# Questions tagged [operads]

The operads tag has no usage guidance.

251
questions

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### Finitary monads on $\operatorname{Set}$ are substitution monoids. Finitary monads on $\operatorname{Set}_*$ are…?

$\DeclareMathOperator\Fin{Fin}\DeclareMathOperator\Lan{Lan}\DeclareMathOperator\Set{Set}$It is well known that the category of functors $\Fin \to \Set$ is equivalent to the category of finitary ...

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61 views

### Why is an operad of associative algebras Koszul?

Let $Assoc$ be an operad of associative algebras. What does it mean for $A$ to be a Koszul operad? Is it related to standard Koszul duality for algebras? As far as I understand, if $Assoc_{\infty}$ ...

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82 views

### Is there a n-category structure on algebras for $e_n$-like operads?

I'm fishing in troubled waters here and therefore the question is vague and meant to be as general as possible. In particular "$e_n$-like operad" can be an algebraic or topological $e_n$ operad, as ...

**3**

votes

**1**answer

171 views

### Semi-cocartesian operads

Context: In this interesting blog post, Mike Shulman indicates an approach for defining generalized types of operads. If I interpret the details correctly, (edit: which I apparently did not,) the idea ...

**5**

votes

**1**answer

120 views

### $G_{\infty}$ (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 ...

**6**

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310 views

### 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 ...

**2**

votes

**1**answer

149 views

### An exercise from Loday and Vallette about Koszul morphism

I tried to solve the following exercise from Loday and Vallette's Algebraic Operad. The first three parts are straightforward, however I have no idea how to solve the last part. I can't find any ...

**8**

votes

**1**answer

261 views

### 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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141 views

### The notion of $\infty$-Cooperads for which Bar-Cobar duality is an equivalence

In the paper Bar-Cobar Duality by Michael Ching, he proves that the category of operads in spectra is equivalent via the Bar-Cobar adjunction to some model category of co-operads defined in the paper. ...

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162 views

### Lie algebras in braided monoidal categories

Let $\mathcal{C}$ be a braided (not necessarily symmetric) monoidal category. Then we can define what monoids and commutative monoids in $\mathcal{C}$ are. What is the correct definition of a Lie ...

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173 views

### Can an algebra over an operad be described by generators and relations?

Experts in operads, please be gentle to a beginner.
Suppose I have a collection of generators $\{x_i\}_{i\in I}$ and some relations like
$$ x_1x_2 -x_5x_6x_7=0\qquad x_4x_8+2x_{11}x_{12}=3x_5x_6$$
...

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196 views

### For which $k$-types are $E_{n,m}$-algebras automatically $E_{n+1}$ algebras?

Recall that an $E_{n,m}$ algebra is an $A_m$ algebra in $E_n$ algebras. Here I index my $A_m$ algebras so that an $A_1$ algebra is pointed, an $A_2$ algebra has a unital multiplication, $A_3$ is ...

**1**

vote

**1**answer

141 views

### Algorithm for identifying reducible braids

If $\vec{n} = (n_1,...,n_k)$ is a vector of integers, there seems to be a well-defined homomorphism
$B_k \ltimes \left(B_{n_1} \times \cdots \times B_{n_k}\right) \to B_N$
where $N = \sum n_i$ and $...

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251 views

### 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 ...

**3**

votes

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267 views

### Homology of a loop-suspension space and action of $\mathcal{D}_1$-operad

If $X$ is a based connected topological space, it is well-known what the homology of $\Omega\Sigma X$ is: according to the Bott-Samelson theorem, it is a tensor algebra over reduced homology of $X$. (...

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**1**answer

141 views

### Group completion of $E_k$-algebras

Let $X$ be an $E_k$-algebra. We can form the delooping $BX$, which is a $E_{k-1}$-algebra. The space $\Omega B X$ is again an $E_k$-algebra, which is grouplike (i. e. $\pi_0(\Omega B X)=\pi_1(B X)$ is ...

**3**

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93 views

### Are these two natural $A_\infty$-structures on the realization of a cosimplicial commutative algebra isomorphic?

Given a cosimplicial commutative algebra $A^\bullet$ over a field of characteristic zero, there are two ways of producing an $A_\infty$-structure on its realization $|A^\bullet| := \int^\Delta C^*(\...

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97 views

### Homotopy transfer of cyclic L-infinity algebras

Suppose $W$ is a cyclic $L_\infty$ algebra, i.e. $W$ has a non-degenerate, symmetric, invariant pairing $\langle\cdot,\cdot\rangle_W$. Let $V$ be a cochain complex, and suppose given the data of a ...

**10**

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269 views

### Functoriality of infinite loop space machines?

If $C$ is a symmetric monoidal category, then $BC$ is canonically an algebra over a certain $E_\infty$ operad, but if $F: C \to D$ is a symmetric monoidal functor then (as far as I can see) $BF: BC \...

**3**

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61 views

### Computing weak operadic colimits as colimits

I am trying to reduce the computation of weak operadic colimits to colimits. Let me introduct some notation. Let $q:C^{\otimes} \to N(Fin_*)$ be a symmetric monoidal category.
Let $p: K \to C^{\...

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**1**answer

215 views

### A model for the framed little disks operad $f{\cal D}_n$ with arity one *equal* to $SO(n)$?

The framed little disks operad $f{\cal D}_n$ can be described as the semidirect product ${\cal D}_n \rtimes SO(n)$, where ${\cal D}_n$ is the little disks operad and $SO(n)$ is the special orthogonal ...

