The operads tag has no wiki summary.

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### Repertory of the different sorts of operads

Many different types of operads have emerged in recent years (symmetric, shuffle, cyclic, anticyclic, coloured, etc.).
I would like, for any of these, list the following data:
Description of the ...

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**1**answer

243 views

### Non-$\Sigma$ $E_n$ algebras?

Any symmetric operad can be considered as an non-$\Sigma$ operad by throwing away permutations. Does anyone know what sort of structure one gets for algebras over $C_n$ the little n-cubes operad, or ...

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708 views

### On Tamarkin's proof of Etingof-Kazhdan quantization of Lie bialgebra

This question is motivated by the desire to understand better the interplay between Drinfeld associators, algebraic structures up to homotopy and related results.
Recall that a Lie bialgebra is a Lie ...

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**1**answer

190 views

### PROPs representations, free module analog

Ordinary operad with one ouput can be obviously regarded as free module on itself. Is there are analogous construction for operad with many outputs (PROP)? This must be difficult question, but what is ...

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297 views

### A (too?) simple notion of “closed multicategory”

Suppose I define a multicategory $M=(Ob(M),Hom_M)$ to be simply closed if
for every sequence $S=(b_1,\ldots,b_n;x)$ of $n+1$ objects in $M$, we provide an object $Exp(S)\in Ob(M)$, and
for every ...

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**1**answer

201 views

### Maps between operads of different arities

Is there a category of operads which allows morphisms which take operations to operations with more or fewer arguments? One example should be when you fix arguments to obtain maps with fewer inputs. ...

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525 views

### Delooping and unreduced operads

Suppose I have an operad $P(n)$ in topological spaces in the sense of the book by Markl, Shnider and Stasheff, i.e. there is no space $P(0)$. In agreement with some of the answers to this question, I ...

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617 views

### How to define the equivalence of Maurer-Cartan elements in an $L_{\infty}$-algebra?

First let $L^{\bullet}$ be a pro-nilpotent differential graded Lie algebra (dgla). We have the set of Maurer-Cartan elements in $L^{\bullet}$ ($MC(L^{\bullet})$) which are $\alpha \in L^1$ such that ...

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**3**answers

345 views

### Koszul duality for modular operads

Has anyone defined what it means for a modular operad to be Koszul, or what the Koszul dual of a modular operad is? In particular, is it meaningful to say that a modular operad is quadratic? Merkulov, ...

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**1**answer

183 views

### What is the definition of “the $L_\infty$ part of a $G_\infty$ morphism”?

We know that in Tamarkin's proof of Kontsevich's formality theorem, he defined the $G_\infty$ structure on the Hochschild cochain complex $C^\cdot(A,A)$ and constructed a $G_\infty$ morphism from ...

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### Are pivotal categories the algebras for a cartesian monad?

It seems to be "known" but not written down that the following are more-or-less equivalent:
...

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**1**answer

217 views

### Generating the graded S_n-module associated to an operad

Suppose I have a symmetric operad $\mathcal{P}$ defined over $\text{Vect}_{\mathbb{K}}$ with generators and relations in degrees at most $l$. Now suppose I already know $\mathcal{P}(k)$ as an ...

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**1**answer

319 views

### How strong is the condition that an operad splits, i.e. O(n)=O(s)xO(n-s)?

I recently proved something for the operad $Comm$ valued in a model category $M$ and am trying to generalize it to other symmetric operads. I'm very new to operads, so please forgive me if there are ...

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200 views

### Boardman Vogt W construction for modules over an operad

The W construction of Boardman and Vogt gives a cofibrant replacement for operads. In http://arxiv.org/abs/math/9907073, Salvatore describes a cofibrant replacement for algebras over an operad. Is ...

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**1**answer

620 views

### Is there a cheap proof that (homotopy) endomorphisms are functorial?

This is, in some sense, the homotopy version of this question.)
If $C$ is a category with $iC$ the subcategory of isomorphisms, there is a functor $X \mapsto End(X)$ from $iC$ to the category of ...

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419 views

### $A_{\infty}$ structure questions

Hello,
I would like explanation or clear source for some things related to $A_{\infty}$-spaces, via Stasheff's polytopes.
I tried not to think about them, because they seem too complicated for me; I ...

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643 views

### Compatibility of the KZ connection with operadic composition

In what sense is the Knizhnik-Zamolodchikov connection compatible with the operadic composition in the little discs operad and/or the operad of $\overline{M}\_{0,n}$'s?
Here are (some) details, ...

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290 views

### 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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352 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

489 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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383 views

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

Informally, an $A_\infty$-space is a monoid whose laws are only satisfied up to homotopy.
Let’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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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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### 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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406 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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906 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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530 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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543 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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### 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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### 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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558 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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### 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 ...

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548 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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945 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, ...

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652 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 ...

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273 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 ...

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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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490 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

458 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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400 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-Teichmüller group in chapter 4. (Also, where ...

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604 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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695 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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961 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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925 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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453 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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368 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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633 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 ...