My book denotes by σ:Δ(adsbygoogle = window.adsbygoogle || []).push({}); ^{k}→X for some suitable topological space X a standard k-simplex of X. It then describes the free abelian group generated by such σ's as the group of k-chains on X. It is not clear to me what is meant by a chain for a map σ. I understand a chain in R^{n}to be sums of integer linear combination of simplicies, but I cannot wrap my head around what is meant by chains of mappings of Δ^{k}. Is each σ mapping Δ^{k}to different neighborhoods on X? Is the group action for chains function composition of each σ? Currently, I am understanding a chain to be a mapping of Δ^{k}multiple times into X such that they connect at their boundaries (or perhaps overlap?), but I feel like this is not the correct view. Any insight would be greatly appreciated.

EDIT: I incorrectly put R^n in the title, it should be R^k to match the dimension of the simplex

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# Mapping of the standard k-simplex in R^n to X

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