Questions tagged [operads]

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).

Filter by
Sorted by
Tagged with
1 vote
1 answer
42 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_{...
user avatar
  • 383
1 vote
1 answer
34 views

Symmetric lax monoidal functor takes operads to operads

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 ...
user avatar
  • 6,013
2 votes
1 answer
60 views

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 ...
user avatar
  • 12.2k
1 vote
0 answers
16 views

Koszul dual of a quadratic operad

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 ...
user avatar
  • 1,281
0 votes
0 answers
31 views

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-...
user avatar
  • 383
0 votes
0 answers
35 views

Proof that the coaugmented cobar construction of a cooperad is acyclic

I am reading the book Algebraic Operads by Loday-Vallette and I am stuck on a part left to the reader in lemma 6.5.14. It is stated that the coaugmented cobar construction $C \circ_\iota \Omega C$ is ...
user avatar
  • 383
3 votes
1 answer
91 views

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 ...
user avatar
  • 383
1 vote
0 answers
49 views

When does the left adjoint of the base change functor between categories of algebras over operads preserve quasi-isomorphisms?

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 \...
user avatar
  • 383
3 votes
1 answer
57 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 ...
user avatar
  • 383
2 votes
0 answers
99 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 $...
user avatar
  • 23
3 votes
1 answer
87 views

Koszul dual cooperad of the associative operad

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 ...
user avatar
  • 6,013
1 vote
1 answer
72 views

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 ...
user avatar
  • 79
1 vote
1 answer
131 views

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 $P(V)=\bigoplus_{r=0}^\infty (P(r)\otimes V^{\otimes r})_{\Sigma_r}$,...
user avatar
  • 6,013
1 vote
0 answers
47 views

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 $\mathbb{D}$...
user avatar
0 votes
0 answers
27 views

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 ...
user avatar
1 vote
0 answers
62 views

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 ...
user avatar
0 votes
1 answer
52 views

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 ...
user avatar
  • 1,846
1 vote
1 answer
61 views

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 ...
user avatar
  • 1,846
4 votes
1 answer
173 views

Higher (chain) homotopies

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_\...
user avatar
  • 1,846
2 votes
1 answer
35 views

A chain complex corresponding to the associahedron

A definition often found is that an $A_\infty$-algebra is an algebra over an $A_\infty$-operad. here, a non-symmetric operad $O$ is $A_\infty$ if all $O(n)$ are contractible symmetric operad $O$ is $...
user avatar
  • 1,846
0 votes
0 answers
48 views

Equivariance in definition of symmetric operads

I know that there are some questions regarding symmetric operads, but I feel like asking yet another question. Consider the operad $A$ with $A(n)=k[S_n]$. Let us denote the simple transpositions in $...
user avatar
  • 1,846
0 votes
0 answers
87 views

Free algebra over an operad

So i read somewhere that "The free dendriform algebra can also be considered as the free algebra over the Dendriform operad." Now i am familiar with free algebras and free operads, but i am not sure ...
user avatar
  • 345
1 vote
1 answer
95 views

How to prove that $\operatorname{Ass}$ is an operad

The definition of an operad requires that that the associativity condition holds: $$f \circ (g_1(x_1^1, \dotsc, x_{n_1}^1), \dotsc, g_r(x_1^r, \dotsc, x_{n_r}^r)) = (f \circ (g_1, \dotsc, g_r)) (x_1,\...
user avatar
  • 345
2 votes
1 answer
58 views

Is the pre-Lie operad a nonsymetric operad

Can a pre-Lie algebra be modeled as a nonsymmetric operad? Looking at the relation $(x \circ y) \circ z - x\circ (y \circ z) = (x \circ z) \circ y - x \circ (z \circ y ) $ it doesn't look like it ...
user avatar
  • 345
0 votes
1 answer
23 views

The associative operad with more elements still associative?

So the associative operad is defined as $P(n) = a_n$ for each $n$. What happens if we set $P(3) = a_3$ but add more elements to the other $P(n)$ for example if $P(5) = \{x,y,z\} $. Will an algebra ...
user avatar
  • 345
0 votes
2 answers
49 views

An algebra over the associative operad

So an algebra over the Associative operad is an operad morphisme $Ass \rightarrow End_V $. And is an associative algebra. Are there any nice examples of an associative algebras definined this way? I ...
user avatar
  • 345
0 votes
0 answers
20 views

Proving an algebra over the Ass operad is the associative algebra.

So i know how to prove that an algebra over the Ass operad is associative. But that doesn't mean that this algebra is the associative algebra as it might have some more structure. How would you prove ...
user avatar
  • 345
4 votes
1 answer
123 views

$\Omega X$-modules are functors from $X$

Let $X$ be a connected (nice) space, $x\in X$ and $\Omega X$ the loopspace at $x$. Then $\Omega X$ has an $E_1$-structure, and so we may consider left $\Omega X$-modules. There are a few ways to do ...
user avatar
  • 41.7k
0 votes
0 answers
90 views

What are $E_1$ spaces?

