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Assume everything is over a field of characteristic zero and one has a family of chain maps $f_t\,:\,V_\bullet \rightarrow W_\bullet$ where $t\in [0,\,1]$. I was wondering if there is a way to construct a chain homotopy between $f_0$ and $f_1$?

One option is constructing a chain map from $V_\bullet$ to $W_\bullet\otimes \Omega^\bullet_{[0,\,1]}$, where $\Omega\bullet_{[0,\,1]}$ is the de Rham algebra of $[0,\,1]$ (homologically graded), then integrate from $0$ to $1$. But I haven't worked this out.

Any suggestions or references would be really appreciated.

Yining Zhang
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    The family must satisfy some extra condition, I suppose? Just parameterizing 'discretely' a collection of chain maps won't yield a homotopy. – Pedro May 19 '21 at 22:58
  • If you assume what you have is a chain map to the tensor product with the de Rham algebra, then that's a different story! – Pedro May 19 '21 at 23:01
  • @PedroTamaroff $f_t$ varies 'continuously' on $t$ if it makes any sense. – Yining Zhang May 19 '21 at 23:07
  • @PedroTamaroff The chain map to the tensor product is what I want to construct, but I wasn't able to – Yining Zhang May 19 '21 at 23:08
  • It is not clear to me what "varies continuously" means here. Could you clarify? Maybe if you add the context surrounding this question/problem, it will be easier to obtain an answer. – Pedro May 21 '21 at 13:28

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