# Questions tagged [operads]

The operads tag has no usage guidance.

314
questions

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### Tensor product of $L_\infty$-algebra with DG commutative algebra

For my thesis, I'm trying to put the structure of an $L_\infty$-algebra on a tensor product $A\otimes L$ of a $L_\infty$-algebra $L$ with a differential, graded commutative algebra $A$. I know that ...

7
votes

1
answer

248
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### Non(skew)commutative Lie algebras?

The Lie operad $\text{Lie}$ is generated by a binary operator $[\ ,\ ]$, modulo a degree two relation (skew commutativity $[x,y]=-[y,x]$) and a degree three relation (Jacobi $[x,[y,z]]+[y,[z,x]]+[z,[x,...

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2
answers

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### Degree 8 multilinear operations on Jordan algebras

I am interested in the dimension, or, even better, in the $S_8$-module structure of the space of degree 8 multilinear operations on Jordan algebras.
Recall that a Jordan algebra is a commutative but ...

5
votes

0
answers

278
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### What is an $\infty\text{-}E_{\infty}$ morphism?

My question is essentially what the title says, but here is some background that I have gathered from skimming through the literature. Please feel free to correct me if my understanding is wrong at ...

6
votes

1
answer

204
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### $(\infty,n)$-operads?

I wonder whether there is (or should be) a theory of colored $(\infty,n)$-operads or multicategories?
We know that multicategories are generalizations of categories, and nonsymmetric colored $\infty$-...

4
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0
answers

104
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### For which operads $O$ does $\operatorname{coAlg}_O(C) = C$ whenever $C$ is cartesian monoidal?

Let $O$ be an operad, and let $D$ be a symmetric monoidal category. Then there is a forgetful functor $\operatorname{Alg}_O(D) \to D$. This functor is an equivalence in either of the following cases:
...

15
votes

2
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### Why are operads sometimes better than algebraic theories?

Question 1: Are there any contexts in which replacing the category of (non-symmetric or symmetric) operads (in some monoidal category or symmetric monoidal category, respectively) with the category of ...

5
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### Grothendieck group of coconnective dg-algebra

Is there an example of an $E_{2}$-coconnective differential graded algebra $A$ (with unit) such that $K_{0}(A)=0$ ?

3
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105
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### Bar constructions of $A_\infty$-algebras and rectifications

Let $\mathscr{C}_1$ be the little 1-cubes operad. If $X$ is an algebra over $\mathscr{C}_1$, I can think of (at least) two ways how to deloop it:
I can consider its two-sided bar construction $B_\...

3
votes

0
answers

132
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### Transporting $\mathbb E_n$-monoidal structures between categories

Suppose given an $\mathbb E_n$-monoidal presentable $\infty$-category $\mathcal C$ (wrt. the Lurie tensor product $\otimes$), and $\mathcal D$ a presentable $\infty$-category. Suppose given a pair of ...

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3
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491
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### How now to study operads in homotopy theory?

There is a great introduction by May, "The Geometry of Iterated Loop Spaces". I really enjoy reading it, but it was written 50 years ago and contains outdated technical details related to ...

3
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1
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181
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### Is the normalized simplicial bar construction of an operad a cooperad?

Suppose that $\mathcal{P}$ is a connected, unital operad in $\mathbb{k}$-vector spaces (or complexes), i.e. $\mathcal{P}(1)=\mathbb{k}$ and the unit map for $\mathcal{P}$ is the identity. One may form ...

3
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0
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75
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### Which positive flat stable model structures on (flavors of) spectra have the property that cofibrant operad-algebras forget to cofibrant spectra?

Let $M$ be a monoidal model category and $O$ an operad valued in $M$, and the category of $O$-algebras inherits a model structure from $M$ where a map $f$ is a weak equivalence (resp. fibration) if ...

5
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1
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263
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### Is there a model structure for S-modules such that cofibrant operad-algebras forget to cofibrant S-modules?

In 1997, Elmendorf, Kriz, Mandell, and May wrote a book Rings, Modules, and Algebras in Stable Homotopy Theory in which they introduced the category of $S$-modules as a model for the stable homotopy ...

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0
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115
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### The S-module Ass is same as the composite of Com and Lie

It has been cited in several places (eg. https://arxiv.org/pdf/1912.05519.pdf) that the S-module Ass is isomorphic to the composite of the S-modules Com and Lie. Is there a reference which gives the ...

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### Another model for $\infty$-operads?

There are several well-developed notions of $\infty$-operad in the literature, which are nowadays known to be equivalent (see e.g. the introduction of Chu-Haugseng-Heuts. However, another model is ...

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### Riemann-Hilbert-type correspondence for locally constant factorization algebras

This is related to a previous post, but a bit softer and should probably stand on its own.
In Appendix A of "Higher Algebra", Lurie shows that for a reasonably good topological space, there ...

6
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1
answer

304
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### $\mathbb{E}_M$ as colimit of little cubes operads

In Lurie's "Higher Algebra", Remark 5.4.5.2 towards the end, there is the following statement: "It follows that $\mathbb{E}_M$ can be identified with the colimit of a diagram of $\infty$...

