The operads tag has no wiki summary.

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### When can I assure that the representation theory of a PROP is faithful?

Recall that a PROP is a symmetric monoidal category whose object-set is freely $\otimes$-generated by a single object $x\in P$. A representation or algebra for a prop $P$ in a symmetric monoidal ...

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338 views

### []-infinity algebra and Projective representation

This is a very vague question.
We know that some algebra structures can be viewed as modules of some fantastic stuff, call T. Such examples include: Abelian groups are $\mathbb{Z}$-modules, chain ...

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**1**answer

458 views

### Where does the definition of “Tower of Algebras” come from?

A tower of algebras is a sequence of algebras
$$A_0 \hookrightarrow A_1 \hookrightarrow \cdots \hookrightarrow A_n \hookrightarrow \cdots$$
with embeddings $A_n \otimes A_m \hookrightarrow A_{n+m}$ ...

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370 views

### Homotopic monoids and $A_\infty$ spaces

Informally, an $A_\infty$-space is a monoid whose laws are only satisfied up to homotopy.
Letâ€™s define now what I will call a "homotopic monoid" to be a space $M$ together with a point $e\in{}M$ and ...

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**4**answers

1k views

### Why are operads so closely connected to mathematical physics?

Mark Sapir's question inspired me to ask the question in the title. A lot of mathematicians who have done work related to mathematical physics (e.g Kontsevich, Stasheff, Getzler, Manin, etc.) have ...

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2k views

### Why are operads useful?

The question is not about where operads are used, I know that. It is about what makes them useful. For example, van Kampen diagrams are useful in combinatorial group theory because these are planar ...

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**1**answer

391 views

### Is there a “derived” Free $P$-algebra functor for an operad $P$?

Recall that an operad (in vector spaces, say) $P$ consists of a collection of vector spaces $P(n)$ for $n\geq 0$, such that $P(n)$ is equipped with an action by the symmetric group $S_n$, with maps ...

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1k views

### Relation between monads, operads and algebraic theories

I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. Is there any ...

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**3**answers

862 views

### Monad arising from operad

It's known that from every operad arises a cartesian monad whose algebras are the algebras for the operad. Leinster proved that there are different operads from which arise the same monad, in this way ...

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**2**answers

512 views

### GL(V)-representation theory for a Lie bracket kernel

Let $V$ be a vector space over a field of characteristic $0$, and let $L_k(V)$ be the degree $k$ part of the free Lie algebra over $V$. There is an exact sequence
$$0\to D_n(V)\to L_1(V)\otimes ...

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531 views

### Splitting of homomorphism from cactus group to permutation group

We all learned in kindergarten that the category of finite-dimensional (type I, say) $U_q(\mathfrak{g})$-modules is braided monoidal for $\mathfrak{g}$ a complex semisimple Lie algebra. This gives an ...

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2k views

### Shuffle Hopf algebra: how to prove its properties in a slick way?

Let $k$ be a commutative ring with $1$, and let $V$ be a $k$-module. Let $TV$ be the $k$-module $\bigoplus\limits_{n\in\mathbb N}V^{\otimes n}$, where all tensor products are over $k$.
We define a ...

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**1**answer

585 views

### Do I need to know what an infinity-Gerstenhaber algebra is, and if so, what is it?

I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R ...

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**1**answer

534 views

### PBW theorem over a Q-algebra, without freeness or flatness

Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-Lie algebra, which is not necessarily free as a $k$-module. Let $S\left(L\right)$ denote the symmetric algebra of $L$ (over $k$), constructed ...

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**1**answer

287 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 ...

**5**

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**1**answer

522 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 ...

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**1**answer

925 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, ...

**13**

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**1**answer

623 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**

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**0**answers

258 views

### Question about $A_\infty$ maps

Given $A_\infty$-spaces $X$ and $Y$, Boardman and Vogt defined an $A_\infty$-mapÂ 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 ...

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**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 ...

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448 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 ...

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**1**answer

419 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 ...

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**1**answer

384 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:
...

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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-TeichmĂ¼ller group in chapter 4. (Also, where ...

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**1**answer

583 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 ...

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**2**answers

688 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 ...

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**1**answer

908 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 ...

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**1**answer

883 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 ...

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440 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 ...

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### 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 ...

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**1**answer

365 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 ...

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616 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 ...

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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 ...

**12**

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**1**answer

564 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 ...

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614 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 ...

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538 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 ...

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**1**answer

321 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 ...

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**2**answers

285 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 ...

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**1**answer

431 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 ...

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1k 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 ...