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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### Homotopy in the space of linear isometries

I wanted to understand the linear isometries operad $\mathscr{L}$ as a model for $E_{\infty}$-operads for which I wanted to show that $\mathscr{L}(n) \simeq *$. This all reduces down to some ...
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### Internal Homs of (Higher) Operads and $(\infty, 2)$-Categories

While $(\infty, 1)$-categories continue to scare me (but also bring me joy!), it is almost frightening how naturally $(\infty, 2)$-categories seem to pop up if you're interested in $(\infty, 1)$-...
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Is there a nice reference which explains in detail how composition works for the $A_\infty$-operad (associahedra operad)? It would be neat to have a visualization for low dimensions of what's ...
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### Sanity Check: Monoids as Algebras over an Operad

I should probably study the classical viewpoint first. I haven't yet and I will eventually but let us stick to $\infty$-operads for this question. I'm following the lecture notes from Hebestreit-...
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Let $P$ be a Koszul operad. In the book of Loday-Vallette "Algebraic Operads", an $\infty$-morphism between algebras $A$, $B$ over a $P_\infty$-operad is defined as a dg-morphism $P^\textrm{¡... • 422 0 votes 0 answers 91 views ### Decomposable differential of operads To my understanding, a degree$p$derivation (for$p\in \mathbb{N}_{>0}$) on a graded reduced (meaning$P(0)=0$) operad$P$of vector spaces is given by a sequence of equivariant degree$pk$-... • 1,769 4 votes 1 answer 139 views ### Construction of the free operad I am reading Fresse's "Koszul duality of operads and homology of partition posets", and trying to understand the construction of the free operad$F(M)$on a symmetric sequence$M$as ... • 1,769 1 vote 0 answers 60 views ### (Why) is the category of$k$-linear operads not abelian? For$k$a field, a$k$-linear operad is defined to be a symmetric operad in the category of$\mathbb{Z}$-graded$k$-vector spaces (that is, a symmetric sequence of$k$-vector spaces such that certain ... • 1,769 2 votes 0 answers 55 views ### On categorical operads A symmetric$1$-coloured operad can be defined in any symmetric monoidal category. In particular, it can be defined for the cartesian monoidal category$\mathsf{Cat}$of small categories and functors. ... • 1,769 3 votes 1 answer 75 views ### Derivations of the free operad generated by a polynomial functor The following section in Kontsevich' and Soibelman's paper I don't understand: Let$F$be a polynomial functor [this is the author's terminology for analytic functor as explained here],$P= Free_{\...
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In their paper BOUSFIELD LOCALIZATION AND ALGEBRAS OVER COLORED OPERADS David White and Donald Yau, write: Colored operads encode even more general algebraic structures, inluding the category of ...
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### On the relationship of Hopf-algebras and (co)Operads

In the paper Axiomatic Homotopy Theory for Operads Berger and Moerdijk construct a cooperad out of a commutative bimonoid (I have no idea, why they call the latter a Hopf-object, since in the whole ...
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I'm following Loday-Vallette's book Algebraic Operads, and I'm having some trouble understanding the quadratic cooperad associated to a quadratic data $(E, R)$. They define the quadratic cooperad $\... • 422 2 votes 0 answers 98 views ### Reference for Stasheff's Associahedra Operad I am currently reading up on operads and am more than confused about the Stasheff operad. It is completely unplausible that I am the first student, who feels this way, so I hope to find a reference ... • 11.3k 3 votes 0 answers 44 views ### Reference request: Tensoring with Barratt-Eccles is$\Sigma_* $-cofibrant replacement In some different places (e.g. in Berger & Fresse, Combinatorial operad actions on cochains), it is stated as a classical result that the Barratt-Eccles operad$\mathcal{E}$(in chain complexes) ... 4 votes 0 answers 94 views ### Anticommutative operad Let$(C, \otimes, 1)$be a additive symmetric monoidal category with internal hom (well, lets think about$\mathrm{Mod}_R$with$R$a commutative ring), I am trying to figure out what would be the ... • 925 1 vote 1 answer 87 views ### Koszul Duality and Andre-Quillen homology relationship This may be a very open ended question but what is the relationship between koszul duality and Andre-Quillen homology? For example if I consider the André-Quillen homology$AQ_k(B)$of an associative ... • 157 1 vote 1 answer 103 views ### Map between endomorphism operads Let$A$and$B$be alegebras of any kind (the example that I have in mind is the underlying module of a graded operad) and consider its endomorphism operads$\mathrm{End}_A$and$\mathrm{End}_B$, ... • 6,383 2 votes 1 answer 89 views ### The category of graded$\mathbb S$-modules form a monoidal category I am reading paragraph 6.2 in Algebraic Operads by Jean-Louis Loday and Bruno Vallette. Proposition 6.2.2 states: The category of graded$\mathbb S$-modules, with the (composite) product$\circ$and ... • 1,323 0 votes 1 answer 53 views ### An explicit formula for group action on an operad I read Algebraic Operads by Loday and Vallette, namely paragraph 5.3.7 Partial definition of an operad. It's not clear to me how does$\sigma''$act (the phrase "acting identically on ... with ... • 1,323 1 vote 1 answer 100 views ### Cohomology of algebra over an operad as naive Ext-functor instead of with Kähler module of differentials Let$\mathcal{P}$be an operad and$A$a$\mathcal{P}$-algebra. In Algebraic Operads(AO) by Loday-Vallette there are some cohomology theories defined for$A$: Operadic cohomology = cohomology of$C_{...
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I am looking for a reference to the following theorem. If $F:\mathcal{C}\to\mathcal{D}$ is a symmetric lax monoidal functor between symmetric monoidal categories, then it induces a well-defined ...
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### Is There a Notion of Diagram in Multicategories and/or Operads?

