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Actually, I think I’ve heard it used more generally for a functor from a totally ordered poset.
The use of chain as in chain complexes derives from the order theoretic use I suspect, but I am not sure. Does anyone know?
@Todd:
That seems like a straighforward generalization (remove the requirement of injectivity, generalize the codomain), although not one that I've seen that I can recall.
@Tim:
How do you see that coming about? I don't see anything ordered there.
The notation ,for example for chains of open sets in the nerve of a cover, when defining Cech homology etc., uses implicitly orders the elements making up the group of chains in the space. Perhaps even probably I should have written ’ from the use of order in their definition’, rather than saying ‘order theoretic’. It was coming from the intuition based on simplexes and the combinatorial formulae for their boundaries.
Looking back at EIlenberg’s original paper on singular homotopy, (first page of introduction only as it is behind a paywall!!!), it seems that he is thinking of mapping polyhedra into the space, and the geometric intuition behind simplices and other polyhedra has a good deal of the intuition of order in it. However I do no know how people thought of the early chains, whether it was as a sequence of edges or whatever or as a collection of edges (with no order). Does any one know when ‘chain’ was first used in this contect? I think my point is that a physical chain is not a collection of rings, rather it is a sequence of linked rings, hence order is involves in its very existence as a chain.
Interesting question, Tim! There is some information here.
Thanks, Richard. There are some interesting points made in the replies to that question.
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