**10**

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238 views

### 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}) = \...

**5**

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169 views

### Topological category of topological monoids / operads

The category of topological monoids can be made into a topological category in a naive way. Namely, the space of all continuous homomorphisms between two topological monoids is a closed subspace of ...

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**2**answers

304 views

### Obstructions to $E_2$-algebra structure on $E_1$-algebra

Let $A$ be an $E_1$-algebra in chain complexes over $\mathbb Q$.
Is there an easy way to check if $A$ admits the structure of an $E_2$-algebra (or $E_\infty$-algebra)?

**3**

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54 views

### Generating an enriched multicategory

Let $C$ be an $(M,\otimes,1)$-enriched category. I am looking for a reference for a notion of “generating the morphisms of $C$” (for ordinary categories, but also for multicategories, see below).
My ...

**6**

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**1**answer

277 views

### 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 \...

**4**

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158 views

### Dyer–Lashof operations for more than 2 inputs

Let $\mathcal{O}$ be a topological operad and $X$ an algebra over it. Let the base ring be $\mathbb{Z}_2$. If $C_*$ denotes the singular chain complex over $\mathbb{Z}_2$, the action of $\mathcal{O}$ ...

**5**

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**1**answer

136 views

### Graded commutativity of the $n$th Browder bracket

Let $\mathcal{O}$ be a topological operad and $X$ an algebra over $\mathcal{O}$. Then $H_*(X)$ is an algebra (in the category of $\mathbb{Z}$-graded $R$-modules) over $H_*(\mathcal{O})$. Each $e\in ...

**6**

votes

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177 views

### 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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34 views

### On the notation $C[\lambda]$ where $C$ is a free cooperad in a proof of formality (and other details)

In his paper where details for Tamarkin's proof of formality are given, Hinich considers a Koszul quadratic operad $P$, a graded $P$-algebra $H$, a $P_\infty$-algebra $X$ with $HX=H$ (as $P$-algebras ...

**5**

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480 views

### 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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126 views

### The table reduction morphism of operads from Barratt-Eccles to Surjection

The Barratt-Eccles operad $E$ in the category of simplicial sets is obtained by applying the nerve functor to the canonical operad $\{\Sigma_n\}_{n>0}$ in groups. Berger-Fresse defined here an ...

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172 views

### “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 ...

**3**

votes

**1**answer

82 views

### $H$-space structure on coloured algebras

If $\mathcal{O}$ is a (classical) topological operad with unit $1\in \mathcal{O}(1)$, $\mathcal{O}(0)=\{0\}$ and multiplications $(m;a_1,\dotsc,a_r)\mapsto m(a_1,\dotsc,a_r)$. Let $X$ be an algebra ...

**5**

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195 views

### 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 ...

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283 views

### 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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134 views

### $E_\infty$-algebras and Tor-unital rings

Recall that a non-unital ring $R$ is called Tor-unital if $Tor^1_{R_+}(\mathbb Z,\mathbb Z) \cong 0$ where $R_+$ is the unitalization of $R$. See e.g. https://arxiv.org/pdf/1610.04998.pdf. If $R$ is ...

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245 views

### 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 ...

**18**

votes

**3**answers

947 views

### 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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359 views

### 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 $H(A)$...

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votes

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174 views

### 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 ...

**8**

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**1**answer

165 views

### 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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**1**answer

527 views

### “Exactness” of operadic cohomology

There are two somewhat widely known theorems which say
if $A$ is a nonnegatively graded commutative algebra in char $0$, then forgetful map on operadic cohomology $H^*_{Harr}(A, A) \to H^*_{Hoch}(A, ...

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263 views

### 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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622 views

### 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
$$F_{Comm_\ast}(z) = \exp(z)-1 \qquad F_{Lie}(z) = \ln(1-z)$$
...

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229 views

### Monoidal structures on modules over derived coalgebras

Given a Hopf-algebra $H$ (over a commutative ring), it is a classical fact that its category of (left) modules is monoidal, even if $H$ is not commutative. Given two left modules $M$ and $N$, we can ...

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votes

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187 views

### Coefficient (or target) category for factorization homology

In the article "Factorization homology of topological manifolds" by Ayala and Francis, a symmetric monoidal $\infty$-category $\mathcal{V}$ is fixed as the target or coefficient category. This ...

**8**

votes

**1**answer

544 views

### What is the interpretation of the Gerstenhaber bracket?

The homology of an $E_2$-algebra is a Gerstenhaber algebra.
How precisely is the Gerstenhaber structure related to the $E_2$-structure?
Obviously, the Gerstenhaber product is the commutative product ...

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178 views

### Higher Braces algebra and operads

1) In [HIGHER OPERATIONS ON HOCHSCHILD COMPLEX], Gerstenhaber and Voronov showed that the Hochschild complex $C_1(\mathcal A)$ of any associative algebra (or e_1 algebra) $\mathcal A$ is naturally ...

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136 views

### Generalisation of the notion of operad

Let $\mathscr P$ be an operad in the category of vector spaces. An algebra (of the type encoded by $\mathscr P$) on the vector space $V$ is a morphism of operads $\mu:\mathscr P\to End_V$ with $End_V$ ...