The original definition of operads involves maps $$ \gamma\colon C(n) \times \prod_{i=1}^n C(k_i) \to C(\sum_ik_i) $$ There is an alternative definition in terms of maps $$ \circ_i \colon C(n) \times C(m) \to C(n+m-1). $$ It is not hard to outline an argument that the two definitions are equivalent (subject to some assumptions about $C(0)$ and $C(1)$; various choices of details are possible). Have the details been spelled out carefully somewhere? I have seen this called an "observation", but that seems a little blasé to me.

  • $\begingroup$ I've seen it in the Ginzburg & Kapranov Koszul duality paper. Not making this an answer since I am not sure what is your rigor standard :D (The paper has a subsequent erratum but it concerns a different place) $\endgroup$ Sep 2 '16 at 9:33
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    $\begingroup$ Phil Hackney just mentioned that they are not equivalent if $C(1)$ is not the unit of the monoidal structure. Maybe that is relevant? $\endgroup$ Sep 2 '16 at 9:57
  • $\begingroup$ @SeanTilson you are right that this is an issue; I have added a parenthetical comment. $\endgroup$ Sep 2 '16 at 10:17
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    $\begingroup$ I put a proof in one of my papers, but the referee told me to take it out. $\endgroup$ Sep 2 '16 at 11:16
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    $\begingroup$ @GregoryArone If you still have the draft of that paper you could copy-paste that proof here as an answer and finally get it out in the open. (A reference to a MO post is better than nothing...) $\endgroup$ Sep 2 '16 at 12:31

There is a detailed discussion of the "partial" composition operations of an operad in Volume 1, Chapter 2 of Fresse's monograph:

The equivalence of the two definitions of operad is stated as Theorem 2.1.10, and the proof is worked out in Appendix A (see A.2.10).


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