 4. Deőne tnk ∈ Z∆([n+ 1], [n]) for n ⩾ 0 as
tnk =
{
0 if n ⩽ k + 1,
∑n
j=k+2(−1)n+1−jσj · πk−1 if n ⩾ k + 2.
Then tk is the homotopy Πk ∼ Πk+1.
Proof. We have to prove for n ⩾ 0 that
Πk+1 −Πk = ∂ · tnk + tn−1
k · ∂ : [n] → [n]. (4.1)
where ∂ denotes the sum
∑n
i=0(−1)n−i∂i ∈ Z∆([n − 1], [n]), n > 0 and
t−1
k = 0.
Assume that k ⩾ 0. Notice that, for k = 0 the following holds as well
taking into account that π−1 = 1 by convention. Consider empty sums to
be 0.
For n ⩽ k + 1 the sought equation (4.1) is obvious.
For n ⩾ k + 2 notice that
Πk+1 −Πk = πk+1 − πk = −P k+1 ·
(
1− P k
)
· πk−1
=
(
∂k · σk+2 − ∂k+1 · σk+2
)
· πk−1.
Then for n = k + 2 we have
k+3
∑
i=0
(−1)k+3−i∂i · tk+2
k + 0
=
k+3
∑
i=0
(−1)k+3−i∂i · (−σk+2 · πk−1) =
k+3
∑
i=0
(−1)k−i∂i · σk+2 · πk−1
= Πk+1 −Πk +
k−1
∑
i=0
(−1)k−iσk+1 · ∂i · πk−1 = Πk+1 −Πk.
Let n ⩾ k + 3. Then
n+1
∑
i=0
(−1)n+1−i∂i · tnk + tn−1
k ·
n
∑
i=0
(−1)n−i∂i
262 Homotopy equivalence of some complexes, revisited
=
n+1
∑
i=0
(−1)n+1−i∂i ·
n
∑
j=k+2
(−1)n+1−jσj · πk−1
+
n−1
∑
j=k+2
(−1)n−jσj · πk−1 ·
n
∑
i=0
(−1)n−i∂i. (4.2)
Rewrite the őrst sum in the right-hand side as
n
∑
j=k+2
n+1
∑
i=0
(−1)i+j∂i · σj · πk−1
=
n
∑
j=k+2
k−1
∑
i=0
(−1)i+j∂i · σj · πk−1 +
n
∑
j=k+2
n+1
∑
i=k
(−1)i+j∂i · σj · πk−1
=
n
∑
j=k+2
k−1
∑
i=0
(−1)i+j∂i · σj · πk−1 +
n
∑
j=k+3
n+1
∑
i=k
(−1)i+j∂i · σj · πk−1
+ (∂k · σk+2 − ∂k+1 · σk+2) · πk−1 +
n+1
∑
i=k+2
(−1)k+i∂i · σk+2 · πk−1
=
n
∑
j=k+2
k−1
∑
i=0
(−1)i+jσj−1 · ∂i · πk−1 +
n
∑
j=k+3
j−1
∑
i=k
(−1)i+j∂i · σj · πk−1
+
n
∑
j=k+3
n+1
∑
i=j
(−1)i+j∂i · σj · πk−1 +Πk+1 −Πk
+
n+1
∑
i=k+4
(−1)k+i∂i · σk+2 · πk−1
+ (−∂k+3 · σk+2 + ∂k+2 · σk+2) · πk−1
=
n
∑
j=k+3
j−1
∑
i=k
(−1)i+jσj−1 · ∂i · πk−1 +
n
∑
j=k+3
n+1
∑
i=j+2
(−1)i+j∂i · σj · πk−1
−
n
∑
j=k+3
∂j+1 · σj · πk−1 +
n
∑
j=k+3
∂j · σj · πk−1 +Πk+1 −Πk
+
n+1
∑
i=k+4
(−1)k+iσk+2 · ∂i−1 · πk−1
V. Lyubashenko, A. Matsui 263
=
n
∑
j=k+3
j−1
∑
i=k
(−1)i+jσj−1 · ∂i · πk−1 +
n
∑
j=k+3
n+1
∑
i=j+2
(−1)i+j∂i · σj · πk−1
+
n+1
∑
i=k+4
(−1)k+iσk+2 · ∂i−1 · πk−1 +Πk+1 −Πk.
