Talk:Homotopy category of chain complexes
Latest comment: 5 years ago by Lilalas in topic Why cochain instead of chain?
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Notation
editWhat does the double arrow mean in the chain homotopy diagram. Thanks! Randomblue (talk) 18:38, 27 August 2008 (UTC)
- It means that there are two maps: see the letters next to each arrow? This diagram does not entirely follow the usual conventions for commutative diagrams, as you can see from the formula above it. Ryan Reich (talk) 02:27, 28 August 2008 (UTC)
Notations are troubling since A is both an additive category and a complex in section 1! Olivier Peltre (talk) 12:14, 2 April 2018 (UTC)
- Maybe we could omit to name the category altogether and say something like "The homotopy category (over an additive category) is defined by ..." This should work since all objective live in the same category. --Lilalas (talk) 19:58, 20 January 2019 (UTC)
Why cochain instead of chain?
editIn the case of chain complexes, the derivation (boundary operation) goes to descending direction of the indices, while in the case of cochain, in the opposite. Here goes in ascending direction, so here are cochains. Why? Why not chains? 89.135.8.194 (talk) 06:28, 30 January 2016 (UTC)
- I think you are right. A _chain_ homotopy should be h_n : A_n -> B_(n+1) such that f_n - g_n = d^B_(n+1) h_n + h_(n-1) d^A_n . Note that the index is written in the subscript, as opposed to the _cochain_ complex where indices are usually written in the superscript. --Lilalas (talk) 20:07, 20 January 2019 (UTC)