Questions tagged [operads]
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328
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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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7
answers
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Why are operads useful?
The question is not about where operads are used, I know that. It is about what makes them useful. For example, van Kampen diagrams are useful in combinatorial group theory because these are planar ...
user6976
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5
answers
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What are natural questions to ask about an operad?
I just discovered that something I've been working with has the structure of an operad. So I'm wondering what natural basic questions does one ask about operads? For example, if I knew I had the ...
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votes
3
answers
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Algebras over the little disks operad
Hello,
The so-called "recognition principle" of Boardman-Vogt and May leaves me unsatisfied.
My problem is the following:
The "recognition principle" says that every "group-like" algebra over the ...
- 2,501
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votes
7
answers
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Shuffle Hopf algebra: how to prove its properties in a slick way?
Let $k$ be a commutative ring with $1$, and let $V$ be a $k$-module. Let $TV$ be the $k$-module $\bigoplus\limits_{n\in\mathbb N}V^{\otimes n}$, where all tensor products are over $k$.
We define a $k$...
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1
answer
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What's up with Wick's theorem?
Sorry about the dumb title.
I'd like to understand Wick's theorem. More specifically, I have seen it pop up in several different contexts and I am really puzzled by the different statements of it ...
- 39.3k
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3
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Is it possible to construct an action of an $E_\infty$ operad on $BU$ that respects filtration by $BU(n)$?
It is well known that $BU$ is an infinite loop space, and as such it has an action of an $E_\infty$ operad. An explicit construction of such an action is given, for example, in an answer to this MO ...
- 10.7k
22
votes
1
answer
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Little disks operad and $Gal (\bar {Q}/Q)$
My question is simple:
How do the little disks operad and $Gal (\bar {Q}/Q)$ relate?
I realize that a huge amount of heavy-machinery can be brought into an answer to this, but I'm struggling with ...
- 2,714
21
votes
3
answers
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Relation between monads, operads and algebraic theories
I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. Is there any ...
- 3,299
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5
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Good reference for studying operads?
Can you, please, recommend a good text about algebraic operads?
I know the main one, namely, Loday, Vallette "Algebraic operads". But it is very big and there is no way you can read it fast. Also ...
21
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answer
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On Tamarkin's proof of Etingof-Kazhdan quantization of Lie bialgebra
This question is motivated by the desire to understand better the interplay between Drinfeld associators, algebraic structures up to homotopy and related results.
Recall that a Lie bialgebra is a Lie ...
- 8,244
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2
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Kontsevich's conjectures on the Grothendieck-Teichmüller group?
Reading Kontsevich's "Operads and Motives in Deformation Quantization", I was wondering about the state of the many conjectures concerning the Grothendieck-Teichmüller group in chapter 4. (Also, where ...
- 10.7k
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Are G_infinity algebras B_infinity? Vice versa?
What is the relationship between $G_\infty$ (homotopy Gerstenhaber) and $B_\infty$ algebras?
In Getzler & Jones "Operads, homotopy algebra, and iterated integrals for double loop spaces" (a paper ...
- 1,008
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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 ...
- 1,635
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2
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Does Koszul duality between $Comm$ and $Lie$ imply the power series identity $\exp(\ln(1-z))-1 = -z$?
To a symmetric sequence $V_\bullet$ of vector spaces, associate the generating function $F_V(z) = \sum_n \frac{\dim(V_n)}{n!} z^n$. Then
$$F_{Comm_\ast}(z) = \exp(z)-1 \qquad F_{Lie}(z) = \ln(1-z)$$
...
- 61.5k
18
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answer
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Higher homotopy algebraic structure on the homology of an operad
Given a DGA $A$, then by standard techniques such as homological perturbation theory, the ring structure on the homology $H(A)$ extends to a minimal $A_\infty$-algebra structure such that $H(A)$ is ...
