The operads tag has no usage guidance.

**6**

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306 views

### Semidirect products and PROPs

$\newcommand{\p}{\mathcal{P}}$Let G be a group, and let $G_n$ denote the wreath product $G^n \rtimes \Sigma_n$.
There seems to be a notion of a PROP $\p$ where the role of the symmetric groups ...

**6**

votes

**1**answer

562 views

### Classifying spaces of E_1 - spaces

Hello,
I try to understand aspects of homotopy coherence, in particular "recognition principle" of May.
About the following I did not think a lot, but I decided to ask here anyway, so to save ...

**4**

votes

**1**answer

984 views

### A-infinity structure of E-infinity algebras

This is perhaps somewhat related to this question. Fix a field $k$ of characteristic $p>0$. Suppose that $A$ is an $E_\infty$-algebra over $k$. Then $A$ also has an $A_\infty$-algebra structure, ...

**16**

votes

**1**answer

690 views

### Higher homotopy algebraic structure on the homology of an operad

Given a DGA $A$, then by standard techniques such as homological perturbation theory, the ring structure on the homology $H(A)$ extends to a minimal $A_\infty$-algebra structure such that $H(A)$ is ...

**8**

votes

**0**answers

279 views

### Question about $A_\infty$ maps

Given $A_\infty$-spaces $X$ and $Y$, Boardman and Vogt defined an $A_\infty$-map from $X$ to $Y$ to be a map $f: X \to Y$ of underlying based spaces and an $A_\infty$-structure on the reduced mapping ...

**18**

votes

**1**answer

3k views

### What's up with Wick's theorem?

Sorry about the dumb title.
I'd like to understand Wick's theorem. More specifically, I have seen it pop up in several different contexts and I am really puzzled by the different statements of it ...

**1**

vote

**0**answers

531 views

### Recognition principle

Hello,
The classical statement of the recognition principle (after Boardman, Vogt, Milgram and May) that I know is:
Let $X$ be a (group-like) topological space acted on by the little $n$-discs ...

**6**

votes

**1**answer

490 views

### How does Berger-Moerdijk's relative Boardman-Vogt work?

In "The Boardman-Vogt resolution of operads in monoidal model categories," the authors construct factorizations of sufficiently nice operad maps $P\to Q$ into a cofibration followed by a weak ...

**4**

votes

**1**answer

423 views

### Repairing the Lie operad in characterstic 2?

Recall: We present an operad (with $S_n$-action) in $R-Mod$ for any commutative ring $R$ not of characteristic 2 generated by a single element in degree $2$ satisfying the following identities:
...

**15**

votes

**2**answers

3k views

### Kontsevich's conjectures on the Grothendieck-Teichmüller group?

Reading Kontsevich's "Operads and Motives in Deformation Quantization", I was wondering about the state of the many conjectures concerning the Grothendieck-Teichmüller group in chapter 4. (Also, where ...

**3**

votes

**1**answer

636 views

### Automorphisms of the rooted tree operad

This follows Ryan Budney's comment to the question asked here.
What is the automorphism group of the rooted tree operad?
(By the rooted tree operad, I just mean the operad with object rooted ...

**1**

vote

**2**answers

711 views

### What are operad automorphisms?

What is the general concept of an *operad automorphism*$?$ Is there a "standard" definition?
[added after comment] If an operad automorphism is an invertible operad endomorphism, how then is operad ...

**16**

votes

**1**answer

1k views

### Little disks operad and $Gal (\bar {Q}/Q)$

My question is simple:
How do the little disks operad and $Gal (\bar {Q}/Q)$ relate?
I realize that a huge amount of heavy-machinery can be brought into an answer to this, but I'm struggling ...

**12**

votes

**1**answer

975 views

### koszul duality and algebras over operads

Given a pair of Koszul dual algebras, say $S^*(V)$ and $\bigwedge^*(V^*)$ for some vector space $V$, one obtains a triangulated equivalence between their bounded derived categories of ...

**6**

votes

**1**answer

465 views

### Do you recognise this variant of the cubes operad?

In the subject of operads one comes across many quirky variants of more or less the same operad. The cubes operad has many incarnations with various interesting properties. In a recent paper I came ...

**26**

votes

**5**answers

2k views

### What are natural questions to ask about an operad?

I just discovered that something I've been working with has the structure of an operad. So I'm wondering what natural basic questions does one ask about operads? For example, if I knew I had the ...

**2**

votes

**1**answer

377 views

### Extending a property of commutative algebras to C infinity algebras

If A is a commutative algebra and B is an X- algebra, then the tensor product $A \otimes B$ is an X-algebra (so for example, $Com \otimes Lie$ is a Lie algebra). This is seen using the language of ...

**5**

votes

**2**answers

653 views

### Operad terminology - Operads with and without O(0).

In the Markl, Schneider and Stasheff text, topological operads are an indexed collection of spaces $O(n)$ for $n \in \{1,2,3,\cdots\}$ satisfying some axioms. In May's text, the index set is allowed ...

**29**

votes

**3**answers

2k views

### Algebras over the little disks operad

Hello,
The so-called "recognition principle" of Boardman-Vogt and May leaves me unsatisfied.
My problem is the following:
The "recognition principle" says that every "group-like" algebra over the ...

**13**

votes

**1**answer

627 views

### Are G_infinity algebras B_infinity? Vice versa?

What is the relationship between $G_\infty$ (homotopy Gerstenhaber) and $B_\infty$ algebras?
In Getzler & Jones "Operads, homotopy algebra, and iterated integrals for double loop spaces" (a paper ...

**9**

votes

**2**answers

643 views

### Right actions of operads and monads

Given an operad $A$, there is an associated monad $M_A$ given by $M_A(X) = \coprod_n A(n) \otimes X^{\otimes n}$ such that being an $A$-algebra and being an algebra over the monad $M_A$ is the same ...

**8**

votes

**2**answers

581 views

### Is the cohomology of a topological operad a cooperad?

For cohomology with coefficients in a field $F$ the map $H^\cdot(X;F) \otimes H^\cdot(Y;F) \to H^\cdot(X \times Y;F)$ of the Kunneth theorem is an isomorphism of algebras over $F$. I am correct in ...

**6**

votes

**1**answer

350 views

### Transmutation versus operads

A while ago, I was reading Majid's book Foundations of quantum group theory, and Section 9.4 has a rather fascinating description of a Tannaka-Krein reconstruction result for quantum groups. In ...

**9**

votes

**2**answers

307 views

### When do PROP-morphisms induce adjunctions?

If (C,tensor,1) is a symmetric monoidal category and f:A-->B is a morphism of PROPs (or monoidal cats = colored PROPs), one gets a forgetful functor f^*:B-Alg(C)-->A-Alg(C) (where ...

**6**

votes

**1**answer

448 views

### Pontryagin product from an operad

For a topological group G, we have a Pontryagin product in homology by multiplying representative cycles. This gives the homology the structure of an associative graded algebra. Am I correct in ...

**15**

votes

**4**answers

2k views

### Deligne's conjecture (the little discs operad one)

Deligne's conjecture states that the Hochschild cochain complex of an A-infinity algebra is an algebra over the operad of chains on the topological little discs operad.
Of course the conjecture has ...