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It appears that the terms "complex" and "sequence" are used synonymously in homological algebra.

But there seem to be collocations (in the linguistic sense) that prefer one of those words. For example, one usually uses

  • "short exact sequences" but not "short exact complexes"
  • "simplicial complex" but not "simplicial sequence"
  • "double complex" but not "double sequence"

Is there any hidden principle when to use which word, or is it perfectly fine in a scientific publication to stick to the common collocations, and decide by gut feeling in all other cases?

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    $\begingroup$ I think there are two clearly distinct terms: "complex" and "exact sequence". The latter happens to be a special case of the latter, but that doesn't mean that "complex" and "sequence" can be used as synonyms. $\endgroup$ Oct 15, 2014 at 17:30
  • $\begingroup$ "The latter happens to be a special case of the latter" is rather tautological. $\endgroup$ Jan 27, 2015 at 15:33

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The word complex can mean two different things: First, it is the short form for chain complex, which is a special case of a sequence of morphisms in some appropriate category. Second, it is the short form for simplicial complex or CW complex, which are special cases of topological spaces and not of sequences or morphisms. If a sequence of morphisms is exact, it is automatically a complex, hence "exact complex" would be redundant.

In view of those facts, I would say that it is not a good idea to use the terms "complex" and "sequence" as synonyms in general.

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