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Questions tagged [operads]

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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 ...
Alessandro Nanto's user avatar
7 votes
1 answer
248 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
  • 5,122
10 votes
2 answers
171 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
278 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
  • 151
6 votes
1 answer
204 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
104 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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15 votes
2 answers
1k 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
144 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
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3 votes
0 answers
105 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
132 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
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0 votes
3 answers
491 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
3 votes
1 answer
181 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
0 answers
75 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
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5 votes
1 answer
263 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
115 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
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5 votes
0 answers
177 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
2 votes
0 answers
115 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
304 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
89 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
  • 213
8 votes
0 answers
107 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
328 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
  • 496
3 votes
0 answers
159 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
94 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
  • 213
2 votes
1 answer
427 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
  • 9,387
4 votes
0 answers
186 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
  • 9,387
6 votes
0 answers
240 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
  • 9,387
5 votes
0 answers
121 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
  • 1,059
2 votes
0 answers
168 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
  • 55.4k
1 vote
1 answer
264 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
124 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
208 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
192 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
77 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
65 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
  • 408
3 votes
1 answer
153 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
223 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
  • 1,613
6 votes
1 answer
200 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
212 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
  • 55.4k
4 votes
1 answer
210 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
  • 408
3 votes
0 answers
222 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
7 votes
0 answers
191 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 ...
Tim Campion's user avatar
  • 55.4k
2 votes
1 answer
72 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 ...
Javi's user avatar
  • 477
6 votes
1 answer
122 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. (...
FKranhold's user avatar
  • 1,613
3 votes
1 answer
205 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 ...
Aidan's user avatar
  • 408
4 votes
1 answer
286 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}...
ABC's user avatar
  • 530
5 votes
1 answer
238 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 ...
Minkowski's user avatar
  • 499
1 vote
0 answers
67 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\, ...
mtraube's user avatar
  • 183
9 votes
1 answer
225 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 ...
skupers's user avatar
  • 7,698
1 vote
0 answers
122 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^{\...
Pavel's user avatar
  • 446
5 votes
1 answer
128 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 ...
Dasha Poliakova's user avatar

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