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79 votes
Accepted

Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop Space"?

I'm not quite certain what Peter May had in mind 40 years ago, but probably he had in mind the fact that pushouts are a lot better behaved in CGWH than in CGH. Specifically, CGWH is closed under ...
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20 votes

Is the Amitsur-Levitzki identity essentially unique?

The answer is No. For $n=2$, there is the Hall identity $[[x,y]^2,z]=0$. Drensky proved in 1981 that these two identities (Hall and standard (=Amitsur-Levitzki)) form a basis of identities of $Mat_2(\...
20 votes
Accepted

Why are operads sometimes better than algebraic theories?

First - yes, for symmetric set-operads this functor is "injective", though it is not fully faithful. It is faithful on general maps and fully faithful on isomorphisms. Its image can easily ...
19 votes
Accepted

Positivity of coefficients of the inverse of a certain power series

Combining Omar and fedja's comments gives a quick solution. By the Lagrange Inversion formula, we want to show that the coefficient of $t^k$ in $(1-t+t^2)^{-(7k+1)}$ is positive. Writing this as a ...
19 votes
Accepted

In what sense are operads "better" than PROPs?

I was friends with Frank Adams and Saunders Mac Lane, who invented PROPs in one of the world's most extensive unpublished collaborations. Saunders once showed me a box full of their correspondence. ...
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18 votes
Accepted

Does Koszul duality between $Comm$ and $Lie$ imply the power series identity $\exp(\ln(1-z))-1 = -z$?

Let me flesh out the answer a little. The general statement is given by Theorem 7.5.1 in the book Algebraic Operads by Loday and Vallette. First a definition. Let $P = P(E,R)$ be a quadratic operad, ...
17 votes
Accepted

Is the Amitsur-Levitzki identity essentially unique?

The answer to your question is "no", as explained by Anton Klyachko in his answer. Let me refer you to a remarkable statement of Razmyslov and Procesi that describes all identities. They proved (...
17 votes
Accepted

Is it possible to construct an action of an $E_\infty$ operad on $BU$ that respects filtration by $BU(n)$?

Unfortunately there is no such filtration. At first, this looks very similar (but not as strong as) asking for a map of $E_\infty$ spaces $\coprod BU(n) \to BU$ which would become a splitting map $ku ...
  • 48.3k
16 votes

In what sense are operads "better" than PROPs?

One thing you can do with an operad that you cannot do with a prop is write down a monad such that algebras over the monad correspond to algebras over the operad. For example, Hopf algebras have a ...
15 votes
Accepted

Homotopy theories of operads

The answer is yes: see the paper of Chu-Haugseng-Heuts, "Two models for the homotopy theory of ∞-operads", arXiv:1606.03826. In brief, already Cisinski and Moerdijk ("Dendroidal sets and simplicial ...
  • 37.9k
14 votes
Accepted

Obstructions for $E_n$-algebras

Let me expand a little on what Qiaochu and Craig mentioned. If you want an obstruction theory for building an uber-gadget, you'll need (i) an algebraic approximation to such gadgets, and (ii) a way ...
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13 votes
Accepted

What is the interpretation of the Gerstenhaber bracket?

I think the most transparent interpretation is by identifying $E_2$ algebras with brace algebras (which was proved by McClure and Smith). Namely, a brace algebra satisfies the relation $$ab - (-1)^{|...
13 votes

Does Koszul duality between $Comm$ and $Lie$ imply the power series identity $\exp(\ln(1-z))-1 = -z$?

Yes. (Mathoverflow won't let me make this my total answer, so ...) Koszul duality says that a certain chain complex of graded vector spaces is acyclic. Thus the alternating sum of the Poincare ...
13 votes
Accepted

"Exactness" of operadic cohomology

The second result you mention is significantly easier than the first. Indeed from the PBW theorem we know that $L$ is a direct summand of $UL$ as an $L$-module, and the second result follows. But in ...
  • 37.9k
12 votes
Accepted

What are algebras for the little n-balls/n-cubes/n-something operads exactly?