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0
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89
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### Koszul complex of the cobar construction is acyclic

This is a follow-up question on my question on math stackexchange (https://math.stackexchange.com/questions/4399553/proof-that-the-coaugmented-cobar-construction-of-a-cooperad-is-acyclic)
I think I ...

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### Identity for the associator involving a third root of unity

This is a reference request. I came across the class of nonassociative algebras satisfying the following identity:
$$
(a,b,c)+\omega(b,c,a)+\omega^2(c,a,b)=0.
$$
Here:
by an "algebra" I mean a ...

7
votes

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328
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### Transfer of E-infinity algebra structures

Skip to the bottom for my questions, first some discussion:
It is a celebrated theorem of Kadeišvili that $A_{\infty}$-algebra structures can be transferred along homotopy equivalences so that the ...

3
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0
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159
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### Augmented algebras over $\infty$-operads via the envelope

Let $\mathcal{O}^\otimes$ be an $\infty$-operad and $\mathcal{C}^\otimes$ a symmetric monoidal $\infty$-category, both in the sense of Lurie's Higher Algebra.
By augmented $\mathcal{O}^\otimes$-...

1
vote

1
answer

94
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### Differential of the Twisted complex for algebraic operads

I have a question about the proof of lemma 6.4.12 in the book Algebraic Operads (Loday-Vallette) which I do not seem to be able to fully complete on my own. Hopefully, somebody here can point out what ...

2
votes

1
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427
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### Is there an operad homotopifying the Koszul rule?

In homotopy theory one has the idea of a homotopy-commutative multiplication, in which one replaces the relation $$ab=ba$$ in a commutative monoid/group/ring/etc. for an unspecified homotopy. One ...

4
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186
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### Is there a 1-categorical treatment of operadic left Kan extensions in the literature?

Lurie develops in Section 3.1.2 of Higher Algebra a notion of operadic left Kan extension used to compute free algebras, giving a left adjoint $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})\to\mathrm{Alg}_{\...

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240
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### Generalising supercommutativity as a grading by the $1$-truncated sphere spectrum

A discussion that has been going recently is that supersymmetry corresponds to grading over the sphere spectrum, coming from an insight due to Kapranov.
To formalise such a statement, one needs a ...

5
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0
answers

121
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### Definition of $E_{n}$-operad in dgCat

In "Derived Algebraic Geometry and Deformation Quantization" Toën defines in 5.1.2 an $E_{n}$-monoidal A-linear dg-category as an $E_{n}$-monoid in the symmetric monoidal $\infty$-category $...

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168
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### What is an invertible operad?

Let $\mathcal V$ be a nice symmetric monoidal ($\infty$-)category, and consider the ($\infty$-)category $Op(\mathcal V)$ of $\mathcal V$-enriched (symmetric) operads, symmetric monoidal under the ...

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vote

1
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264
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### Lawvere theory of Lawvere theories

There is a coloured operad $sOp$ such that $sOp$-algebras are single-coloured operads. This operad has a simple description in terms of generators and relations, say, as an operad $F(X)/R$. There is a ...

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124
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### Can there be a non-trivial $A_\infty$-algebra which is Z/2-graded?

I am not used to $A_\infty$-algebras, so I am sorry if this is a stupid question. It seems that an $A_\infty$-algebra $A$ is typically a $\mathbb Z$-graded vector space $A$ along with morphisms
$$ m_k:...

3
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1
answer

208
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### Functors that preserve monoids

In the comments section of this question there was a question that I don't know if it has been asked on the site. It is well-known and easily proved that lax monoidal functors preserve monoids. So the ...

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### Understanding the disintegration of unital $\infty$-operads

In section 2.3.4 of Higher
Algebra,
Lurie shows that any unital $\infty$-operad (whose underlying
$\infty$-category is an $\infty$-groupoid) can be obtained by gluing
together a family of reduced $\...

2
votes

0
answers

77
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### Are there examples of brace algebras that are not operads?

The most typical example of a brace algebra is the brace algebra structure on the Hochschild complex of an associative algebra. This is a particular case of the following construction applied to the ...

2
votes

1
answer

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### Right action by an operad on a non symmetric collection

Suppose we have a non symmetric operad $\mathcal{O}$, a collection of sets
$\{P(n)\}_{n\geq 0}$ and maps
$$P(n)\otimes \mathcal{O}(k_1)\otimes\cdots \otimes \mathcal{O}(k_n)\to P(k_1+\cdots + k_n)$$
...

3
votes

1
answer

153
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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 $...

5
votes

1
answer

223
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### 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
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1
answer

200
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### 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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### 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 ...

4
votes

1
answer

210
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### 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 ...

3
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0
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222
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### 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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191
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### 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
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### 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 ...

6
votes

1
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122
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### 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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1
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### 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 ...

4
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1
answer

286
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### 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}...

5
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1
answer

238
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### 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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0
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### 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\, ...

9
votes

1
answer

225
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### 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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0
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122
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### 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^{\...

5
votes

1
answer

128
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### 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 ...