# Tagged Questions

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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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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### 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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### $E_{\infty}$ 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 ...