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I'm in an intro algebraic topology class. In the textbook Algebraic Topology (Hatcher), a chain homotopy is defined by saying that a map $P$ is a chain homotopy between two maps $f$ and $g$ if $dP + Pd = g - f$, where $d$ is the boundary map in a chain complex.

What I don't understand is what this has to do with homotopy. In what sense is a chain homotopy a homotopy?

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  • $\begingroup$ An instructive exercise, if you learn about simplicial things, is to see that a simplicial homotopy $X\times \Delta[1] \to Y$ induces a chain homotopy on simplicial chains. That makes the connection between spaces and chains very clear. $\endgroup$ – Justin Young Feb 22 '16 at 12:45
  • $\begingroup$ Possibly related: math.stackexchange.com/questions/132533/… $\endgroup$ – Watson Feb 21 '18 at 15:24
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This is a very good question, so let me give a lowbrow answer and a highbrow answer.

The lowbrow answer is that homotopies of topological spaces induce this relation on their associated chain complexes (the ones you use to define singular homology, for example), so it makes sense to call this relation homotopy. See the proof of homotopy invariance of homology in Hatcher's text, for example.

The highbrow answer is that one can axiomatize homotopy theory, which gives the notion of a model category. In a model category, we use an "interval object" to define the notion of homotopy. For example, in the category of topological spaces, the interval object is $[0,1]$. It turns out one can define an interval object for the category of chain complexes, and if one works out the abstract definition of homotopy in the category of chain complexes, we get exactly the definition you gave.

For more on the interval object, see here.

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  • $\begingroup$ Thank you, this was helpful. Your link seems to be broken though. $\endgroup$ – G Pace Feb 22 '16 at 3:56
  • $\begingroup$ @GPace Thanks for pointing that out. I fixed it. $\endgroup$ – Potato Feb 22 '16 at 12:26

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