# Questions tagged [operads]

The operads tag has no usage guidance.

282
questions

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### Is the category of topological operads left proper?

I just learned that there is a model structure on the category $Op_{Top}$ of topological operads, due to Berger-Moerdjik [1], obtained by right transfer of the Quillen model structure on $Top$.
Since $...

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votes

**1**answer

136 views

### Enriched coends which preserve equivalences

Although this question might be formulated in higher generality, let me try to be concrete:
Let $(\mathbf{Top},\times,*)$ be the monoidal category of compactly generated weak Hausdorff spaces; and let ...

**6**

votes

**1**answer

143 views

### Monochromatic infinity operads as algebras over the “operad operad”

In the "ordinary" operad category, it is known that there is a colored operad $Op$ with set of colors $\mathbb{N}$ corresponding to "degrees" of vertices and with operations ...

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182 views

### What is the operad for homotopy associative, homotopy commutative objects?

There is an operad whose algebras are objects with a homotopy unital multiplication -- the $A_2$ operad.
There is an operad whose algebras are objects with a homotopy unital, homotopy associative ...

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votes

**1**answer

111 views

### Operadic cohomology in terms of infinitesimal composition

Given a non symmetric operad $\mathcal{O}$, is there an explicit description of its (André-Quillen or other) cohomology in low degrees in terms of infinitesimal composition?
I ask because I am ...

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170 views

### Ginzburg Kapranov paper on Koszul duality

I am studying the article "Koszul duality for operads" by Ginzburg and Kapranov, https://arxiv.org/pdf/0709.1228.pdf.
The problem is that this version of the paper contains empty spaces ...

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164 views

### Higher categorical / operadic approach to homotopy associative, homotopy commutative, $H_\infty$ ring spectra?

Let $\mathcal C$ be a symmetric monoidal $\infty$-category, and let $O$ be an operad (for example, $O$ could be an $A_m$ or $E_n$ operad or a tensor product thereof, and $\mathcal C$ could be spaces ...

**2**

votes

**1**answer

53 views

### Bigraded operadic suspension

I know from this paper by Ward that one can obtain the (signs of) the Gerstenhaber bracket using operadic suspension on any operad $\mathcal{O}$. More precisely, the insertion $\tilde{\circ}$ of the ...

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votes

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112 views

### Is there a filtered splitting of product labelling spaces?

For a well-based space $X$ denote by $C(\mathbb{R};X)$ the unordered configuration space of points on the real line with labels in $X$, and a point can vanish if its label reaches the basepoint. (...

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votes

**1**answer

126 views

### Generalised operad structures

We can naively consider an operad as a collection $\{P(n)\}_{n\geq 0}$ of vector spaces $P(n)$ consisting of "functions" with $n$ inputs and one output, equipped with a number of ...

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votes

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229 views

### map of endomorphism operad

Let $\mathbf{Top}$ be the category of (nice) topological spaces.
For any space $Z$, define $\mathbf{End}_{\text{operad}}(Z)$ as the endomorphism operad.
Is there always a map of operads
$$\mathbf{End}...

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votes

**1**answer

173 views

### Boardman-Vogt resolution of the little 2-cubes operad

If $\mathbf{P}$ is a (coloured) operad, one can build a topological operad $W(\mathbf{P})$ called the $W$-construction or the Boardman-Vogt resolution of $\mathbf{P}$. Let me denote the resulting map ...

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vote

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61 views

### Functor from modular operads to (wheeled) properads

In Algebra+Homotopy=Operad in the conclusion it says that there is a commutative square of functors
$\require{AMScd}$
\begin{CD}
modular\, operads @>>> properads\\
@VVV @VVV\\
cyclic\, ...

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votes

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147 views

### Maps between unitary little disks operads and non-unitary little disks operads

Derived mapping spaces between little $d$-disks operads $E_d$ play an important role in embedding calculus. For example, Dwyer-Hess expresses the homotopy of framed long knots as loop spaces such ...

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67 views

### Degree shift of multilinear maps

Let $V$ be a graded vector space over $\mathbb{k}$ and $V[1]$ its odd degree shift.
Given $k$, $l\in \mathbb{N}_0$, is there a natural way to define the following map,
$$
\psi: \hom_{\mathbb{k}}(V^{\...

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votes

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58 views

### Planar dendroidal sets?

The meta picture is: (non-planar) dendroidal sets are to symmetric colored operads as simplicial sets are to categories.
This suggests that one should have the notion of planar dendroidal sets (with a ...

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197 views

### References for Homotopy transfer problem

I am trying to read Algebra+homotopy=operad by Bruno Vallette.
Consider the following set up :
chain complexes $(A,d_A),(H,d_H)$,
a degree $1$ morphism of chain complexes $h:(A,d_A)\rightarrow (A,d_A)...

