The operads tag has no wiki summary.
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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 ...
5
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1answer
325 views
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 ...
14
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1answer
385 views
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 ...
9
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1answer
249 views
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 $A$.
...
7
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206 views
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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2answers
290 views
Has this construction, which builds a symmetric multicategory from a commutative monoid, been described or studied anywhere, and if so, where?
Whenever $R$ is a commutative ring, write $R[x^{(n)}]$ for the set of all $p \in R[x]$ such that $p$ is a monic polynomial of degree $n$. Then $R[x^{(n)}]$ is not closed under sums, nor does it ...
3
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0answers
160 views
Straightening for $\infty$-operads
There is this straightening/unstraightening procedure of Jacob Lurie's which takes a symmetric monoidal $\infty$-category (which is the data of a coCartesian morphism of simplicial sets $C^\otimes\to ...
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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. ...
6
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2answers
289 views
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 ...
4
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2answers
370 views
$E_n$-space and n-connected pointed space
Is it true that the homotopy category of group-like $E_n$-spaces is equivalent to the homotopy category of pointed $n$-connected spaces ? If it is true, what should be the statement when ...
12
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3answers
366 views
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}} ...
3
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1answer
345 views
$E_{\infty}$ spaces are $A_{\infty}$ spaces
While studying the well-known "Geometry of Iterated Loop Spaces", I found this corollary which is not completely clear to me. (By $\mathcal{M}$ is meant the operad given by $\mathcal{M}(j):=\Sigma_j$, ...
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0answers
94 views
Finitely co/continuous monad induced by an operad
It is well known that any operad on a nice monoidal category induces a monad.
I was wondering which conditions can be imposed on the operad $T$ (say, in a monoidal closed $\cal V$) so that the ...
9
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1answer
307 views
$k$-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 ...
4
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1answer
395 views
Is there an analog of the Barratt-Eccles construction for group-like E_∞-spaces and E_∞-ring spaces?
The Barratt-Eccles operad is an operad in simplicial sets
that provides a particularly nice model of an E∞-operad;
algebras in spaces over the Barratt-Eccles operad model E∞-spaces,
i.e., homotopy ...
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0answers
140 views
Reference request: modern proof of the recognition principle [closed]
I'm looking for a modern version of P. May's proof of the recognition principle (in "geometry of iterated loop spaces"). Can someone help me?
10
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1answer
276 views
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 ...
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140 views
What is a morphism of $B_\infty$ algebra
Let $k$ be a commutative ring and $C$ a $ \mathbb{Z}$ graded $k$ module. A $B_\infty$ structure on $C$ is the datum of a differential and a multiplication on $ BC:= \bigoplus^\infty_{i =0} ...
10
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1answer
329 views
Condition on a Hopf operad for tensor product in the base categoy 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. ...
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1answer
459 views
understanding the definition of $\infty$-operad of module objects
I'm just trying to understand the following definition:
Definition 3.3.3.8 in Higher Algebra by J. Lurie defines the $\infty$-operad of $O$-module objects, and says the following:
Let $O^\otimes$ be ...
4
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1answer
281 views
What's a good reference for the following obstruction theory yoga?
Fix a colored operad, which I will leave implicit, and a field $\mathbb K$ of characteristic $0$. By algebra in this post I will mean a dg algebra over $\mathbb K$ for the given colored operad. I ...
1
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1answer
168 views
comparison between two monadic definitions for an operad
According to May, an operad $\mathcal{C}$ valued in sets is equivalent to a monad in Cat on the endofunctor $C\colon X\mapsto \coprod_i \mathcal{C}(i)\times X^i.$
According to Leinster, an operad is ...
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184 views
Where is the Courant operad discussed?
Where is the Courant operad discussed? And hopefully defined precisely.
By the Courant operad or rather a suitable generalization of operad to accommodate the inner product, the operad whose ...
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1answer
225 views
Correspondence between operads and monads requires tensor distribute over coproduct?
In checking the details of the correspondence between operads over a symmetric monoidal category and monads on some associated endofunctor of the category, I cannot make the obvious proof work without ...
9
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2answers
446 views
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 ...
1
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1answer
216 views
A question on the definition of operad
The nlab page says
A (Set-based) operad is a monoid in the monoidal category
$(Psh(ℙ),∘,I)$, where $ℙ$ is the category of $\sqcup_{n\ge 0} S_n$.
The monoidal structure is given by the so ...