- 6,169
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1
answer
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Positivity of coefficients of the inverse of a certain power series
Consider the unique formal power series $g(z)$ with $g(0)=0$ and $g'(0)=1$ satisfying the equation
$$
g(z)-g(z)^8+g(z)^{15}=z,
$$
that is the inverse of
$$
z-z^8+z^{15}
$$
in the group of formal ...
- 16.5k
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2
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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 ...
- 7,902
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1
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Are Lurie's operads special SMCs?
In Higher Operads, Higher Categories, Leinster nicely characterizes operads among monoidal categories (as PROs). Roughly speaking, a monoidal category comes from an operad if its set of objects is ...
- 8,559
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4
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Deligne's conjecture (the little discs operad one)
Deligne's conjecture states that the Hochschild cochain complex of an A-infinity algebra is an algebra over the operad of chains on the topological little discs operad.
Of course the conjecture has ...
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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 ...
- 6,484
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1
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What is this operad-like structure called?
I'd like to know what's the name (if any) of the following categorical structure, and also references where it has been considered.
Given a category $C$, let $O=\{O(n)\}_{n\geq 0}$ be a sequence of ...
- 14.9k
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Free Loop-Space Recognition Principle
It is well-known that one can detect based loopspaces using the machinery of operads. Namely, given a group-like space $X$ with an action of $\mathbb{E}_n$-operad, then it is homotopy equivalent as an ...
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Is the Amitsur-Levitzki identity essentially unique?
Let us consider the matrix algebra. $Mat_n(\mathbb{C})$. The Amitsur-Levitzki identity states that for any matrices $X_1, X_2, ..., X_{2n} \in Mat_n(\mathbb{C})$ the sum $\Sigma_{\sigma \in S_{2n}} ...
- 1,832
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2
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What homotopy classes can attaching an $E_n$-cell kill?
Let $A$ be a connected $E_{n+1}$-ring spectrum and let $\alpha\in\pi_k(A)$. I am having trouble showing that attaching an $E_n$-cell along $\alpha$ will necessarily not kill an element $\beta\in\pi_k(...
- 10.1k
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1
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Homotopy transfer in the opposite direction
Let $X\rightleftarrows Y\circlearrowleft$ be a strong deformation retraction of chain complexes (a.k.a. contraction), i.e. $X\rightarrow Y\rightarrow X$ is the identity, $Y\rightarrow Y$ is a homotopy ...
- 14.9k
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Is this an $E_\infty$-algebra?
I have a particular kind of algebraic structure that's come up in my work. It's basically a chain complex equipped with a multiplication which is commutative and associative up to homotopy in a ...
- 39.3k
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0
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Splitting of homomorphism from cactus group to permutation group
We all learned in kindergarten that the category of finite-dimensional (type I, say) $U_q(\mathfrak{g})$-modules is braided monoidal for $\mathfrak{g}$ a complex semisimple Lie algebra. This gives an ...
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Correspondence between operads and $\infty$-operads with one object
Given a simplicial operad one can form its category of operators. This is a simplicial category with a functor to the category of finite pointed sets which is a bijection on objects and whose hom-...
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Model structure for cooperads and for coalgebras
I am studying the homotopy theory of (algebraic) operads and I came up with several questions I am unable to answer to. I would like to stress that I don't have applications in mind, I just would like ...
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2
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Introducing division strategically in operads to accommodate formulas like the Baker-Campbell-Hausdorff formula
I am not as familiar with operad terminology as I'd like to be, so I might be missing some well-known term in the area. If so, I'd appreciate any pointers to the correct terms.
Consider the following ...
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When does an $E_\infty$ algebra come from a commutative differential graded algebra?
Suppose that $K$ is an $E_\infty$-algebra on a space $X$ (more generally, any ringed topos; also, feel free to assume that $X$ is a point). That is, $K$ is a cochain complex of sheaves on $X$, endowed ...
- 15.3k
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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/...
- 9,236
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2
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GL(V)-representation theory for a Lie bracket kernel
Let $V$ be a vector space over a field of characteristic $0$, and let $L_k(V)$ be the degree $k$ part of the free Lie algebra over $V$. There is an exact sequence
$$0\to D_n(V)\to L_1(V)\otimes L_{n+1}...
