22 votes
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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 ...
Simon Henry's user avatar
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20 votes
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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I was friends with Frank Adams and Saunders Mac Lane, who invented PROPs in one of the world's most extensive unpublished collaborations.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. ...
Peter May's user avatar
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18 votes
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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, ...
Najib Idrissi's user avatar
17 votes
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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 ...
Tyler Lawson's user avatar
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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 ...
Qiaochu Yuan's user avatar
15 votes
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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 ...
Dan Petersen's user avatar
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13 votes
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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)^{|...
Pavel Safronov's user avatar
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 ...
Nicholas Kuhn's user avatar
13 votes
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"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 ...
Dan Petersen's user avatar
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12 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 ...
Tom Leinster's user avatar
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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 ...
Phil Tosteson's user avatar
11 votes
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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 ...
Peter May's user avatar
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11 votes
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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 ...
Dan Petersen's user avatar
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11 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 ...
Phil Tosteson's user avatar
10 votes
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understanding the definition of $\infty$-operad of module objects

I know this question is a little old but I just came across it. Roughly, as you say, from this data you get an object $v$ of $O^\otimes$ and an algebra $A$ of $Alg_{/O}(C)$. However, you also get an ...
Tyler Lawson's user avatar
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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 ...
Najib Idrissi's user avatar
9 votes
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Is there a proof of the formality of configuration spaces of Euclidean spaces that do not involve operads?

When $n$ is even these spaces are complex algebraic varieties, so the cohomology comes with a mixed Hodge structure. Moreover, this mixed Hodge structure is pure: the cohomology ring is generated in ...
Dan Petersen's user avatar
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9 votes
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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 ...
Peter LeFanu Lumsdaine's user avatar
9 votes
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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 ...
Najib Idrissi's user avatar
9 votes

Conceptual (operadic?) reason for the generalized EHP fiber sequence $J_{q-1}(S^{2n}) \to J S^{2n} \to JS^{2nq}$?

In my view, the fact of that this is a fibration sequence is something to be cherished, and I wouldn't think that it generalizes without complication. Regarding your comments that the James-Hopf ...
Nicholas Kuhn's user avatar
9 votes
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How to characterize $E_n$ morphisms from $\mathrm{Mod}(A)$ to $\mathrm{Mod}(B)$?

By Corollary HA.4.8.5.20, the functor from $\mathbb{E}_{n+1}$-algebras to $\mathbb{E}_n$-monoidal categories and colimit-preserving, $\mathbb{E}_n$-monoidal functors is fully faithful. (Notice that ...
Dylan Wilson's user avatar
  • 13.2k
9 votes
Accepted

Non(skew)commutative Lie algebras?

Such objects are known as Leibniz algebras. A Leibniz algebra is a module $M$ together with a bilinear pairing $$[-,-]\colon M⊗M→M$$ that satisfies the Leibniz identity: $$[a,[b,c]]=[[a,b],c]+[b,[a,c]]...
Dmitri Pavlov's user avatar
8 votes

Definition of E-infinity operad

There exists several models of $E_{\infty}$-operads. One model which works in chain complexes over any ring is the Barratt-Eccles operad, defined by applying aritywise the normalized chain complex to ...
Sinan Yalin's user avatar
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8 votes
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Does this notion related to species/operads/FI-modules have a name?

Depending on whether you want it to agree with the symmetric structure or only with monoidal structure, this would be usually referred to, respectively, as twisted commutative algebras or twisted ...
Vladimir Dotsenko's user avatar
8 votes

Suspension operad

$\Lambda$ is correct. $\Lambda'$ is not an operad because the $\circ_i$ maps are not equivariant with respect to the symmetric group actions. If $\sigma$ is the nontrivial element of $S_2$, then for ...
Gabriel C. Drummond-Cole's user avatar
8 votes
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Can operads (or category theoretic structures more generally) be compared?

Yes, operads can be compared. There are lots of kinds of operad (enriched in various categories, symmetric or plain or defined with respect to a monad, one-colored or many-colored, and don't even get ...
Tim Campion's user avatar
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8 votes
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Functoriality of infinite loop space machines?

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 ...
Peter May's user avatar
  • 30.2k
8 votes
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Is there a filtered splitting of product labelling spaces?

The answer to your first question is no. And this can be seen by homology considerations. Note that this equivalence induces an isomorphism of Hopf algebras $$ H_*(C(\mathbb R; X \vee Y \vee (X\...
Nicholas Kuhn's user avatar
8 votes
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Enriched coends which preserve equivalences

One sufficient condition is that either $B$ or both $A$ and $A'$ are cofibrant in the projective model structure. If $C$ is a Reedy category, possibly in the generalized sense defined by Berger and ...
Gregory Arone's user avatar
8 votes
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How now to study operads in homotopy theory?

One of the most comprehensive references today is certainly: B. Fresse, Homotopy of Operads and Grothendieck–Teichmüller Groups, Mathematical Surveys and Monographs 217. https://bookstore.ams.org/...
Najib Idrissi's user avatar

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