Due to (1.2) we may rewrite the second sum in the right-hand side of (4.2)
as
n−1
∑
j=k+2
n
∑
i=0
(−1)i+jσj · ∂i · πk−1 =
n−1
∑
j=k+2
k−1
∑
i=0
(−1)i+jσj · ∂i · πk−1
+
n−1
∑
j=k+2
n
∑
i=k
(−1)i+jσj · ∂i · πk−1 =
n−1
∑
j=k+2
j
∑
i=k
(−1)i+jσj · ∂i · πk−1
+
n−1
∑
j=k+2
n
∑
i=j+1
(−1)i+jσj · ∂i · πk−1 =
n−1
∑
j=k+2
j
∑
i=k
(−1)i+jσj · ∂i · πk−1
+
n−1
∑
j=k+3
n
∑
i=j+1
(−1)i+j∂i+1 · σj · πk−1 +
n
∑
i=k+3
(−1)k+iσk+2 · ∂i · πk−1.
The őrsts, the seconds, and the thirds summands cancel each out, and (4.2)
reduces to Πk+1 −Πk.
For k = −1 the general proof holds, deőning π−2 = 1, ∂−1 = 0 and
σ−1 = 0 as we show below. For n = 0 equation (4.1) obviously holds.
For n ⩾ 1
Π0 −Π−1 = 1− P 0 − 1 = −σ0 · ∂0 = −∂0 · σ1
Then for n = 1 equation (4.1) follows from the identity
(∂2 − ∂1 + ∂0) · (−σ1) = −∂0 · σ1
And for n ⩾ 2 the right-hand side of (4.1) is
n+1
∑
i=0
(−1)n+1−i∂i · tn−1 + tn−1
−1 ·
n
∑
i=0
(−1)n−i∂i (4.3)
=
n+1
∑
i=0
(−1)n+1−i∂i ·
n
∑
j=1
(−1)n+1−jσj +
n−1
∑
j=1
(−1)n−jσj ·
n
∑
i=0
(−1)n−i∂i.
264 Homotopy equivalence of some complexes, revisited
Rewrite the őrst sum as
n
∑
j=1
n+1
∑
i=0
(−1)i+j∂i · σj
=
n
∑
j=2
n+1
∑
i=0
(−1)i+j∂i · σj − ∂0 · σ1 +
n+1
∑
i=1
(−1)1+i∂i · σ1
=
n
∑
j=2
j−1
∑
i=0
(−1)i+j∂i · σj +
n
∑
j=2
n+1
∑
i=j
(−1)i+j∂i · σj
+Π0 −Π−1 +
n+1
∑
i=3
(−1)1+i∂i · σ1 + (−∂2 · σ1 + ∂1 · σ1)
=
n
∑
j=2
j−1
∑
i=0
(−1)i+jσj−1 · ∂i +
n
∑
j=2
n+1
∑
i=j+2
(−1)i+j∂i · σj −
n
∑
j=2
∂j+1 · σj
+
n
∑
j=2
∂j · σj +Π0 − 1 +
n+1
∑
i=3
(−1)1+iσ1 · ∂i−1
=
n
∑
j=2
j−1
∑
i=0
(−1)i+jσj−1 · ∂i +
n
∑
j=2
n+1
∑
i=j+2
(−1)i+j∂i · σj
+
n+1
∑
i=3
(−1)1+iσ1 · ∂i−1 +Π0 − 1.
The second sum in the right-hand side of (4.3) is
n−1
∑
j=1
n
∑
i=0
(−1)i+jσj · ∂i =
n−1
∑
j=1
j
∑
i=0
(−1)i+jσj · ∂i +
n−1
∑
j=1
n
∑
i=j+1
(−1)i+jσj · ∂i
=
n−1
∑
j=1
j
∑
i=0
(−1)i+jσj ·∂i+
n−1
∑
j=2
n
∑
i=j+1
(−1)i+j∂i+1 ·σj +
n
∑
i=2
(−1)1+iσ1 ·∂i.
The őrsts, the seconds, and the thirds summands cancel each out, and (4.3)
reduces to Π0 −Π−1.
