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### The S-module Ass is same as the composite of Com and Lie

It has been cited in several places (eg. https://arxiv.org/pdf/1912.05519.pdf) that the S-module Ass is isomorphic to the composite of the S-modules Com and Lie. Is there a reference which gives the ...
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### Another model for $\infty$-operads?

There are several well-developed notions of $\infty$-operad in the literature, which are nowadays known to be equivalent (see e.g. the introduction of Chu-Haugseng-Heuts. However, another model is ...
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### Riemann-Hilbert-type correspondence for locally constant factorization algebras

This is related to a previous post, but a bit softer and should probably stand on its own. In Appendix A of "Higher Algebra", Lurie shows that for a reasonably good topological space, there ...
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### $\mathbb{E}_M$ as colimit of little cubes operads

In Lurie's "Higher Algebra", Remark 5.4.5.2 towards the end, there is the following statement: "It follows that $\mathbb{E}_M$ can be identified with the colimit of a diagram of $\infty$...
1 vote
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### Koszul complex of the cobar construction is acyclic

This is a follow-up question on my question on math stackexchange (https://math.stackexchange.com/questions/4399553/proof-that-the-coaugmented-cobar-construction-of-a-cooperad-is-acyclic) I think I ...
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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 ...
256 views

### 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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### Augmented algebras over $\infty$-operads via the envelope

Let $\mathcal{O}^\otimes$ be an $\infty$-operad and $\mathcal{C}^\otimes$ a symmetric monoidal $\infty$-category, both in the sense of Lurie's Higher Algebra. By augmented $\mathcal{O}^\otimes$-...
1 vote
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### Differential of the Twisted complex for algebraic operads

I have a question about the proof of lemma 6.4.12 in the book Algebraic Operads (Loday-Vallette) which I do not seem to be able to fully complete on my own. Hopefully, somebody here can point out what ...
1 vote
307 views

### Is there an operad homotopifying the Koszul rule?

In homotopy theory one has the idea of a homotopy-commutative multiplication, in which one replaces the relation $$ab=ba$$ in a commutative monoid/group/ring/etc. for an unspecified homotopy. One ...
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### What is an invertible operad?

Let $\mathcal V$ be a nice symmetric monoidal ($\infty$-)category, and consider the ($\infty$-)category $Op(\mathcal V)$ of $\mathcal V$-enriched (symmetric) operads, symmetric monoidal under the ...
1 vote
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### Lawvere theory of Lawvere theories

There is a coloured operad $sOp$ such that $sOp$-algebras are single-coloured operads. This operad has a simple description in terms of generators and relations, say, as an operad $F(X)/R$. There is a ...
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### Boardman-Vogt resolution of the little 2-cubes operad

If $\mathbf{P}$ is a (coloured) operad, one can build a topological operad $W(\mathbf{P})$ called the $W$-construction or the Boardman-Vogt resolution of $\mathbf{P}$. Let me denote the resulting map ...
1 vote
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In Algebra+Homotopy=Operad in the conclusion it says that there is a commutative square of functors $\require{AMScd}$ \begin{CD} modular\, operads @>>> properads\\ @VVV @VVV\\ cyclic\, ...
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### Maps between unitary little disks operads and non-unitary little disks operads

Derived mapping spaces between little $d$-disks operads $E_d$ play an important role in embedding calculus. For example, Dwyer-Hess expresses the homotopy of framed long knots as loop spaces such ...
1 vote
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### Degree shift of multilinear maps

Let $V$ be a graded vector space over $\mathbb{k}$ and $V$ its odd degree shift. Given $k$, $l\in \mathbb{N}_0$, is there a natural way to define the following map,  \psi: \hom_{\mathbb{k}}(V^{\...
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### Planar dendroidal sets?

The meta picture is: (non-planar) dendroidal sets are to symmetric colored operads as simplicial sets are to categories. This suggests that one should have the notion of planar dendroidal sets (with a ...
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### Recovering operad units from homotopy units

It is my understanding that the $\infty$-category of non-unital connected topological monoids is equivalent to the $\infty$-category of connected topological groups. It follows that the functor from ...
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### Different ways to “deloop” a (topological) $A_\infty$-algebra

Let $\varphi:A\to \mathrm{Ass}$ be an $A_\infty$-operad in topological spaces, and let $X$ be an $A$-algebra. I see three possibilities to construct a delooping out of $X$: Rectify $X$ by taking the ...
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Let $D(\mathbb{F}_p)$ and $\mathcal{D}(\mathbb{F}_p)$ be the derived category and derived infinity-category of cochain complexes of $\mathbb{F}_p$-vector spaces. If $A$ is a sheaf of cdgas over $\... 8 votes 0 answers 139 views ### A question on recognition of equivariant loop spaces I have a question about equivariant loop space that has been bothering me, and that I have not been able to find an answer to in the obvious places. We know from the work of Segal that to give a loop ... 8 votes 2 answers 319 views ### Conceptual explanation for the sign in front of some binary operations In several situations, I've seen that given a binary operation on a graded module$m:A\otimes A\to A$, a new operation$M(x,y)=(-1)^{|x|}m(x,y)$is defined so that it satisfies some properties. One ... 2 votes 1 answer 166 views ### Detailed proof of$\mathfrak{s}^{-1}\mathrm{End}_V\cong \mathrm{End}_{\Sigma V}$I asked this question on MSE but I want to ask it again here with some more context sine it received no answers. In Chapter 3 (Algebra) of the book Operads in Algebra, Topology and Physics by Markl, ... 7 votes 0 answers 628 views ### Beginner's guide to$A_{\infty}$-algebras I have some general questions about$A_{\infty}$-algebras. Altough I understand bare definition from nLab I have no association how to think intuitively about them. Which picture one should have in ... 0 votes 0 answers 100 views ### Is the free algebra over an operad an algebra over that operad? I'm asking here this question I asked on MSE that got no answers. Let$V$be a dg-module and$P$an operad. The free$P$-algebra on$V$is defined by$P(V)=\bigoplus_{r=0}^\infty (P(r)\otimes V^{\...
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 ...
Given a graded vector space $V$ over a field $k$, consider it's suspension $\Sigma V$ such that $(\Sigma V)^i=V^{i-1}$. For an operad of graded vector spaces over a field $\mathcal{O}$, the operadic ...