# Detailed proof that $\circ_i$ operads are the same as operads

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.

• 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) Sep 2 '16 at 9:33
• Phil Hackney just mentioned that they are not equivalent if $C(1)$ is not the unit of the monoidal structure. Maybe that is relevant? Sep 2 '16 at 9:57
• @SeanTilson you are right that this is an issue; I have added a parenthetical comment. Sep 2 '16 at 10:17
• I put a proof in one of my papers, but the referee told me to take it out. Sep 2 '16 at 11:16
• @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...) Sep 2 '16 at 12:31