An $E_n$ algebra (an algebra over the little $n$-cubes operad, etc.) is intuitively an object with $n$ compatible monoid structures. All of the subtlety in this theory lies in making "compatible" ...
12 votes

Homotopy Gerstenhaber algebras: description via operads vs derivations

The equivalence between these two notions is nontrivial, since it amounts to a choice of formality isomorphism for the operad of little disks. Let $D_2$ be the little disks operad. The easy part ...
11 votes
Accepted

Definition of E-infinity operad

Operads $\mathcal{C}$ can be defined in any symmetric monoidal category, and then $E_{\infty}$ operads are specified in accordance with the (or a) notion of equivalence relevant to that category. In ...
  • 29.3k
11 votes
Accepted

Tensor product of a DGA and an $A_\infty$ algebra

In fact the tensor product of two $A_\infty$ algebras can be made into an $A_\infty$ algebra in an explicit way: there are two constructions, one by Saneblidze-Umble and one by Loday. See the paper ...
  • 37.9k
11 votes

Are there prominent examples of operads in schemes?

Mikhail Kapranov's invited lecture at the 1998 ICM was about exactly this: Mikhail Kapranov, Operads and algebraic geometry. Documenta Mathematica Extra Volume ICM II (1998), 277-286. (PDF of the ...
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10 votes
Accepted

$k$-Disk algebras versus $E_k$ algebras

There is an unfortunate clash of terminologies here. Traditionally, the little discs operad comes in two variants: the "usual" $\mathtt{D}_n$: the space of arity $r$ operations consists of embeddings ...
10 votes
Accepted

Reference Request: Grouplike Algebras over the little $n$-cubes operad are $n$-fold loop spaces

May gave a proof in the case $n = \infty$ in this followup paper to "Geometry of iterated loop spaces," which relies on knowing that a certain map (from a free algebra on $X$ to the free infinite loop ...
  • 48.3k
10 votes
Accepted

Boardman-Vogt tensor product

The answer to 2) is yes for cofibrant operads, see http://arxiv.org/abs/1102.1311 by Fiedorowicz and Vogt. The answer to 1) is no; the Boardman-Vogt tensor product does not interact well with ...
10 votes

Are there prominent examples of operads in schemes?

The homology of $M_{0,n}$ is an operad in a natural way that is compatible with AG structure (like weights). The construction uses the operad structure on the compactification plus Gysin maps-- it is ...
10 votes

Are there prominent examples of operads in schemes?

I recommend reading the paper Mixed Hodge structures and formality of symmetric monoidal functors by Cirici and Horel. They give several examples, and the paper is a great motivation for why one would ...
9 votes

Errata for Getzler-Kapranov "Cyclic operads and Cyclic homology"

I am using this version of the paper, and it seems like the published version looks exactly like that. Here is what I have found. First, a warning: Getzler and Kapranov use the term "quadratic operad"...
9 votes
Accepted

$E_{\infty}$ spaces are $A_{\infty}$ spaces

If $X$ and $Y$ are $G$-spaces for any group $G$, then the projection $\pi\colon X\times Y\to Y$ is a $G$-map, trivially: $\pi (gx,gy) = gy$. The map $\pi_2$ of your question is a very special case; ...
  • 29.3k
9 votes
Accepted

Are $E_n$-operads not formal in characteristic not equal to zero?

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 ...
  • 29.3k
9 votes
Accepted

Is there an operad that codifies groupoids?

The question (under the relaxed notion of operad in the original post) will be settled negatively if we show that the category $\mathrm{Grpd}$ (internal groupoids in $\mathrm{Set}$) isn't monadic over ...
  • 51k
9 votes
Accepted

Two monoidal structures and copowering

No. Consider the case where $(M,\otimes,1)$ is $(\mathbf{Set},\times,1)$, so the enrichment is vacuous, and $(C,\oplus,0)$ is $(\mathbf{Set},+,0)$, with copowering $\odot$ given by $\times$. Then ...
9 votes
Accepted

Free operad over a monoid object

Let me mention that this is related to this earlier question of mine (which is unanswered :-( ) and more generally to semi-direct products of operads by bialgebras. You construction is the semi-direct ...

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