In one of the answers to this question, $E_1$ spaces are mentioned. In that context, they are described as a homotopically invariant version of topological group. To me, this would imply that we have ...
user avatar
  • 3,032
1 vote
0 answers
56 views

Defining certain operads for general symmetric monoidal categories

I've been studying operads and something that upsets me is that there are some operads that have the same "meaning" but have a different definition depending on the category we're working on. For ...
user avatar
  • 6,013
1 vote
0 answers
47 views

Truncated action of Stasheff associahedron

I was reading this introduction to operads in which the associahedron operad $\mathcal{K}$ is explained. In section 3 there is an explanation of a truncated action of this operad. Literally it says ...
user avatar
  • 6,013
3 votes
0 answers
74 views

Every monad in Set induces an operad

$\require{AMScd}$I am fighting with an exercise in Leinster's Higher operads: in short, Example 2.2.11 says that every monad on $\bf Set$ gives an operad taking the sequence of $T(n)$, where $n$ is an ...
user avatar
  • 10.9k
1 vote
1 answer
49 views

Homotopy coherence for a $ A_{\infty }$ space

This question came up when I read german article about operads in topology & homotopy theory. translated to english the statement is: a $ A_{\infty }$ space is a topological space $Y$ with a ...
user avatar
2 votes
0 answers
93 views

The role of genus in Tillmann's Surface Operad

I suppose my question has two parts. To begin Tillmann defines the operad $\mathcal{M}$ by its groupoids: $$\coprod_{g\geq 0} B\mathcal{S}_{g,n,1}$$ In the paper she proves that algebras over this ...
user avatar
  • 31
3 votes
1 answer
98 views

What is the free product in the category of operads?

Is the Hadamard product (5.3.3 in Loday-Vallette's book on Algebraic operads) the free product in the category of operads? Is there a free product defined in the category of 1/2props or props?
user avatar
3 votes
1 answer
232 views

Symmetric and non-symmetric operads

I have trouble understanding why they are distinct notions of symmetric operad and non-symmetric operads : are they really both needed ? It seems like symmetric operads are more general, but I do not ...
user avatar
2 votes
2 answers
89 views

"Algebras are to operads as group representations are to groups"

I just read the wikipedia page on operads, and it says: Algebras are to operads as group representations are to groups Based on the definition of operads, I cannot immediately see why this is true....
user avatar
4 votes
0 answers
47 views

How does a group action on a category pass to the geometric realization?

Let $Y_n$ be the category whose objects are pairs $(x,y)$, where $x$ belongs to the braid group $B_n$ and $y$ is a parenthesizing of the non-associative product of $n$ elements, for instance, $(x_1x_2)...
user avatar
  • 6,013
3 votes
0 answers
93 views

Is there an operad whose algebras are homotopy commutative $E_1$-algebras?

I might guess that the Boardman-Vogt tensor product of the $E_1$ operad and the $A_2$ operad might do the trick. That is, I would guess that an $A_2$ object in $E_1$ algebras, or equivalently an $E_1$ ...
user avatar
  • 5,445
2 votes
1 answer
76 views

Notation involving the symmetric group $S_n$

I'm reading this paper by Hinich and he uses two notations involving the symmetric group $S_n$ that he doesn't clarify, so I assume that they are standard, but I don't know what they mean. The first ...
user avatar
  • 6,013
1 vote
1 answer
71 views

Is it possible to "mod" the action of a symmetric group on a symmetric operad?

I am relatively new to category theory, so only have a rough understanding of the technicalities behind operads. My understanding is that symmetric operads are defined so that they are "nicely" acted ...
user avatar
3 votes
1 answer
139 views

Explicit model of the $E_{n}$-operad in simplicial sets

The space of $k$ little $n$-disks, denoted $E_{n}(k)$, is usually constructed in the category of topological spaces as the space of $k$-tuples $(c_{d_{1},p_{1}},\dots, c_{d_{k},p_{k}})$ of disjoint $n$...
user avatar
3 votes
0 answers
70 views

Base change of topological operad to any symmetric monoidal model category and $E_n$-algebras outside of $\textbf{Top}$

I would like to ask about how we can talk about Algebra over little $n$-disk operad $D_n$ in a greater generality outside of $\textbf{Top}$. I know that in the topological context, an $E_n$-operad is ...
user avatar
  • 881
2 votes
1 answer
46 views

Notation: Definition of little $d$-disk operad $D_d$ for $d=\infty$

Let $D_d$ be the little $d$-disk operad as outlined in Fresse's book Homotopy of Operads and Grothendieck-Teichmuller Groups. We have the sequence of inclusions of operads $$D_1 \to D_2 \to \cdots \...
user avatar
  • 881
1 vote
1 answer
112 views

Construction of a monad from an operad is in the CGWH category

If $\mathcal{C}$ is an operad and if $X\in\mathcal{J}$ then $CX\in\mathcal{J}$, where $\mathcal{J}$ is the category of compactly generated weakly Hausdorff spaces well-based. I'm studying the ...
user avatar
1 vote
0 answers
73 views

Clarification on Tillmann's construction of the higher genus surface operad

I'm currently reading Tillmann's paper 'Higher genus surface operad detects infinite loop spaces' but am having trouble understanding the construction of said operad. Specifically, there are two ...
user avatar
  • 480
1 vote
1 answer
154 views

Action of a symmetric group in operad

For a symmetric monoidal category $(V,\otimes, I)$, we have a definition of symmetric operad as: collection of objects $P(n) \in V$ unit map $I \to P(1)$ product map $P (n) \otimes P(j_1)\otimes .......
user avatar
  • 881
3 votes
0 answers
122 views

The construction of free symmetric/non-symmetric operads

I have a couple of questions concerning basic notions in operad theory. What is the ideological difference between symmetric and non-symmetric operads? I think about the difference in the following ...
user avatar
  • 183
2 votes
0 answers
66 views

What is the meaning of some concepts in Operads?

I have already studied the notion of Operads. As I am new in this way, I need some clarification of basic concepts related to Operas through examples. We have the following notation: $\sum$: a set ...
user avatar
  • 1,220
13 votes
2 answers
1k views

Do there exist algebras of more directions of operation than left-right?

Again I am kind of new to most things algebraic, only having learned the very basics about groups. As little I have learned about groups and their operations is that an operation has two arguments, ...
user avatar
  • 24.3k