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Jun 9, 2023 at 7:06 comment added Fernando Muro For non symmetric monoidal categories you have notions of left and right closed. The category whose monoids are operads is left closed but not right closed (or the other way around).
Jun 9, 2023 at 2:23 comment added David Roberts Published version of Street's article: doi.org/10.1016/j.jpaa.2012.09.020
Sep 30, 2015 at 14:46 comment added john (1)Then you get a non-symmetric monoidal category. (2) Yes, non-symmetric promonoidal categories exist - if C is a non-symmetric monoidal category then setting $P(A,B,C)=C(A \otimes B,C)$ gives a non-symmetric promonoidal category.
Sep 29, 2015 at 19:30 comment added SCappella I think I understand now, but a point of clarification, in my question, I'm not talking about symmetric monoidal categories, only monoidal categories. What happens when you use a non-symmetric promonoidal category and demand that it's representable? Do non-symmetric promonoidal categories exist?
Sep 29, 2015 at 19:11 vote accept SCappella
Sep 29, 2015 at 19:27
Sep 28, 2015 at 13:08 history edited Zhen Lin CC BY-SA 3.0
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Sep 28, 2015 at 12:37 history answered john CC BY-SA 3.0