Let P,Q : C → C : sA → Ch⩾0(A) be chain idempotents, P · Q =
P = Q ·P . Then Q−P is an idempotent and there exist i : ImP →֒ ImQ,
p : ImQ ▷ ImP representing ImQ as a direct sum ImP ⊕ Im(Q−P ).
In particular, i · p = 1ImP , p · i = P |ImQ.
V. Lyubashenko, A. Matsui 265
Let chain maps P,Q : CA→ CA be homotopic: there exist hn : An →
An+1 ∈ A, n ⩾ −1, h−1 = 0, such that Qn − Pn = d · hn−1 + hn · d for
n ⩾ 0. Then i, p are homotopy inverse to each other, the subcomplex
ImP ⊂ ImQ is homotopy isomorphic to ImQ, Im(Q− P ) is contractible
in CA and in ImQ. In fact, h can be replaced with h′n = Qn · hn ·Qn+1 :
ImQn → ImQn+1, since
d · h′n−1 + h′n · d = Qn · (Qn − Pn) ·Qn = Qn − Pn,
1ImQ − p · i = Q− P = d · h′ + h′ · d.
Let P,Q,R : C → C : sA → Ch⩾0(A) be chain idempotents, P ·Q =
P = Q · P , Q ·R = Q = R ·Q. Assume that P ∼ Q, Q ∼ R, that is,
Q− P = d · h+ h · d, R−Q = d · k + k · d.
Then P ∼ R with the homotopy l = h+ k,
R− P = d · (h+ k) + (h+ k) · d.
If Qn · hn · Qn+1 = hn, Rn · kn · Rn+1 = kn, then Rn · ln · Rn+1 = ln as
well.
Summing the homotopies obtained in Proposition 4 we get a homotopy
between id :
(
([n])n, ∂
)
→
(
([n])n, ∂
)
and Π∞ :
(
([n])n, ∂
)
→
(
([n])n, ∂
)
in Ch⩾0(Z∆):
tn =
n−2
∑
k=−1
n
∑
j=k+2
(−1)n+1−jσj · πk−1 ∈ Z∆([n+ 1], [n]), n ⩾ 0.
Therefore, the natural transformations ◦p : ◦C → ◦N and ◦i : ◦N → ◦C
are homotopy inverse to each other. In fact, for any B ∈ Ob cosA the
morphisms 1◦CB and B(Π∞) : ◦CB → ◦CB are naturally homotopic via
B(t). Similarly, the natural transformations p : C → N and i : N → C
are homotopy inverse to each other.
References
[Dol58] Albrecht Dold, Homology of symmetric products and other functors of
complexes, Annals of Mathematics, Second Series 68 (1958), no. 1, 54ś80,
http://www.jstor.org/stable/1970043.
[EM53] Samuel Eilenberg and Saunders Mac Lane, On the groups H(Π, n). I, Annals
of Mathematics. Second Series 58 (1953), no. 1, 55ś106, https://doi.org/
10.2307/1969820.
http://www.jstor.org/stable/1970043
https://doi.org/10.2307/1969820
https://doi.org/10.2307/1969820
266 Homotopy equivalence of some complexes, revisited
[Kan58] Daniel M. Kan, Functors involving c.s.s. complexes, Transactions of the
American Mathematical Society 87 (1958), no. 2, 330ś346, https://doi.
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[Lyu21] V. V. Lyubashenko, DoldśKan correspondence, revisited, 2021.
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Wissenschaften. Springer-Verlag, Berlin, Heidelberg (1963)
[McC01] John McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge
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Contact information
Volodymyr
Lyubashenko
Institute of Mathematics NASU, 3 Tereshchenkivska
st., Kyiv, 01024, Ukraine
E-Mail(s): lub@imath.kiev.ua
Web-page(s): https://www.imath.kiev.ua/~lub
Anna Matsui Kyiv National Taras Shevchenko University, Faculty
of Mechanics and Mathematics, 4-e Akademika
Hlushkova Ave, Kyiv, 03127, Ukraine
E-Mail(s): matsuiannam@gmail.com
Received by the editors: 15.08.2021.
https://doi.org/10.1090/S0002-9947-1958-0131873-8
https://doi.org/10.1090/S0002-9947-1958-0131873-8
mailto:lub@imath.kiev.ua
https://www.imath.kiev.ua/~lub
mailto:matsuiannam@gmail.com
V. Lyubashenko, A. Matsui
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