Search Results

A pair of brief papers "Definitions: operads, algebras and modules" and "Operads, algebras and modules", which are available at http://www.math.uchicago.edu/~may/PAPERS/mayi.pdf and http://www.math.uchicago.edu … Benoit Fresse's book "Modules over operads and functors" gives a quite different take on operads, with a focus on modules over algebras over operads. …

let me give a possibly helpful "answer" by ancient history. Since I don't actually have an answer, these should be comments, but they are too long for that. In Section 1 of Chapter I of "$E_{\inft …

Here is a nice gentle old-fashioned answer. Symmetric monoidal categories are functorially equivalent as symmetric monoidal categories to permutative (symmetric strict monoidal) categories, and thos …

I put in a remark about this question back in 2013. I'm going to expand on that for emphasis. Let me shamelessly quote myself, from the expository paper http://www.math.uchicago.edu/~may/PAPERS/mayi. …

Justin, that is a cute question, and I've never thought about it. One starting thought free guess (acting on connected spaces) is that a non-$\Sigma$ $C_n$-space has $n$ possibly inequivalent but def …

Operads are of course equivalent to a special kind of PROP, and the specialization made it very much easier to find operad actions. … The connection with monads was intrinsic to the definition of operads (I convinced Mac Lane to switch from "triples'' to "monads" in Categories for the Working Mathematician in large part in order to make …

Ok, Sean. I'll write in terms of homology operations. Let $\mathcal C$ be any $\Sigma$-free operad, in spaces or in chain complexes, makes no real difference to the answer. For definiteness, take c …

Denis and ``Ring Spectra'', thanks for the references. I did not treat non-connected spaces in "The geometry of iterated loop spaces", which is why you couldn't find that there. It should have been t …

Let me refer you to Section 7 of a recently posted paper by Guillou and myself where certain operads P, Q, and R are defined and related. … I would bet that there are operads like P, maybe your P itself, which act on spaces that even have non-Abelian fundamental groups. Just a hunch. Sorry not to be more helpful. …

The map $\pi_2$ of your question is a very special case; note that $\mathcal M$ as I defined it is an operad as I defined operads, with $\Sigma_j$ acting on $\mathcal M(j)$. … Model categories are extremely important but entirely irrelevant here, and taking cofibrant approximations of operads tends to destroy their relevant individuality: different $E_{\infty}$ operads play …

Operads are definable in any symmetric monoidal category. They always have associated monads in that category such that the categories of algebras over the operad and monad are isomorphic. … That relationship between operads and monads motivated my coining of the word "operad'', back in 1971. …

Consider reduced operads in a cartesian monoidal category, so that $C(0) = \ast$. A monoid $M$ in our category gives a reduced operad $R(M)$ with $j$th object $M^j$. … The isomorphism of hom sets uses the degeneracy operators that are there because we are working with reduced operads. Obviously $LR= id$. Very often $id\to RL$ is an inclusion. …

By the way, there is an inherent flaw in the little discs operads $\mathcal{D}_n$, namely there is no map of operads $\mathcal{D}_n\longrightarrow \mathcal{D}_{n+1}$ that is compatible with suspension … The Steiner operads have all the good properties of both the $\mathcal{C}_n$ and the $\mathcal{D}_n$. …

A 2020 paper Operads, monoids, monads, and bar constructions by Ruozi Zhang, Foling Zou, and J.P. … The cited paper shows that unital operads admit an interesting variant of Kelley's interpretation of operads as monoids in a certain monoidal category, and that reinterpretation is seriously interesting …

From the horse's mouth. I would think a good theory of parametrized $E_1$-spaces should not be too hard to develop, along the general lines of parametrized spaces (and spectra) as developed ad naus …