Can we think of a chain homotopy as a homotopy?

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:XY be continuous functions between topological spaces. A homotopy from f to g is a continuous map H:X×[0,1]Y such that H(,0)=f and H(,1)=g.

Let f,g:AB be chain maps between chain complexes (A,dA) and (B,dB). A chain homotopy from f to g is a sequence of maps hn:AnBn+1 such that fngn=dBhn+hn1dA.

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:AB is the same as a map H:AIB, 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:AnBn+1 corresponds to my H:AnI1Bn+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 AAA 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 AAA 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:AB is a map H:AB where the composition AAAB is fg. You’ll see this pattern again and again.

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

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