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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJun 20th 2016
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   Author: Mike Shulman
   Format: MarkdownItexThe domains of morphisms in a multicategory are usually defined to be finite lists of objects. However, in the symmetric case one can also define them to be families of objects indexed by an arbitrary finite set. A natural functorial action by isomorphisms of finite sets then encodes the symmetries in a symmetric multicategory without having to arbitrarily choose any orderings. (One can do something similar for non-symmetric multicategories by using totally ordered finite sets, but the payoff is less.) I am sure I've seen this written down somewhere, but I can't think of where. Does anyone know a reference for this?

   The domains of morphisms in a multicategory are usually defined to be finite lists of objects. However, in the symmetric case one can also define them to be families of objects indexed by an arbitrary finite set. A natural functorial action by isomorphisms of finite sets then encodes the symmetries in a symmetric multicategory without having to arbitrarily choose any orderings. (One can do something similar for non-symmetric multicategories by using totally ordered finite sets, but the payoff is less.)

   I am sure I’ve seen this written down somewhere, but I can’t think of where. Does anyone know a reference for this?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 20th 2016
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   Author: Urs
   Format: MarkdownItexThat sounds like the definition of operads via monoidal categories suitably fibered over Segal's category $\Gamma$ of (pointed) finite sets, as in "Higher Algebra".

   That sounds like the definition of operads via monoidal categories suitably fibered over Segal’s category $\Gamma$ of (pointed) finite sets, as in “Higher Algebra”.

3. • CommentRowNumber3.
   • CommentAuthorKarol Szumiło
   • CommentTimeJun 20th 2016
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   Author: Karol Szumiło
   Format: MarkdownItexIf I understand your question correctly, then Appendix A.2 in Leinster's _Higher Operads, Higher Categories_ is exactly about that.

   If I understand your question correctly, then Appendix A.2 in Leinster’s Higher Operads, Higher Categories is exactly about that.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJun 20th 2016
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   Author: Mike Shulman
   Format: MarkdownItexThanks Karol, that's what I was thinking of! I looked in HOHC, but the unhelpful title of appendix A threw me off.

   Thanks Karol, that’s what I was thinking of! I looked in HOHC, but the unhelpful title of appendix A threw me off.