The operads tag has no usage guidance.

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323 views

### Galois action on $E_n$-operads

Let $E_n$ be the little $n$-cubes operad which acts on $n$-fold loop spaces (up to group completion, an $E_n$-action is precisely the data needed to perform an $n$-fold delooping). I am looking for ...

**2**

votes

**1**answer

216 views

### Embedding e_n -> e_m

Let $e_n$ be the operad of (say, rational) chains of the operad of little n-disks. Consider the natural embedding $e_n\to e_m$, where $n<m$ and $n,m>1$ induced by the embedding $\mathbb{R}^n\to ...

**9**

votes

**0**answers

276 views

### The spheres operad

I have a rather naive question. Consider the space of all maps
$$ S^{j_1} \times S^{j_2} \times \cdots \times S^{j_k} \to S^n $$
for all possible natural numbers $n, k, j_1, \cdots , j_k$.
This ...

**2**

votes

**2**answers

336 views

### Reference for Stasheff Operad

I want a reference for Stasheff operad, where operad maps are defined explicitly at the point-set level. I would also like to ask the question that what exactly do one mean by Stasheff operad? Is ...

**7**

votes

**4**answers

401 views

### How universal is operadic approach to studying algebras?

I have just started to read about operads, so this question might be silly.
So it seems to me that any "reasonable" class of algebras can actually be defined as a class of all algebras over a certain ...

**4**

votes

**1**answer

270 views

### A_n operad as configuration spaces

$A_\infty$ operad can be described both in terms of Stasheff polytopes and configuration spaces.$A_n$ operad can be described as subspace of Stasheff operad described using Stasheff polytope. Is there ...

**3**

votes

**1**answer

197 views

### Is the definition of Gerstenhaber bracket related to operads?

I was reading these notes by Keller. On the page 19 he defines the Gerstenhaber bracket on the Hochschild cochain complex. But first of all he defines the operation $\bullet$ on the cochains by
$$
...

**6**

votes

**2**answers

539 views

### Aspherical operads

Let $O$ be an operad in $\mathtt{SETS}$. Assume that $O(0)$ is empty and $O(1)$ only consists of the identity. Assume for simplicity that $O$ is monochromatic, i.e. we have no labels on the ...

**15**

votes

**3**answers

1k views

### Good reference for studying operads?

Can you, please, recommend a good text about algebraic operads?
I know the main one, namely, Loday-Vallette "Algebraic operads". But it is very big and there is no way you can read it fast. Also ...

**1**

vote

**0**answers

109 views

### Exact sequence of L-infinity-algebras

We call a sequence of $L_\infty$-algebras (weak) maps
$$0\to L\xrightarrow{f} M\xrightarrow{g} N\to 0$$
is exact if it is exact on the the underlying chain complexes level.
Thought I don't know ...

**4**

votes

**0**answers

287 views

### Does abstract nonsense of model categories determine the “nonlinear” morphisms of $L_\infty$ algebras?

Recall that a Lie algebra is a module for the operad $\mathrm{Lie}$, which is freely generated by a binary antisymmetric operation $\beta$ modulo an equation that is quadratic in $\beta$. There is a ...

**7**

votes

**1**answer

305 views

### Identifying the little disk operad with parenthesized braids

Let $D_2$ be the topological operad of little disks. This operad can be modelled "combinatorially" in terms of an operad of groupoids called $\newcommand{\PaB}{\mathbf{PaB}}\PaB$, the operad of ...

**11**

votes

**2**answers

531 views

### Correspondence between operads and $\infty$-operads with one object

Given a simplicial operad one can form its category of operators. This is a simplicial category with a functor to the category of finite pointed sets which is a bijection on objects and whose ...

**12**

votes

**2**answers

393 views

### Introducing division strategically in operads to accommodate formulas like the Baker-Campbell-Hausdorff formula

I am not as familiar with operad terminology as I'd like to be, so I might be missing some well-known term in the area. If so, I'd appreciate any pointers to the correct terms.
Consider the following ...

**8**

votes

**1**answer

571 views

### Seeking errata for Berger-Moerdijk Axiomatic Homotopy Theory for Operads

The paper I'm referring to can be found here. It came out in 2003. I've been told by professors before that I should be careful relying on things from this paper, because it contained errors.
Is ...

**1**

vote

**0**answers

85 views

### Generating series of free PROs

Let
\begin{equation}
G := \biguplus_{p \geq 0} \: \biguplus_{q \geq 0} G(p, q)
\end{equation}
be a bigraded set of generators and $\mathcal{F}(G)$ be the free PRO generated by $G$ (see [1] for a net ...

**12**

votes

**1**answer

725 views

### Are Lurie's operads special SMCs?

In Higher Operads, Higher Categories, Leinster nicely characterizes operads among monoidal categories (as PROs). Roughly speaking, a monoidal category comes from an operad if its set of objects is ...

**4**

votes

**1**answer

482 views

### Which E_∞-spaces are homotopy colimits of k-truncated E_∞-spaces?

This question is closely related to my previous question about modules over truncated sphere spectra, in particular, it has the same motivation.
Recall that every space (or ∞-groupoid) can be ...

**8**

votes

**1**answer

291 views

### Is the operadic butterfly symmetric?

The operadic butterfly is a diagram in the category of operads in vector spaces. It extends the short exact sequence relating commutative, associative and Lie operads.
$$\begin{array}{ccccc}
& ...

**5**

votes

**1**answer

363 views

### 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 ...

**9**

votes

**1**answer

260 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 ...

**13**

votes

**1**answer

878 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 ...

**3**

votes

**1**answer

200 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 ...

**4**

votes

**2**answers

306 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 ...

**2**

votes

**1**answer

210 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. ...

**8**

votes

**2**answers

551 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 ...

**3**

votes

**2**answers

746 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 ...

**7**

votes

**3**answers

378 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, ...

**2**

votes

**1**answer

184 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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180 views

### 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:
...

**8**

votes

**1**answer

219 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 ...

**3**

votes

**1**answer

348 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 ...

**3**

votes

**0**answers

214 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 ...

**5**

votes

**1**answer

664 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 ...

**4**

votes

**1**answer

456 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 ...

**10**

votes

**1**answer

689 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, ...

**8**

votes

**0**answers

309 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 ...

**0**

votes

**0**answers

363 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 ...

**5**

votes

**1**answer

522 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 $A_n \otimes A_m \hookrightarrow A_{n+m}$ ...

**2**

votes

**2**answers

405 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 ...

**8**

votes

**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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votes

**7**answers

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 ...

**8**

votes

**1**answer

435 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 ...

**15**

votes

**3**answers

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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votes

**3**answers

976 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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votes

**2**answers

538 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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**0**answers

568 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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**7**answers

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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votes

**1**answer

654 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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votes

**1**answer

602 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 ...