New stub for the notion of clone in universal algebra, including a reference relating it to operads.

]]>The definition and the annotated bibliography are given for Feynman category.

I wonder how useful this could be in related to elucidate the cohomological and motivic quantization via correspondences (Kan extensions in the setup of Feynman categories can help getting the pushforwards, Connes-Kreimer Hopf algebra, Feynman transform (which in some cases gives coefficients in the formal development of the Feynman integal, basically being partition functions, hence connection to graphs).

]]>Is there a term to denote collectively all the constructions of “theories” of some sort, without giving rise to ambiguities?

What I exactly mean is a collective term for monads, operads, clubs, Lawvere theories, and in general all categorical constructions that may have “algebras”, or which describe formal operations in some (possibly generalized) way. In my mind they all encode some sort of “theory”, but the word “theory” seems to be reserved for “algebraic theory”/”Lawvere theory”.

Which term would you use instead?

(I hope it’s clear enough what I’m asking.)

]]>Added a page about a colored generalization of the notion of a symmetric sequence at symmetric colored sequence. I’m happy to merge this (or some heavily edited and corrected version of it) with the page on symmetric sequences. Also open to massive edits or whatever. Just feel like *something* like this should be on here.

Couldn’t find a latest changes discussion for symmetric sequence so I am just reporting that I added a little bit to that page. In particular, I added another slicker definition in the case that we are interested in a symmetric sequence for the sequence of symmetric groups.

]]>Hey everyone,

I feel like this must be written down somewhere, and you all are probably the most knowledgeable about such things. Given a monoidal category, if I take its underlying multicategory, take the free symmetric multicategory thereon, followed by the category of operators, does this thing admit a Grothendieck opfibration to the category of operators of the free symmetric multicategory associated to the associative operad (i.e. the category of operators of the multicategory with one object and mapping sets given by the symmetric groups)? This seems certainly true, but I’m more interested in finding a place where this is written down so I can cite it directly (trying to avoid going into too significant detail on multicategories in a paper I’m writing).

Thanks for any kind of references to check out!

-Jon

]]>The nLab article A-∞-operad has a section “The standard categorical A_∞ operad”. Is there a reference for this construction?

]]>Small A_∞-dg-categories and small dg-categories admit a model structure and the forgetful functor from dg-categories to A_∞-dg-categories is a right Quillen functor so that the resulting Quillen adjunction is actually a Quillen equivalence, which can be seen as a many-objects version of the Quillen equivalence between A_∞-dg-algebras and dg-algebras, as described, for example, by Berger and Moerdijk, “Axiomatic homotopy theory for operads”.

Is there a written source for this Quillen equivalence?

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