8
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0answers
285 views
Galois action on $E_n$-operads
Let $E_n$ be the little $n$-cubes operad which acts on $n$-fold loop spaces (up to group completion, an $E_n$-action is precisely the data needed to perform an $n$-fold delooping). I am looking for ...
2
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1answer
147 views
Embedding e_n -> e_m
Let $e_n$ be the operad of (say, rational) chains of the operad of little n-disks. Consider the natural embedding $e_n\to e_m$, where $n<m$ and $n,m>1$ induced by the embedding $\mathbb{R}^n\to ...
8
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The spheres operad
I have a rather naive question. Consider the space of all maps
$$ S^{j_1} \times S^{j_2} \times \cdots \times S^{j_k} \to S^n $$
for all possible natural numbers $n, k, j_1, \cdots , j_k$.
This ...
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2answers
277 views
Reference for Stasheff Operad
I want a reference for Stasheff operad, where operad maps are defined explicitly at the point-set level. I would also like to ask the question that what exactly do one mean by Stasheff operad? Is ...
6
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4answers
371 views
How universal is operadic approach to studying algebras?
I have just started to read about operads, so this question might be silly.
So it seems to me that any "reasonable" class of algebras can actually be defined as a class of all algebras over a certain ...
4
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1answer
240 views
A_n operad as configuration spaces
$A_\infty$ operad can be described both in terms of Stasheff polytopes and configuration spaces.$A_n$ operad can be described as subspace of Stasheff operad described using Stasheff polytope. Is there ...
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1answer
178 views
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
$$
...
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2answers
522 views
Aspherical operads
Let $O$ be an operad in $\mathtt{SETS}$. Assume that $O(0)$ is empty and $O(1)$ only consists of the identity. Assume for simplicity that $O$ is monochromatic, i.e. we have no labels on the ...
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3answers
865 views
Good reference for studying operads?
Can you, please, recommend a good text about algebraic operads?
I know the main one, namely, Loday-Vallette "Algebraic operads". But it is very big and there is no way you can read it fast. Also ...
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0answers
100 views
Exact sequence of L-infinity-algebras
We call a sequence of $L_\infty$-algebras (weak) maps
$$0\to L\xrightarrow{f} M\xrightarrow{g} N\to 0$$
is exact if it is exact on the the underlying chain complexes level.
Thought I don't know ...
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0answers
241 views
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 ...
6
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1answer
261 views
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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2answers
429 views
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 ...
12
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2answers
380 views
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 ...
7
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1answer
506 views
Seeking errata for Berger-Moerdijk Axiomatic Homotopy Theory for Operads
The paper I'm referring to can be found here. It came out in 2003. I've been told by professors before that I should be careful relying on things from this paper, because it contained errors.
Is ...
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0answers
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Generating series of free PROs
Let
\begin{equation}
G := \biguplus_{p \geq 0} \: \biguplus_{q \geq 0} G(p, q)
\end{equation}
be a bigraded set of generators and $\mathcal{F}(G)$ be the free PRO generated by $G$ (see [1] for a net ...
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1answer
637 views
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 ...
4
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1answer
454 views
Which E_∞-spaces are homotopy colimits of k-truncated E_∞-spaces?
This question is closely related to my previous question about modules over truncated sphere spectra, in particular, it has the same motivation.
Recall that every space (or ∞-groupoid) can be ...
8
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1answer
275 views
Is the operadic butterfly symmetric?
The operadic butterfly is a diagram in the category of operads in vector spaces. It extends the short exact sequence relating commutative, associative and Lie operads.
$$\begin{array}{ccccc}
& ...
3
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1answer
238 views
Repertory of the different sorts of operads
Many different types of operads have emerged in recent years (symmetric, shuffle, cyclic, anticyclic, coloured, etc.).
I would like, for any of these, list the following data:
Description of the ...
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1answer
240 views
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 ...
11
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1answer
685 views
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 ...
3
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1answer
186 views
PROPs representations, free module analog
Ordinary operad with one ouput can be obviously regarded as free module on itself. Is there are analogous construction for operad with many outputs (PROP)? This must be difficult question, but what is ...
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2answers
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A (too?) simple notion of “closed multicategory”
Suppose I define a multicategory $M=(Ob(M),Hom_M)$ to be simply closed if
for every sequence $S=(b_1,\ldots,b_n;x)$ of $n+1$ objects in $M$, we provide an object $Exp(S)\in Ob(M)$, and
for every ...
