Homotopic Chain Maps Have Equal s-Homology and d-Homology

It is known that the homotopic chain maps of abelian groups or more generally of R-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 category C, following the notation of [3], we recall the following.


Introduction and Preliminaries
It is known that the homotopic chain maps of abelian groups or more generally of -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 -homologies and the same -homologies.
To this end, for a pointed category C, following the notation of [3], we recall the following. are, respectively, the kernel, the cokernel, and the kernel pair of ; see [4,5].
(ii) The image   of  is the coequalizer of the kernel pair of .In a homological category,   ≅ ker(  ); see [4].
(iii) For a pair of maps    → Coe(, ) are, respectively, the equalizer and the coequalizer of (, ).
(iv) Given the diagram below in which the squares are commutative and the rows are coequalizers,  is the unique map making the right square commute.Furthermore,  is a regular epi.
(1) (v) For a category C 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  :  →  to   :   →   the pairs (, ) of morphisms of C, making the following square commutative: International Journal of Mathematics and Mathematical Sciences And the pair-chain category Ĉ of C has as objects the pairchains, that is, the composable pairs, (, ), of morphisms of C, such that  = 0, and as morphisms from (, ) to (  ,   ) the triple (, , ) of morphisms of C, making the following squares commutative: Being functor of the following items are investigated and established in [3].
(vi) The kernel functor  : C → C takes (, ) :  →   to the left vertical arrow in the following commutative diagram:    →  2 , a kernel transformation is a natural transformation  : ∘ →  : C → C such that for all (, ) in Ĉ, the pullback,  *  :   →  2  , of   along   and the coequalizer of the pair  1 = pr 1  *  and  2 = pr 2  *  exist, where pr 1 and pr 2 are the projection maps.

Lemma 3. Let C be a pointed category with pullbacks and pushouts and let 𝑑 be a kernel transformation in C.
There is a natural transformation  :   →   : Ĉ → C. Furthermore,  is pointwise regular epic.
Definition 5. Let C be a pointed category.A chain complex in C is a differential graded object of C of degree −1 as in which ]  ] +1 = 0 for all  ∈  and a chain map  * :  * →  * is a graded map {  :   →   :  ∈ } as in the following diagram, in which all the squares commute: These chain complexes and chain maps form a category that is denoted by  * (C).Let Sim be simplicial complexes and let  * : Sim → Set Δ  be the simplicial functor.If (, ) ∈ Sim, then the simplicial -homology of (, ) is   * () =   *  *  Δ   * (, ).