Research Article | Open Access
Homotopic Chain Maps Have Equal -Homology and -Homology
The homotopy of chain maps on preabelian categories is investigated and the equality of standard homologies and -homologies of homotopic chain maps is established. As a special case, if and are the same homotopy type, then their th -homology -modules are isomorphic, and if is a contractible space, then its th -homology -modules for are trivial.
1. 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 , following the notation of , we recall the following.
(ii) The image of is the coequalizer of the kernel pair of . In a homological category, ; see .
(iii) For a pair of maps , the maps and 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.
(v) For a category with a zero object, kernels, kernel pairs, and coequalizers of kernel pairs, the arrow category of has as objects the morphisms of and as morphisms from to the pairs of morphisms of , making the following square commutative:And the pair-chain category of has as objects the pair-chains, that is, the composable pairs, , of morphisms of , such that , and as morphisms from to the triple of morphisms of , making the following squares commutative:
Being functor of the following items are investigated and established in .
(vi) The kernel functor takes to the left vertical arrow in the following commutative diagram:
(vii) The image functor takes to the left vertical arrow in the following commutative diagram:
(viii) The functor takes the object to and the morphism to , and we have the following commutative diagram:
(ix) With , the squaring functor, taking to , a kernel transformation is a natural transformation such that for all in , the pullback, , of along and the coequalizer of the pair and exist, where and are the projection maps.
(x) The -homology functor takes to and the morphism to . We have the following commutative diagram:
(xi) Let be a commutative ring with unity. Any kernel transformation in is of the form , for some . In particular, any kernel transformation in is of the form , for some .
(xii) Let and with . Let be a pair-chain. Then, , is the inclusion, and , where is the equivalence class under the equivalence relation if and only if such that and .
(xiii) As a special case of above example, for or with , we have .
(xiv) The standard homology or -homology functor takes to , and for a pair-chain map , we have the following commutative diagram:where and .
Definition 1. Let be an additive category. Two morphismsin are said to be homotopic whenever there is a pair of morphisms in , as in the diagram below, such that ; see ,
Theorem 2. Let be a preabelian category. If the maps and in are homotopic, then .
Proof. Considersince we have .
, since is a regular epi. Therefore, .
Lemma 3. Let be a pointed category with pullbacks and pushouts and let be a kernel transformation in . There is a natural transformation . Furthermore, is pointwise regular epic.
Proof. Let be a natural transformation. For and , . Factoring and using the morphism , naturality of yields, , and so . Setting , we get . So, there exists a unique map , such that the triangles in the following diagram commute:Therefore, and so that . Then, there is a unique morphism such that ; that is, the following diagram commutes:Furthermore, by a mentioned point in preliminaries, is regular epic.
To show naturality, given in , consider the following diagram that, by the above diagram the triangles commute, with the functors and implying the square and the left parallelogram are commutative:We have . Since is epic, and the right parallelogram commutes. Then, the following diagram commutes:and so is a natural transformation.
Theorem 4. Let be a preabelian category. If and are homotopic, then .
Proof. The equalities , , , and and the facts that is regular epic and standard homologies are equal imply as desired.
Definition 5. Let be a pointed category. A chain complex in is a differential graded object of of degree −1 asin which 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 .
Definition 6. Let be an additive category and two maps in ; one says is chain homotopic to if there is a morphism of degree 1 assuch that .
Corollary 7. Let be a preabelian category and let and be homotopic in . Then, and .
Proof. Suppose is a homotopy of the pair ; that is, there is the following diagram in which for all we have :By the above theorems, and , where .
Let , , , , and be, respectively, categories of topological spaces, simplicial sets, simplicial -modules, chain complexes of -modules, and graded -modules, and let , , , and be, respectively, singular functor, induced functor by free generator functor , chain complexes generator functor, and -homology functor. If , then the singular -homology of is . is PID.
Example 8. If is homotopic continuous maps, then .
Example 9. Let 1 be a terminal object in . Since for and the natural transformation is pointwise regular epic, for and in which .
Let be simplicial complexes and let be the simplicial functor. If , then the simplicial -homology of