Questions tagged [operads]
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327
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Transmutation versus operads
A while ago, I was reading Majid's book Foundations of quantum group theory, and Section 9.4 has a rather fascinating description of a Tannaka-Krein reconstruction result for quantum groups. In ...
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544
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Identifying the little disk operad with parenthesized braids
Let $D_2$ be the topological operad of little disks. This operad can be modelled "combinatorially" in terms of an operad of groupoids called $\newcommand{\PaB}{\mathbf{PaB}}\PaB$, the operad of ...
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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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427
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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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2
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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 ...
8
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238
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Classification of formality morphisms for chains and Drinfel'd associators
In his 1997 preprint q-alg/9709040, M. Kontsevich proved constructively the existence of a $L_\infty$-quasi-isomorphism between the differential graded algebra structure on the deformation complex of ...
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0
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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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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 ...
8
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313
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Did the Goerss-Hopkins manuscript "Multiplicative stable homotopy theory" ever appear?
A citation to "M. J. Hopkins and P. Goerss, Multiplicative stable homotopy theory, unpublished manuscript, 1996" appears in the Hill, Hopkins, Ravenel Annals paper on the Kervaire invariant. It was ...
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232
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Clarification of Tillmann's construction of the higher genus surface operad
Sorry if this question is inappropriate for overflow. I tried asking on stackexchange yesterday but didn't get any responses, so I thought that this site might be better. Anyway, my question is as ...
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Framed higher Hochschild cohomology
Given an $E_n$-algebra $A$, one can define its $E_n$-Hochschild complex $CH_{E_n}(A,A)$ by the formula $$Ch_{E_n}(A,A)=RHom_{Mod_A^{E_n}}(A,A)$$ where $Mod_A^{E_n}$ is the category of $A$-modules over ...
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When are the categories of algebras over props (co)complete?
Suppose P is a (colored) prop in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable?
It seems that one can relatively easy ...
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312
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A model category for E-infty algebras in a non-monoidal model category?
Given a suitable nice symmetric monoidal category $C$, symmetric monoidally enriched, tensored, and cotensored over a symmetric monoidal category $S$, and an operad $\mathcal{O}$ in $S$, we can ...
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Does abstract nonsense of model categories determine the "nonlinear" morphisms of $L_\infty$ algebras?
Recall that a Lie algebra is a module for the operad $\mathrm{Lie}$, which is freely generated by a binary antisymmetric operation $\beta$ modulo an equation that is quadratic in $\beta$. There is a ...
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390
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When can I assure that the representation theory of a PROP is faithful?
Recall that a PROP is a symmetric monoidal category whose object-set is freely $\otimes$-generated by a single object $x\in P$. A representation or algebra for a prop $P$ in a symmetric monoidal ...
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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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Classifying spaces of E_1 - spaces
Hello,
I try to understand aspects of homotopy coherence, in particular "recognition principle" of May.
About the following I did not think a lot, but I decided to ask here anyway, so to save ...
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393
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Tensor product of a DGA and an $A_\infty$ algebra
In general there seems no way to naturally define the tensor product of two $A_\infty$ algebras $A$ and $B$. But, if $(A, m^A_1,m^A_2)$ is only a DGA(differential graded algebra) and $(B, m^B_k, k\ge ...
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3
answers
666
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Koszul duality for modular operads
Has anyone defined what it means for a modular operad to be Koszul, or what the Koszul dual of a modular operad is? In particular, is it meaningful to say that a modular operad is quadratic? Merkulov, ...
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261
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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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Errata for Getzler-Kapranov "Cyclic operads and Cyclic homology"
Do you know if anyone made an errata for the Getzler-Kapranov paper "Cyclic operads and Cyclic homology"? I was trying to read it, and I have found (at least I think I did) quite a few typos not all ...
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Does every equivalence of operads in the category of small categories have a weak inverse?
Call a map of operads $\mathcal{O}\rightarrow \mathcal{U}$ in the category of small categories an equivalence, if each functor $\mathcal{O}(n)\rightarrow \mathcal{U}(n)$ is an equivalence of ...
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Is the definition of Gerstenhaber bracket related to operads?
I was reading these notes by Keller. On the page 19 he defines the Gerstenhaber bracket on the Hochschild cochain complex. But first of all he defines the operation $\bullet$ on the cochains by
$$
\...
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903
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Is there a cheap proof that (homotopy) endomorphisms are functorial?
This is, in some sense, the homotopy version of this question.)
If $C$ is a category with $iC$ the subcategory of isomorphisms, there is a functor $X \mapsto End(X)$ from $iC$ to the category of ...
