22
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 ...

20
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I was friends with Frank Adams and Saunders Mac Lane, who invented PROPs in one of the world's most extensive unpublished collaborations.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. ...

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 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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

10
votes

Accepted

### 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 ...

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

Accepted

### 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 ...

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 ...

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 ...

9
votes

Accepted

### 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 ...

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]]...

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 ...

8
votes

Accepted

### 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 ...

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 ...

8
votes

Accepted

### 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 ...

8
votes

Accepted

### 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 ...

8
votes

Accepted

### 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\...

8
votes

Accepted

### 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 ...

8
votes

Accepted

### 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/...

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