- 4,838
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1
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koszul duality and algebras over operads
Given a pair of Koszul dual algebras, say $S^*(V)$ and $\bigwedge^*(V^*)$ for some vector space $V$, one obtains a triangulated equivalence between their bounded derived categories of finitely-...
- 2,696
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What are types of coalgebras that are more naturally described by cooperads?
Some background. Let $\mathsf{C}$ be a symmetric monoidal category. An object $X \in \mathsf{C}$ has two operads "naturally" (the two constructions aren't functorial) associated to it: the operad of ...
- 4,840
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votes
1
answer
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Tensor product of dendroidal sets: counter-examples
For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$.
...
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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 "...
- 1,641
12
votes
1
answer
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Compatibility of the KZ connection with operadic composition
In what sense is the Knizhnik-Zamolodchikov connection compatible with the operadic composition in the little discs operad and/or the operad of $\overline{M}_{0,n}$'s?
Here are (some) details, ...
- 407
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1
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Condition on a Hopf operad for tensor product in the base category to be a (categorical) coproduct for algebras
A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. "Hopf"...
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Does the $(\mathbb Z/2)$-graded isomorphism $E_n \cong E_{n+2}$ have any nice properties?
This question assumes everything is dg. Let's decide to work over the "field" $\mathbb Q[\mu,\mu^{-1}]$ where $\mu$ has homological degree $+2$. Then chain complexes are just $\mathbb Z/2$-graded. ...
- 53.1k
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Are $E_n$-operads not formal in characteristic not equal to zero?
This is a short question:
Is it just unproven folklore (yet), or is it definitively known that $E_n$-operads are not formal, if the characteristic of the underlying field is not equal to zero?
- 2,014
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votes
1
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824
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$k$-Disk algebras versus $E_k$ algebras
Background:
The little $k$-cubes operad is the $(\infty,1)$-operad defined by embedding disjoint unions of $k$-dimensional open cubes rectilinearly into one another, that is using maps $(0,1)^k\...
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votes
1
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Conceptual (operadic?) reason for the generalized EHP fiber sequence $J_{q-1}(S^{2n}) \to J S^{2n} \to JS^{2nq}$?
Let $q$ be a prime and $q=p^r$ a power. Then there is a $p$-local fiber sequence from the $q-1$st stage of the James construction on $S^{2n}$, to $J(S^{2n}) = \Omega \Sigma S^{2n}$, to $J(S^{2nq}) = \...
- 61.5k
11
votes
1
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388
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Non-$\Sigma$ $E_n$ algebras?
Any symmetric operad can be considered as an non-$\Sigma$ operad by throwing away permutations. Does anyone know what sort of structure one gets for algebras over $C_n$ the little n-cubes operad, or ...
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1
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Do I need to know what an infinity-Gerstenhaber algebra is, and if so, what is it?
I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R \...
- 53.1k
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1
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PBW theorem over a Q-algebra, without freeness or flatness
Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-Lie algebra, which is not necessarily free as a $k$-module. Let $S\left(L\right)$ denote the symmetric algebra of $L$ (over $k$), constructed ...
- 33.2k
11
votes
1
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508
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On the coalgebraic homotopy transfer theorem
Let $A$ be a dg algebra, say over a field. The Homotopy Transfer Theorem says that $H(A)$ can noncanonically be given the structure of $A_\infty$-algebra, extending the induced multiplication on $H(A)$...
- 39.3k
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votes
1
answer
588
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Dioperads vs polycategories
As defined by Gan, a dioperad consists of sets of operations $P(n,m)$ with "$n$ inputs and $m$ outputs", which can be composed by joining one output of one operation to one input of another, giving ...
- 65.1k
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Infinity-homotopies
Koszul duality for operads allows for straightforward generalizations of $A$-infinity algebras and $A$-infinity morphisms for the so called Koszul operads $\mathcal{O}$, among which we find the ...
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