Questions tagged [operads]
The operads tag has no usage guidance.
328
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A "lax Boardman-Vogt tensor product," or what object represents duoidal categories?
Let me preface this by saying I'm not sure what the fundamental examples should be, and perhaps that's part of my question.
The Boardman-Vogt tensor product of $\infty$-operads $\mathcal{O}$ and $\...
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Quasi-isomorphisms of P-algebras
In the paper "Homotopy algebras are homotopy algebras" from Markl a notion of strong homotopy morphism between strong homotopy P-algebras is defined. The author restricts to the case where $...
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Isn't every algebraic operad equipped with a trivial weight?
In Loday–Vallette "Algebraic Operads" they state the following result (Theorem 6.6.2, Operadic twisting morphism fundamental theorem):
Let $P$ be a connected weight graded differential ...
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Reference on Operads
I was reading 'Category for Scientists' by David Spivak and I'd like some references on operads with that kind of approach, using only "basic category theory", nothing too advanced.
I ...
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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 ...
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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 ...
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Different notions of equivalences of $\mathcal{O}$-monoidal $\infty$-categories
I am currently reading Higher Algebra by Jacob Lurie and I have a question regarding equivalences of $\mathcal{O}$-monoidal categories.
Let $\mathcal{O}$ be an $\infty$-operad. Suppose that I have two ...
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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 ...
CommunityBot
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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 ...
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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 ...
CommunityBot
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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 ...
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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 ...
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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 "...
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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 ...
CommunityBot
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$\mathbf{E}_n$-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 ...
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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 ...
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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. ...
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Vertex algebras and factorization algebras
It is often said that vertex algebras are a special case of factorization algebras. In particular, in their book "FAs in QFT" Costello/Gwilliam construct a functor from a certain class of 2d "...
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Is the operadic nerve functor an equivalence of ∞-categories?
It is now known that the $\infty$-category of $\infty$-operads as defined by Lurie is equivalent to the underlying $\infty$-category of the model category of simplicial operads, see http://arxiv.org/...
CommunityBot
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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 $k$-...
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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 ...
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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 ...
CommunityBot
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One colored infinity operads via symmetric sequences?
The question
One standard approach to the theory of 1-colored (symmetric) operads in classical 1-categorical theory is via monoids in symmetric sequences with respect to the composition product. Has ...
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Permutations and framed braids as cyclic operads
The symmetric groups and the framed braid groups form an operad (in sets, not groups). It is straightforward to see this structure using the string diagrams.
It is also known that these operads are ...
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Homotopy theories of operads
I know of three homotopy theories of colored operads.
The (derived) localization category of Berger-Moerdijk's model structure on the category of strict simplicial (or topological) operads, with weak ...
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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$-...
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3
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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 ...
- 1,635
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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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Question about terminology, and reference request related to the braid operad
Let $\Delta_n$ stand for the Garside element of the braid group $B_n$. It turns out that the family of all Garside elements have the following ``operadic'' property:
$$
\Delta_n\left[ \Delta_{k_1},\...
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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 ...
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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 ...
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Are there prominent examples of operads in schemes?
There is an abundance of examples of operads in topological spaces, chain complexes, and simplicial sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even ...
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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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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 ...
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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:
...
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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 $K_{0}(A)=0$ ?
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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_\...
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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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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 ...
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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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Are there any errors in Axiomatic Homotopy Theory for Operads?
The paper I'm referring to can be found here, and is absolutely seminal work in area of homotopy theory for operads. It came out in 2003. I remember someone once told me there was a mistake somewhere ...
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Operad terminology - Operads with and without O(0).
In the Markl, Schneider and Stasheff text, topological operads are an indexed collection of spaces $O(n)$ for $n \in \{1,2,3,\cdots\}$ satisfying some axioms. In May's text, the index set is allowed ...
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Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop Space"?
This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).
The weak Hausdorff rather ...
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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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$\mathbb{E}_M$ 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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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 ...
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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 ...
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Monad induced by actegory
It seems to be folklore that if we have an actegory, i.e. a monoidal functor from a monoidal category $C$ to an endofunctor category $Cat(D,D)$, we can obtain from it a monad on $D$. This appears for ...
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