# Homotopic Chain Maps Have Equals-Homology and d-Homology

Authors: M. Z. Kazemi-Baneh
Journal: International Journal of Mathematics and Mathematical Sciences
 (2016)

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## Abstract

The homotopy of chain maps on preabelian categories is investigated and the equality of standard homologies andd-homologies of homotopic chain maps is established. As a special case, if X and Y are the same homotopy type, then their nth d-homology R-modules are isomorphic, and if X is a contractible space, then its nth d-homology R-modules for n≠0 are trivial.


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## Body

## 1. Introduction and Preliminaries

It is known that the homotopic chain maps of abelian groups or more generally ofR-modules have the same homologies; see [1, 2]. In this paper, the homotopy of chain maps on preabelian categories is investigated and it is proved that homotopic chain maps have the same s-homologies and the same d-homologies.To this end, for a pointed categoryC, following the notation of [3], we recall the following.(i) Forf:A→B, the maps Kf→kfA, B→cfCf, and Pf⇉π2π1A are, respectively, the kernel, the cokernel, and the kernel pair of f; see [4, 5].(ii) The imageIf of f is the coequalizer of the kernel pair of f. In a homological category, If≅Cker(kf); see [4].(iii) For a pair of mapsA⇉gfB, the maps Equ(f,g)→equ(f,g)A and B→coe(f,g)Coe(f,g) are, respectively, the equalizer and the coequalizer of (f,g).(iv) Given the diagram below in which the squares are commutative and the rows are coequalizers,i is the unique map making the right square commute. Furthermore, i is a regular epi.(v) For a categoryC with a zero object, kernels, kernel pairs, and coequalizers of kernel pairs, the arrow category C¯ of C has as objects the morphisms of C and as morphisms from f:A→B to f′:A′→B′ the pairs (α,β) of morphisms of C, making the following square commutative:And the pair-chain category C^ of C has as objects the pair-chains, that is, the composable pairs, (f,g), of morphisms of C, such that gf=0, and as morphisms from (f,g) to (f′,g′) the triple (α,β,γ) of morphisms of C, making the following squares commutative:Being functor of the following items are investigated and established in [3].(vi) The kernel functorK:C¯→C takes (α,β):f→f′ to the left vertical arrow in the following commutative diagram:(vii) The image functorI:C¯→C takes (α,β):f→f′ to the left vertical arrow in the following commutative diagram:(viii) The functorj:C^→C¯ takes the object (f,g)∈C^ to jfg and the morphism (α,β,γ) to (I(α,β),K(β,γ)), and we have the following commutative diagram:In a pointed regular (homological, semiabelian, or abelian) category,jfg is monic; see [3, 4].(ix) WithS:C→C, the squaring functor, taking a→fb to a2→f2b2, a kernel transformation is a natural transformation d:S∘K→K:C¯→C such that for all (f,g) in C^, the pullback, jfg∗:Rfg→Kg2, of jfg along dg and the coequalizer of the pair j1=pr1jfg∗ and j2=pr2jfg∗ exist, where pr1 and pr2 are the projection maps.(x) Thed-homology functor Hd:C^→C takes (f,g)∈C^ to Hfgd=Coe(j1,j2) and the morphism (α,β,γ) to Hd(α,β,γ). We have the following commutative diagram:(xi) LetR be a commutative ring with unity. Any kernel transformation in Rmod is of the form d=+(r×s), for some r,s∈R. In particular, any kernel transformation d in Abgrp is of the form d=+(r×s), for some r,s∈Z.(xii) LetC=Rmod and d=rpr1+spr2=+(r×s) with r,s∈R. Let (f,g) be a pair-chain. Then, Rfg={(a,b)∈Kg2∣ra+sb∈If}, j∗ is the inclusion, and Hfg=Kg/(j1-j2)(Rfg)={[a]:a∈Kg}, where [a]={b∣r(a-b)∈(r+s)Kg+If}={b∣s(a-b)∈(r+s)Kg+If} is the equivalence class under the equivalence relation a~b if and only if ∃m,n∈Kg such that a-b=m-n and rm+sn∈If.(xiii) As a special case of above example, ford=+(r×1) or d=+(1×r) with r∈R, we have Hfgd=Kg/((1+r)Kg+If).We call the homology which is defined in [2, 4] the standard homology.(xiv) The standard homology ors-homology functor Hs takes (f,g)∈C^ to Coker(jfg), and for a pair-chain map (α,β,γ):(f,g)→(f′,g′), we have the following commutative diagram:where q=coker(jfg) and q′=coker(jf′g′).


