Questions tagged [operads]

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Characterize algebras of the "topological simplices" operad

The operad of topological simplices, which I'll denote $\Delta$, has as $n$-ary operations the set $$ \Delta_n:=\left\{P\colon\{1,\ldots,n\}\to[0,1]\;\middle|\;1=\sum_{i=1}^n P(i)\right\} $$ of ...
David Spivak's user avatar
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5 votes
0 answers
127 views

Classifying the algebraic structure on endomorphism sets

This is motivated by defining modules in a general sense, which is an appropriate homomorphism from $R$ to $\textrm{End}(X)$. If $X$ comes from different categories, the endomorphism set will have ...
Trebor's user avatar
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3 votes
2 answers
215 views

Is the free algebra functor over an $\infty$-operad symmetric monoidal?

Suppose $F: \mathcal{O}^\otimes \to \mathcal{P}^\otimes$ is a map of $\infty$-operads, and $\mathcal{C}$ is a symmetric monoidal $\infty$-category that admits small colimits, such that the tensor ...
Markus Zetto's user avatar
3 votes
1 answer
177 views

On infinity-morphisms between algebras over algebraic operads

I posted this question in the "Mathematics" stack exchange, but it hasn't got much attention... I hope it will get more here. Let $P$ be a Koszul operad. In the book of Loday-Vallette "...
groupoid's user avatar
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4 votes
2 answers
268 views

On a generalized homotopy transfer theorem

In the book of Loday and Vallette "Algebraic Operads" a necessary condition for the Homotopy Transfer Theorem is that the starting operad is Koszul. I am interested in a generalization of ...
groupoid's user avatar
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3 votes
0 answers
124 views

$\mathbf{E}_n$-algebras in nerves of 2-categories

In Example 5.1.2.4 of Higher Algebra, Lurie explains how there is a bijection between equivalence classes braided monoidal structures on one-category $\mathcal{C}$ and $\mathbf{E}_2$-algebra ...
Pulcinella's user avatar
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4 votes
0 answers
114 views

Tensor product - Vertex / Chiral algebras

Two questions regarding tensor product of modules over vertex / chiral algebras: First question: For (rational?) vertex operator algebras there is a notion of fusion product of modules inducing a ...
E. KOW's user avatar
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7 votes
0 answers
114 views

On the invariance of the Kaledin class

In Formality of DG algebras (after Kaledin), Lunts introduces an $A_\infty$-Hochschild cohomology class, called the Kaledin class, controlling formality of an $A_\infty$-algebra up to a certain order. ...
domenico fiorenza's user avatar
6 votes
2 answers
263 views

Poisson and homotopy Poisson operads

$\newcommand{\Pois}{\mathit{Pois}}\newcommand{\Comm}{\mathit{Comm}}\newcommand{\Lie}{\mathit{Lie}}$For my thesis, I'm trying to understand the Poisson operad (I'll call it $\Pois$) and its homotopy ...
Alessandro Nanto's user avatar
2 votes
1 answer
142 views

Left Proper model structure on the category of non-symmetric operads in chain complexes

It is shown in Moriya (Multiplicative formality of operads and Sinha’s spectral sequence for long knots, 2.1) that there exists a left proper model category structure on non-symmetric operads over $k$-...
Andrea Marino's user avatar
4 votes
1 answer
171 views

Delooping a weak $E_1$ map by bar construction

Consider based maps $f : X \to Z$ and $g : Y \to Z$, which induces the following map at the based loop space level : $$\theta := \mu_Z \circ \big(\Omega f \times \Omega g) : \Omega(X\times Y) = \Omega ...
ChesterX's user avatar
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7 votes
1 answer
260 views

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,...
Pulcinella's user avatar
  • 5,487
10 votes
2 answers
204 views

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 ...
Vladimir Dotsenko's user avatar
5 votes
0 answers
307 views

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 ...
ChesterX's user avatar
  • 193
7 votes
1 answer
243 views

$(\infty,n)$-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 $\infty$-...
Z. M's user avatar
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4 votes
0 answers
110 views

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: ...
Tim Campion's user avatar
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16 votes
2 answers
2k views

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 ...
Arshak Aivazian's user avatar
5 votes
0 answers
148 views

Grothendieck group of coconnective dg-algebra

Is there an example of an $E_{2}$-coconnective differential graded algebra $A$ (with unit) such that $K_{0}(A)=0$ ?
LGO's user avatar
  • 169
3 votes
0 answers
115 views

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_\...
FKranhold's user avatar
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3 votes
0 answers
133 views

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 ...
W. Rether's user avatar
  • 445
1 vote
3 answers
587 views

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 ...
Arshak Aivazian's user avatar
4 votes
1 answer
199 views

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 ...
Eugene Rabinovich's user avatar
3 votes
1 answer
148 views

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 ...
David White's user avatar
  • 27.7k
5 votes
1 answer
299 views

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 ...
David White's user avatar
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1 vote
0 answers
120 views

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 ...
ani's user avatar
  • 101
5 votes
0 answers
187 views

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 ...
user124543's user avatar
3 votes
0 answers
127 views

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 ...
Markus Zetto's user avatar
6 votes
1 answer
347 views

$\mathbb{E}_M$ 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 $\infty$...
Markus Zetto's user avatar
1 vote
0 answers
106 views

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 ...
Lilolance's user avatar
  • 223
8 votes
0 answers
110 views

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 ...
Vladimir Dotsenko's user avatar
7 votes
0 answers
383 views

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 ...
J Cameron's user avatar
  • 551
3 votes
0 answers
176 views

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 $\mathcal{O}^\otimes$-...
Jan Steinebrunner's user avatar
1 vote
1 answer
96 views

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 ...
Lilolance's user avatar
  • 223
2 votes
1 answer
538 views

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 ...
Emily's user avatar
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5 votes
0 answers
206 views

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}_{\...
Emily's user avatar
  • 9,877
6 votes
0 answers
247 views

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 ...
Emily's user avatar
  • 9,877
5 votes
0 answers
128 views

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 $...
AT0's user avatar
  • 1,437
2 votes
0 answers
174 views

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 ...
Tim Campion's user avatar
  • 60.2k
1 vote
1 answer
294 views

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 ...
Sergei Burkin's user avatar
2 votes
0 answers
125 views

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:...
Ainfty's user avatar
  • 21
3 votes
1 answer
254 views

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 ...
Javi's user avatar
  • 477
5 votes
0 answers
238 views

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 $\...
David Kern's user avatar
2 votes
0 answers
81 views

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 ...
Javi's user avatar
  • 477
2 votes
1 answer
67 views

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 $$P(n)\otimes \mathcal{O}(k_1)\otimes\cdots \otimes \mathcal{O}(k_n)\to P(k_1+\cdots + k_n)$$ ...
Aidan's user avatar
  • 480
3 votes
1 answer
157 views

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 $...
Tommaso Rossi's user avatar
5 votes
1 answer
242 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 ...
FKranhold's user avatar
  • 1,623
6 votes
1 answer
221 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 ...
Dmitry Vaintrob's user avatar
9 votes
0 answers
220 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 ...
Tim Campion's user avatar
  • 60.2k
4 votes
1 answer
237 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 ...
Aidan's user avatar
  • 480
3 votes
0 answers
246 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 ...
Tommaso Rossi's user avatar

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