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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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### 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 $Assoc_{\infty}$ ...
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 ...
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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 ...
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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 ...
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 ...
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### 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 ...
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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### 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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### 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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### 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 $$x_1x_2 -x_5x_6x_7=0\qquad x_4x_8+2x_{11}x_{12}=3x_5x_6$$ ...
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 ...
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### 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 ...
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### 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 ...
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### 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 $\mathcal{O}$ ...
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### 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 ...
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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### 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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### “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 ...
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### $H$-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 ...
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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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### 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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### $E_\infty$-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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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 ...
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 $End_V$ ...