Operads are structures encoding the properties of algebras (in a very general sense), for example associativity, commutativity, unitality, and the relations between them. Their main uses lie in (abstract-algebra), (category-theory) or (algebraic-topology).

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Operad Associative and operad PreLie

We know that all associative algebra is a pre-Lie algebra, but I could not find an operad morphism between the operad Associative and the operad PreLie. So, this morphism exist? If it exists, can ...
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PreLie Operads and Free PreLie Algebras

Is there any relationship between Free Pre-Lie Algebras and Free Algebras over the Operad Pre-Lie (like in KOSZUL DUALITY FOR OPERADS in pg 13 )? If there is, can someone indicate a reference for this?...
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The Modules over Algebras over Operads are not what they seem.

Operads are a nice framework to model all kinds of different algebras, i.e. Monoids are algebras over the operad Assoc in the category of sets Associative algebras are algebras over the operad Assoc ...
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Maximal subcategory inside a multicategory

Let $\mathcal M$ be a multicategory. Let $C(\mathcal M)$ be a category consisting of all objects and all unary multimorphisms of $\mathcal M$. Is there a standard name for $C(\mathcal M)$?
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Monoidal and classical definition of algebraic operads- equivariance

I am studying Algebraic Operads, following the book Algebraic Operads by Jean-Louis Loday and Bruno Vallette. In this book they provide many equivalents definitions for algebraic operads, and I am ...
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Algebraic operads and block permutations

I am studying Algebraic Operads, my reference is Algebraic Operads by Jean-Louis Loday and Bruno Vallette. There, they use a particular type of permutation, named block permutation. Unfortunately I ...
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$C_\infty$ analog of the correspondence between $A_\infty$-alg. structures on $A$ and dg coalg. strucures on $(\bar T(sA),\Delta)$

There is a 1-1-correspondence between $A_\infty$-algebra structures on a graded vector space $A$ and dg. coalgebra structures on the bar construction $(\bar T(sA),\Delta)$. My question: Is there any ...
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An interpretation of this construction giving an operad from a bialgebra?

Let $A$ be a cocommutative bialgebra object (or even a Hopf algebra) in a symmetric monoidal category. Define an operad $\mathtt{P}_A$ by $\mathtt{P}_A(r) = A^{\otimes r}$ (so that $\mathtt{P}_A(0) = ...
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Graded Vector Spaces (definition)

I am studying Algebraic Operads with the book Algebraic Operads, by Jean-Louis Loday and Bruno Vallette and I'm having a little problem with the definition of graded vector space. My advisor and I ...
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What are types of coalgebras that are more naturally described by cooperads?

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 endomorphisms and ...
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$E_{\infty}$ algebra in characteristic zero

Let $A^{\bullet}$ be a cosimplicial commutative algebra over a field $\Bbbk$. Denote with $N(A)^{\bullet}$ the conormalized Moore complex. Since $A^{\bullet}$ is equipped with a product, the Alexander&...
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Operads are to multicategories as what are to polycategories

I have been reading a little about operads and their cousins multicategories. I am wondering what the cousins to polycategories are and why these mysterious cousins aren't popular in literature. ...
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Relations between the 2-disc operad and fractals?

As you can see, as of late I opened a thread on n-disc operads: Clarification regarding little n-discs operads The thing is, those drawings there could somehow be construed in the real world as ...
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Clarification regarding little n-discs operads

I am reading the wiki page on operad theory and I am trying to figure out how exactly those "Little something" operads work which are mentioned there. Specifically, I am having a hard time, despite ...
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Projection onto the Homology of an operad

Suppose that $\mathcal{O}$ is a differential graded operad over a field and that $H(\mathcal{O})$ (i.e. taking the arity wise homology) is an operad, too. (If possible, I would avoid to restrict to ...
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Symmetric tensor powers as tensors over symmetric group algebra

Let $V$ be a $k$-vector space and $V^{\otimes n}$ the $n$-fold tensor power of $V$ and let $\mathbb{S}_n$ be the symmetric group of an n-element set, with its signum representation denoted by $(-1)^\...
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The algebra of natural transformations of the n-th power tensor functor

Let $k$ be a $0$ characteristic field, $n$ an positive integer and $S_n$ the $n$-th symmetric group. Let's work in the symmetric monoidal category of $k$-vector spaces and linear maps that we denote ...
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Multicategories with out-arities

Basically, my question is: Why the emphasis on domains in the notion of multicategory? I will now give the formal framework to state it correctly. Passing from categories to multicategories ...
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iterated loop spaces and configuration spaces

In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map $$ \phi: Hom_T(X,\Omega Y)\to Hom_T(SX,Y) $$ is defined. And a map $$ \eta_n=\phi^{-...
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Associativity of the tensor product of dendroidal sets

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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Why do configuration spaces not form an operad?

I have seen in multiple sources that the collection of configuration spaces $\{Conf(n,m)\}_{n \in \mathbb{N}}$ where $$Conf(n,m)=\{(x_1, \ldots ,x_n) \in (\mathbb{R}^m)^n | x_i \neq x_j \text{ if } i \...
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$E_{\infty}$ 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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Colored operads as finitely essentially algebraic theory.

I call a planar operad what is also called planar (multi-)coloured operad or multicategory and symmetric operad a symmetric multicategory or symmetric (multi-)colored operad. I have two questions ...
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What is the meaning of “Homotopy of Little disc Operads”

I want to understand what means the homotopy of the little discs operad. I'm starting to research in this area and I have some questions. 1) I don't understand why little discs operad is a ...
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S-modules and Schur functors

I am reading the book "Algebraic Operads" by Loday and Vallette. (I will refer to their version 0.999 here : http://math.unice.fr/~brunov/Operads.pdf) In Chapter 5, they define an $\mathbb{S}$-module ...
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Spaces homotopy equivalent to $A_{\infty}$-spaces

I ask this question after reading Peter May's "Geometry of Iterated Loop Spaces", where the problem is definitely hinted at but I couldn't find a definite answer. Recall a symmetric operad $\mathcal{...
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Why are the algebras of the associative operad unital?

According to the n-lab page: The associative operad Assoc is an operad which is generated by a binary operation $\Theta$ satisfying $$\Theta\circ(\Theta,1)=\Theta\circ(1,\Theta)$$ It then ...
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There is no “operad of fields”

I've read the following proof-less claim: there is no operad such that the algebras over it are fields. We can make that precise by asking whether there's an operad $\mathcal{P}$ in abelian groups ...
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Why is an $E_\infty$-operad a kind of ''strictification'' for a non-commutative operation?

A topological space $X$ which is an algebra over an $E_\infty$-operad $E$ consists of a sequence of maps $$ \mu_n':E(n)\times X^n\to X $$ with compatibility conditions. The spaces $E(n)$ are ...
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Γ-spaces and operads

I'm looking for a comprehensible reference that explains how $\Gamma$-spaces are related to $E_{\infty}$-operads. I've found some old publications but was hoping there are better references out there. ...
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Why is the recognition principle important?

The recognition principle basically states that (under some conditions) a topological space $X$ has the weak homotopy type of some $\Omega^k Y$ iff it is an $E_k$-algebra (ie. an algebra over the ...
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Are groups algebras over an operad?

I'm trying to understand a little bit about operads. I think I understand that monoids are algebras over the associative operad in sets, but can groups be realised as algebras over some operad? In ...