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I'm currently reading through Ed Brown's paper "Twisted tensor products, I", (MR105687, Zbl 0199.58201) and I couldn't find any simple examples of twisting cochains. I understand all algebraic parts of the paper, but I can't find any intuition on twisting cochains.

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    $\begingroup$ what sort of intuition are you looking for? e.g. if you're happy with vector bundles and dg-categories, then there is the (formally true) analogy [twisting cochains : vector bundles :: dg-nerve : nerve]. you can find this sort of scattered through the papers by Wei (and Block et al.) on twisting cochains and perfect complexes $\endgroup$
    – Tim
    Commented Jul 15 at 19:42
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    $\begingroup$ @Tim, I thought about something explicit, like suppose I have Hopf fibration, and I want to describe explicit twisted cochain which can be used to compute homology of $S^3$ as a total space $\endgroup$
    – VadimKSt
    Commented Jul 15 at 23:03
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    $\begingroup$ For the simplicial version, you might like the nLab entry on twisting functions: given a simplicial group $G$, principal $G$-bundles over a simplicial set $X$ can be classified by simplicial maps $X\to\bar W(G)$ to the delooping of $G$. Such maps amount to what is called twisting functions on $X$ with values in $G$. Then further for a $G$-space $F$, total spaces of $G$-bundles with fibre $F$ over $X$ can be constructed using these twisting functions. Passing to (co)chains is more or less straightforward. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jul 16 at 6:56
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    $\begingroup$ For example, the Hopf bundle is a principal circle group bundle over the 2-sphere. Correspondingly, you can take the standard $K(\mathbb Z,1)$ for $G$, the simplicial set with single vertex and single nondegenerate simplex in dimension 2 for $X$, the embedding of generators for $X\to K(\mathbb Z,2)=\bar W(G)$, and easily compute the corresponding twisting function $\phi$. It is determined by $\phi_2:X_2\to G_1$ sending the nondegenerate simplex to the generator of $G_1=\mathbb Z$. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jul 16 at 7:07

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