I’m taking a course in algebraic topology, which includes an introduction to (simplicial) homology, and I’m looking for a bit of intuition regarding chain homotopies.

The definitions I’m using are:

Let f,g:X→Y be continuous functions between topological spaces. A

homotopyfrom f to g is a continuous map H:X×[0,1]→Y such that H(⋅,0)=f and H(⋅,1)=g.Let f∙,g∙:A∙→B∙ be chain maps between chain complexes (A,dA) and (B,dB). A

chain homotopyfrom f to g is a sequence of maps hn:An→Bn+1 such that fn−gn=dBhn+hn−1dA.I’m aware of the properties of a chain homotopy and how they are similar to those of a homotopy, but I still find the definition quite opaque and the notion quite hard to picture − it would help me a lot if I could think of a chain homotopy in a similar way to how I think of a homotopy.

Or, to make my question a bit less vague, I’d like to know:

- What is the rationale behind the definition of a chain homotopy?
- Is there a fundamental similarity between a chain homotopy and a homotopy, beyond their further consequences?
(General waffle would also be appreciated; I’d really like to develop a good understanding.)

**Answer**

If I is a chain complex representing an interval, with I0=Z2 and I1=Z, with ∂(x)=(x,−x), then a chain homotopy between two maps f,g:A→B is the same as a map H:A⊗I→B, where H(a⊗(1,0))=f(a) and H(a⊗(0,1))=g(a). This explains the “shift” up a dimension in the usual definition you’d see of chain homotopy, since your hn:An→Bn+1 corresponds to my H:An⊗I1→Bn+1.

In general, one kind of homotopy in a model category involves what are called cylinder objects. These are functorial factorizations of the fold map A∐A→A through an object A′ that is weakly equivalent (for spaces, an isomorphism on homotopy groups — for chain complexes, a homology isomorphism) to A; the inclusion of A∐A→A′ should also be particularly well-behaved (a cofibration). Effectively, you’re guaranteeing that two copies of A can play nicely in A′, and that A′ is a “thickened up” version of A rather than something pathological.

Then a homotopy between two morphisms f,g:A→B is a map H:A′→B where the composition A∐A→A′→B is f∐g. You’ll see this pattern again and again.

**Attribution***Source : Link , Question Author : Clive Newstead , Answer Author : Thomas Belulovich*