## 2. Homotopy

Definition 1.
LetC be an additive category. Two morphismsin C^ are said to be homotopic whenever there is a pair of morphisms (h,h′) in C, as in the diagram below, such that f′h+h′g=β-β′; see [2],Theorem 2.
LetC be a preabelian category. If the maps (α,β,γ) and α′,β′,γ′ in C^ are homotopic, then Hs(α,β,γ)=Hsα′,β′,γ′.Proof.
Consider(11)kg′jf′g′ef′h+h′gkg=β-β′kg,kg′jf′g′ef′hkg=kg′kβ,γ-kβ′,γ′jf′g′ef′hkg=kβ,γ-kβ′,γ′since q′jf′g′=0 we have q′kβ,γ-kβ′,γ′=0.
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0, Hsα,β,γ-Hsα′,β′,γ′=0 since q is a regular epi. Therefore, Hsα,β,γ=Hsα′,β′,γ′.Lemma 3.
LetC be a pointed category with pullbacks and pushouts and let d be a kernel transformation in C. There is a natural transformation p:Hs→Hd:C^→C. Furthermore, p is pointwise regular epic.Proof.
Letd:S∘K→K be a natural transformation. For (f,g)∈C^ and h:If→0, If=Kh. Factoring f=mfef and using the morphism (mf,0):h→g, naturality of d yields, jdh=dgj2, and so dg〈j,0〉=dgj2〈1,0〉=jdh〈1,0〉. Setting r=dh〈1,0〉, we get dg〈j,0〉=jr. So, there exists a unique map ψ, such that the triangles in the following diagram commute:
Therefore,j=pr1〈j,0〉=pr1j∗ψ=j1ψ and 0=pr2〈j,0〉=pr2j∗ψ=j2ψ so that qj=qpr1j∗ψ=qpr2j∗ψ=q0=0. Then, there is a unique morphism pfg:Hfgs→Hfgd such that pfgcj=q; that is, the following diagram commutes:
Furthermore, by a mentioned point in preliminaries,pfg is regular epic.
To show naturality, given(σ,δ,ζ):(f,g)→(f′,g′) in C^, consider the following diagram that, by the above diagram the triangles commute, with the functors Hs and Hd implying the square and the left parallelogram are commutative:
We havepf′g′Hs(σ,δ,ζ)cj=pf′g′cj′Kδ,ζ=qf′g′K(δ,ζ)=Hd(σ,δ,ζ)qfg=Hd(σ,δ,ζ)pfgcj. Since cj is epic, pf′g′Hs(σ,δ,ζ)=Hd(σ,δ,ζ)pfg and the right parallelogram commutes. Then, the following diagram commutes:and so p:Hs→Hd:C^→C is a natural transformation.Theorem 4.
LetC be a preabelian category. If (α,β,γ) and α′,β′,γ′ are homotopic, then Hd(α,β,γ)=Hdα′,β′,γ′.Proof.
The equalitiespfgcj=q, pf′g′cj′=q′, Hd(α,β,γ)q=q′Kα,β, and Hdα′,β′,γ′q=q′Kα′,β′ and the facts that p is regular epic and standard homologies are equal imply Hd(α,β,γ)=Hdα′,β′,γ′ as desired.Definition 5.
LetC be a pointed category. A chain complex in C is a differential graded object of C of degree −1 as(16)⋯⟶Cn+1→νn+1Cn→νnCn-1⟶⋯in which νnνn+1=0 for all n∈Z and a chain map f∗:C∗→D∗ is a graded map fn:Cn→Dn:n∈Z as in the following diagram, in which all the squares commute:These chain complexes and chain maps form a category that is denoted byC∗(C).Definition 6.
LetC be an additive category and C∗⇉g∗f∗D∗ two maps in C∗(C); one says f∗ is chain homotopic to g∗ if there is a morphism h∗:C∗→D∗ of degree 1 assuch that μ∗h∗+h∗ν∗=f∗-g∗.Corollary 7.
LetC be a preabelian category and let f∗ and g∗ be homotopic in C∗(C). Then, Hs(f∗)=Hs(g∗) and Hd(f∗)=Hd(g∗).Proof.
Supposeh∗:C∗→D∗ is a homotopy of the pair C∗⇉g∗f∗D∗; that is, there is the following diagram in which for all n we have μn+1hn+hn-1νn=fn-gn:
By the above theorems,Hns(f∗)=Hns(g∗) and Hnd(f∗)=Hnd(g∗), where Hn(f∗)=H(fn+1,fn,fn-1).
LetTop, SetΔop, RmodΔop, C∗Rmod, and gRmod be, respectively, categories of topological spaces, simplicial sets, simplicial R-modules, chain complexes of R-modules, and graded R-modules, and let SΔ∗:Top→SetΔop, FΔop:SetΔop→RmodΔop, C∗:RmodΔop→C∗Rmod, and H∗d:C∗Rmod→gRmod be, respectively, singular functor, induced functor by free generator functor F:Set→Rmod, chain complexes generator functor, and d-homology functor. If X∈Top, then the singular d-homology of X is H∗d(X)=H∗doC∗oFΔopoSΔ∗(X). R is PID.Example 8.
IfX⇉gfY is homotopic continuous maps, then H∗d(f)=H∗d(g):H∗d(X)→H∗d(Y).Example 9.
Let 1 be a terminal object inTop. Since Hns(1)=0 for n≠0 and the natural transformation p:Hs→Hd:C^→C is pointwise regular epic, Hnd(1)=0 for n≠0 and H0d(1)=R/(j1-j2)(R00) in which R00=Ker(d0).LetSimCx be simplicial complexes and let S∗:SimCx→SetΔop be the simplicial functor. If (K,S)∈SimCx, then the simplicial d-homology of (K,S) is H∗d(X)=H∗doC∗oFΔopoS∗(K,S).Example 10.
Let(1,1) be a terminal object in SimCx. Then, Hnd(1,1)=0 for n≠0.



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