**3**

votes

**1**answer

108 views

### Boardman-Vogt construction for PROP(erads)

Let $\left\lbrace \mathsf{O}(n)\right\rbrace_{n\in \mathbb{N}} $ be an operad in a symmetric monoidal category $(\mathsf{C},\otimes, \mathbf{1})$ which in addition has the structure of a model ...

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votes

**2**answers

298 views

### How to characterize $E_n$ morphisms from $\mathrm{Mod}(A)$ to $\mathrm{Mod}(B)$?

Suppose $A$ and $B$ are $E_{\infty}$ rings, then $\mathrm{Mod}(A)$ and $\mathrm{Mod}(B)$ are $E_{\infty}$ monoidal categories (left modules over those rings). We can ask about $E_n$ colimit-preserving ...

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114 views

### How does a map from permutahedra to associahedra factor through multiplihedra?

Let $P_i$ denote permutahedra, $K_i$ associahedra and $J_i$ multiplihedra. In their famous paper on operadic diagonals, Saneblidze and Umble use a projection $p_i: P_i \to K_{i+1}$ which factors as $...

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113 views

### Recovering operad units from homotopy units

It is my understanding that the $\infty$-category of non-unital connected topological monoids is equivalent to the $\infty$-category of connected topological groups.
It follows that the functor from ...

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votes

**1**answer

240 views

### Different ways to “deloop” a (topological) $A_\infty$-algebra

Let $\varphi:A\to \mathrm{Ass}$ be an $A_\infty$-operad in topological spaces, and let $X$ be an $A$-algebra. I see three possibilities to construct a delooping out of $X$:
Rectify $X$ by taking the ...

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209 views

### $E_{\infty}$-algebras à la Lurie

Let $D(\mathbb{F}_p)$ and $\mathcal{D}(\mathbb{F}_p)$ be the derived category and derived infinity-category of cochain complexes of $\mathbb{F}_p$-vector spaces.
If $A$ is a sheaf of cdgas over $\...

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131 views

### A question on recognition of equivariant loop spaces

I have a question about equivariant loop space that has been bothering me, and that I have not been able to find an answer to in the obvious places.
We know from the work of Segal that to give a loop ...

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votes

**2**answers

280 views

### Conceptual explanation for the sign in front of some binary operations

In several situations, I've seen that given a binary operation on a graded module $m:A\otimes A\to A$, a new operation $M(x,y)=(-1)^{|x|}m(x,y)$ is defined so that it satisfies some properties.
One ...

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**1**answer

141 views

### Detailed proof of $\mathfrak{s}^{-1}\mathrm{End}_V\cong \mathrm{End}_{\Sigma V}$

I asked this question on MSE but I want to ask it again here with some more context sine it received no answers. In Chapter 3 (Algebra) of the book Operads in Algebra, Topology and Physics by Markl, ...

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289 views

### Beginner's guide to $A_{\infty}$-algebras

I have some general questions about $A_{\infty}$-algebras. Altough I
understand bare definition from nLab I have no association how to think
intuitively about them. Which picture one should
have in ...

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92 views

### Is the free algebra over an operad an algebra over that operad?

I'm asking here this question I asked on MSE that got no answers.
Let $V$ be a dg-module and $P$ an operad. The free $P$-algebra on $V$ is defined by $P(V)=\bigoplus_{r=0}^\infty (P(r)\otimes V^{\...

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**1**answer

192 views

### Infinity-homotopies

Koszul duality for operads allows for straightforward generalizations of $A$-infinity algebras and $A$-infinity morphisms for the so called Koszul operads $\mathcal{O}$, among which we find the ...

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votes

**1**answer

168 views

### Is operadic desuspension inverse to operadic suspension?

Given a graded vector space $V$ over a field $k$, consider it's suspension $\Sigma V$ such that $(\Sigma V)^i=V^{i-1}$. For an operad of graded vector spaces over a field $\mathcal{O}$, the operadic ...

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votes

**1**answer

108 views

### Substitution structure on pointed sets

$\def\Fin{\text{Fin}_*}
\def\Set{\text{Set}_*}
\def\dd{\mathop{\diamond_\land}}$
The present question is intimately related to another question.
Let $\Fin$ be the category of pointed sets.
The ...

**6**

votes

**1**answer

247 views

### Finitary monads on $\operatorname{Set}$ are substitution monoids. Finitary monads on $\operatorname{Set}_*$ are…?

$\DeclareMathOperator\Fin{Fin}\DeclareMathOperator\Lan{Lan}\DeclareMathOperator\Set{Set}$
The present question is intimately related to another question.
It is well known that the category of ...

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88 views

### Why is an operad of associative algebras Koszul?

Let $Assoc$ be an operad of associative algebras. What does it mean for $A$ to be a Koszul operad? Is it related to standard Koszul duality for algebras? As far as I understand, if $Assoc_{\infty}$ ...

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87 views

### Is there a n-category structure on algebras for $e_n$-like operads?

I'm fishing in troubled waters here and therefore the question is vague and meant to be as general as possible. In particular "$e_n$-like operad" can be an algebraic or topological $e_n$ operad, as ...