In ordinary category theory there is a notion of a diagram in a category $\mathsf{C}$ which is usually described as a functor $F: \mathsf{J \to C}$ where $\mathsf{J}$ is some small category. Based on ...
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In Proposition 7.2.4 of Loday and Vallette's Algebraic Operads, they prove that the Koszul dual operad of a quadratic operad $\mathcal{P}(E,R)$, with generators E finite dimensional in each arity and ...
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### Why does the differential of a minimal model (of an algebra, operad) need be decomposable?

In operad theory, a minimal model of a dg-operad $\mathcal{P}$ (or dga A) is given by a quasi-free resolution $(\mathcal{T}(E),\partial) \overset{f}{\longrightarrow} P$ where $f$ is a quasi-...
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### Is there some practical intuition when working with a cooperad given by cogenerators and corelations?

In the case of algebras and operads, a description by generators and relations is common practice and I have a good understanding of this. A non-symmetric operad $\mathcal{P}$ given by a linear space ...
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### Partial composition of Swiss-Cheese Operad

I recently came across an operad called the swiss-cheese operad. I am trying to understand its definition through partial composition. I am having trouble checking the axioms (This makes me think that ...
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I have been thinking about the following question: given a morphism of coloured dg-operads $$\phi: P \longrightarrow Q$$ we derive a lax morphism between their respective monads $T= T_P \... • 413 3 votes 1 answer 73 views ### Differential of the Twisted complex for algebraic operads I have a question about the proof of lemma 6.4.12 in the book Algebraic Operads (Loday-Vallette) which I do not seem to be able to fully complete on my own. Hopefully, somebody here can point out what ... • 413 2 votes 0 answers 103 views ### What is the meaning of this product notation? Let$X,Y$be topological spaces, both of which admit an action by a group$G$. What is the definition of the space denoted by$ X \times_G Y $? In particular, let$C_n(k)$denote the space of ordered$...
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I am trying to compute the $k$-modules of $\mathcal{As}^¡$, the Koszul dual cooperad of the associative operad $\mathcal{As}$. I am using sections 7.1.4 and 7.2.1 of Algebraic Operads to try to do ...
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### Wedge product of permutations in definition of pairing?

In May's definition of a pairing of operads, he states that a pairing of operads $\tau: (A,O)\rightarrow C$ consists of a collection of maps $\tau: A(i)\times O(j)\rightarrow C(ij)$ that satisfies the ...
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### Free algebra over an operad is an algebra over that operad

Let us focus on operads of vector spaces and let $V$ be a vector space and $P$ an operad. The free $P$-algebra on $V$ is defined by $P(V)=\bigoplus_{r=0}^\infty (P(r)\otimes V^{\otimes r})_{\Sigma_r}$,...
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### Generalized multicategories to a $\Sigma$-free operad

According to Elmendorf in his paper "Left Adjoints for Generalized Multicategories", generalized multicategories associated to a $\Sigma$-free operad $\mathcal{D}$, which we call $\mathbb{D}$...
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### How does a map of two categorical operads induce a map of categories of generalized multicategories?

I'm working on a project and I have two categorical operads with a free $\Sigma_n$ action and there exists a map of operads between them. I know via a paper by Elmendorf and others titled "Left ...
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### What is the name of this operad and proof verification.

Let $P_k$ denote the $k$-th permutoassociahedron and define an operad $\Xi$ by $\Xi(k)=V(P_k)$ where the composotion is given by replacement of variables and renaming. An exanmple of replacement of ...
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### Universal dg-algebra of an $A_\infty$-algebra

In this document by Keller, proposition 2.1, it is stated that for every $A_\infty$-algebra $A$ there is a universal dg-algebra $U(A)$ w.r.t. the existence of an $A_\infty$-morphism $A\to U(A)$, and ...
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### Questions wrt. definition of $L_\infty$- and dg-Lie-algebras

I am trying to understand this definition from nLab (Def. 3.2) of $L_\infty$-algebras, and the following example that is supposed to boil down to dg-Lie algebras. What is the difference between ...
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I am aware of this question, which unfortunately doesn't help me enough. Recall that a (chain) homotopy between maps $f, g\colon X_\bullet\to Y_\bullet$ of chain complexes is a collection of maps \$h_\...