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Gijs Heuts
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The answer to 2) is yes for cofibrant operads, see http://arxiv.org/abs/1102.1311 by Fiedorowicz and Vogt. The answer to 1) is no; the Boardman-Vogt tensor product does not interact well with cofibrations.

Gijs Heuts
  • 336
  • 2
  • 5