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votes

**1**answer

212 views

### Semi-cocartesian operads

Context: In this interesting blog post, Mike Shulman indicates an approach for defining generalized types of operads. If I interpret the details correctly, (edit: which I apparently did not,) the idea ...

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138 views

### $G_{\infty}$ (also known as $E_2$)-operad in terms of trees

It's well known that the $A_{\infty}$ and $L_{\infty}$ operads, being resolutions of the associative and Lie operads, admit descriptions as free operads of certain trees.
The description I am ...

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389 views

### Homotopy Gerstenhaber algebras: description via operads vs derivations

There are at least a couple of definitions in the literature for an $E_2$-algebra, also known as a homotopy Gerstenhaber algebra, also known as $G_{\infty}$-algebra.
Suppose $V$ is a graded vector ...

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votes

**1**answer

169 views

### An exercise from Loday and Vallette about Koszul morphism

I tried to solve the following exercise from Loday and Vallette's Algebraic Operad. The first three parts are straightforward, however I have no idea how to solve the last part. I can't find any ...

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323 views

### Is there something “Koszul dual” to formal groups?

The Lie operad is Koszul dual to the commutative operad. In some sense, the data of a formal group is an "elaboration" of the data of a Lie algebra. Is there some corresponding "elaboration" of the ...

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156 views

### The notion of $\infty$-Cooperads for which Bar-Cobar duality is an equivalence

In the paper Bar-Cobar Duality by Michael Ching, he proves that the category of operads in spectra is equivalent via the Bar-Cobar adjunction to some model category of co-operads defined in the paper. ...

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179 views

### Lie algebras in braided monoidal categories

Let $\mathcal{C}$ be a braided (not necessarily symmetric) monoidal category. Then we can define what monoids and commutative monoids in $\mathcal{C}$ are. What is the correct definition of a Lie ...

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**1**answer

205 views

### Can an algebra over an operad be described by generators and relations?

Experts in operads, please be gentle to a beginner.
Suppose I have a collection of generators $\{x_i\}_{i\in I}$ and some relations like
$$ x_1x_2 -x_5x_6x_7=0\qquad x_4x_8+2x_{11}x_{12}=3x_5x_6$$
...

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197 views

### For which $k$-types are $E_{n,m}$-algebras automatically $E_{n+1}$ algebras?

Recall that an $E_{n,m}$ algebra is an $A_m$ algebra in $E_n$ algebras. Here I index my $A_m$ algebras so that an $A_1$ algebra is pointed, an $A_2$ algebra has a unital multiplication, $A_3$ is ...

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votes

**2**answers

264 views

### Algorithm for identifying reducible braids

If $\vec{n} = (n_1,...,n_k)$ is a vector of integers, there seems to be a well-defined homomorphism
$B_k \ltimes \left(B_{n_1} \times \cdots \times B_{n_k}\right) \to B_N$
where $N = \sum n_i$ and $...

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257 views

### Is it a coincidence that $Gal(\mathbb C / \mathbb R) \cong C_2 \cong Aut(E_1)$? (Or: why are $\mathbb C$-algebras with involution so useful?)

The automorphism group $Aut(E_1)$ of the $E_1$ operad is the cyclic group of order 2, $C_2$, and thus $C_2$ acts on any category of algebras (by reversing the multiplication). The seeming coincidence ...

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votes

**2**answers

342 views

### Homology of a loop-suspension space and action of $\mathcal{D}_1$-operad

If $X$ is a based connected topological space, it is well-known what the homology of $\Omega\Sigma X$ is: according to the Bott-Samelson theorem, it is a tensor algebra over reduced homology of $X$. (...

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votes

**1**answer

159 views

### Group completion of $E_k$-algebras

Let $X$ be an $E_k$-algebra. We can form the delooping $BX$, which is a $E_{k-1}$-algebra. The space $\Omega B X$ is again an $E_k$-algebra, which is grouplike (i. e. $\pi_0(\Omega B X)=\pi_1(B X)$ is ...

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**1**answer

174 views

### Are these two natural $A_\infty$-structures on the realization of a cosimplicial commutative algebra isomorphic?

Given a cosimplicial commutative algebra $A^\bullet$ over a field of characteristic zero, there are two ways of producing an $A_\infty$-structure on its realization $|A^\bullet| := \int^\Delta C^*(\...

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147 views

### Homotopy transfer of cyclic L-infinity algebras

Suppose $W$ is a cyclic $L_\infty$ algebra, i.e. $W$ has a non-degenerate, symmetric, invariant pairing $\langle\cdot,\cdot\rangle_W$. Let $V$ be a cochain complex, and suppose given the data of a ...

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282 views

### Functoriality of infinite loop space machines?

If $C$ is a symmetric monoidal category, then $BC$ is canonically an algebra over a certain $E_\infty$ operad, but if $F: C \to D$ is a symmetric monoidal functor then (as far as I can see) $BF: BC \...