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It is well known that the nerve (or Čech complex) of a covering by metric balls is nicely approximated by the Vietoris-Rips complex. Being a flag complex on its 1-simplices, the latter is computationally much more tractable than the nerve itself.

Does anybody know of any other classes of "nice" coverings whose nerves have an approximation à la what Vietoris-Rips provides for coverings by balls? (I'm quite open to vague answers, heavy restrictions on the covers, and poor approximations.)

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