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Monomorphisms in operad algebras
Setup: Let $\mathcal{O}$ be an operad in the category of sets, and let $\mathcal{O}\text{-Alg}$ denote the category of algebras on it (i.e., operad functors $\mathcal{O}\to\mathbf{Set}$. This category ...
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$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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248
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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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426
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When does the enveloping algebra functor lift to the category of bialgebras?
Let $\mathrm{Ass} $ denote the operad, whose algebras are associative unital algebras, considered as a dg-operad.
Denote $\mathrm{Ch} $ the category of chain complexes over a commutative ring $\...
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Construction for algebras over little cubes operad
Recently I came across the following construction:
Fix a dimension $k$. Let $C$ denote the space whose points are disjoint rectilinear embeddings $c\colon I^k\to \mathbb R^k$ of the (closed) $k$-...
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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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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 ...
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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 ...
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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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What is the endomorphism cooperad?
In Loday and Vallette's book on algebraic operads, they use the "Endomorphism cooperad $End^c_{s\mathbb{K}}$", where $s\mathbb{K}$ is the base field, shifted into (homological) degree one. This is an ...
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Monad arising from operad
It's known that from every operad arises a cartesian monad whose algebras are the algebras for the operad. Leinster proved that there are different operads from which arise the same monad, in this way ...
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Definition of E-infinity operad
What is the definition of $E_\infty$-operad in the category of chain complexes over $\mathbb{Z}/p\mathbb{Z}$? J. Smith in http://arxiv.org/abs/math/0004003 define it for complexes over $\mathbb{Z}$ (...
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Obstructions for $E_n$-algebras
In Alan Robinson's paper, Classical Obstructions and S-algebras, he provides conditions for a ring spectrum to have an $A_n$ and $\mathbb{E}_\infty$-structure.
Have the obstructions for an object ...
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640
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Repairing the Lie operad in characterstic 2?
Recall: We present an operad (with $S_n$-action) in $R-Mod$ for any commutative ring $R$ not of characteristic 2 generated by a single element in degree $2$ satisfying the following identities:
$\...
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3
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321
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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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Boardman-Vogt tensor product
Let $\mathbf{sSet}$ be the model category of simplicial sets and $\mathbf{Op}$ the model category of symmetric operads. Equipped with Boardman-Vogt tensor product $ \otimes_{BV}$, the category $\...
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Free operad over a monoid object
Let $\mathcal{O}$ be an operad in the monoidal category $M$. Then $\mathcal{O}(1)$ together with the morphisms
$$\mathcal{O}(1)\otimes \mathcal{O}(1)\to \mathcal{O}(1)$$
and the unit $\eta:1\to \...
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Two monoidal structures and copowering
Let $(\mathbf{M},\otimes,1)$ be a closed monoidal category and $(\mathbf{C},\oplus,0)$ an $\mathbf{M}$-enriched monoidal category. Furthermore, assume that we have a copowering $\odot:\mathbf{M}\times\...
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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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Operad structure on Kontsevich's admissible graphs
In his celebrated 97' preprint q-alg/9709040, M. Kontsevich constructs a $L_\infty$-quasi-isomorphism $\mathcal U:\mathcal D_{\rm poly}\to\mathcal T_{\rm poly}$ between the differential graded algebra ...
6
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260
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An operad-like structure, is there a name for it?
Here is an example which I'd like to have a name for.
Let $P$ be a compact smooth manifold of dimension $p$, possibly with non-empty boundary.
Define $E(k,P)$ to be the space of smooth (codimension ...
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Knot Factorization Homology inputs
Following the paper by Ayala, Francis, and Tanaka: https://arxiv.org/pdf/1409.0848.pdf
If we are talking about knots we are talking about framed 3-manifolds with a framed 1-dimensional sub-manifold ...
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574
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Pontryagin product from an operad
For a topological group G, we have a Pontryagin product in homology by multiplying representative cycles. This gives the homology the structure of an associative graded algebra. Am I correct in ...
6
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659
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Do you recognise this variant of the cubes operad?
In the subject of operads one comes across many quirky variants of more or less the same operad. The cubes operad has many incarnations with various interesting properties. In a recent paper I came ...
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651
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Bracket systems (generalization of Poisson brackets)
Related to Why symplectic geometry gives Poisson geometry by coming at it from the other side.
This isn't as fully formalized as it probably should be, but I think enough of the idea is there to ask ...
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"Left Brace Module"
Let $A$ be an algebra over the brace tree operad and $M$ a module over some base ring.
Is there a good notion of a "left brace module" over a brace algebra?
I do not think the definition of a module ...