Questions tagged [operads]
The operads tag has no usage guidance.
8
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96 views
Vertex algebras and factorization algebras
It is often said that vertex algebras are a special case of factorization algebras. In particular, in their book "FAs in QFT" Costello/Gwilliam construct a functor from a certain class of 2d "...
2
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0answers
108 views
$E_\infty$-algebras and Tor-unital rings
Recall that a non-unital ring $R$ is called Tor-unital if $Tor^1_{R_+}(\mathbb Z,\mathbb Z) \cong 0$ where $R_+$ is the unitalization of $R$. See e.g. https://arxiv.org/pdf/1610.04998.pdf. If $R$ is ...
5
votes
1answer
204 views
An operad-like structure, is there a name for it?
Here is an example which I'd like to have a name for.
Let $P$ be a compact smooth manifold of dimension $p$, possibly with non-empty boundary.
Define $E(k,P)$ to be the space of smooth (codimension ...
16
votes
3answers
792 views
Are there prominent examples of operads in schemes?
There is an abundance of examples of operads in topological spaces, chain complexes, and simplicial sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even ...
5
votes
0answers
171 views
On the coalgebraic Homotopy Transfer Theorem
Let $A$ be a dg algebra, say over a field. The Homotopy Transfer Theorem says that $H(A)$ can noncanonically be given the structure of $A_\infty$-algebra, extending the induced multiplication on $H(A)$...
7
votes
1answer
118 views
Tensor product of a DGA and an $A_\infty$ algebra
In general there seems no way to naturally define the tensor product of two $A_\infty$ algebras $A$ and $B$. But, if $(A, m^A_1,m^A_2)$ is only a DGA(differential graded algebra) and $(B, m^B_k, k\ge ...
8
votes
1answer
136 views
Permutations and framed braids as cyclic operads
The symmetric groups and the framed braid groups form an operad (in sets, not groups). It is straightforward to see this structure using the string diagrams.
It is also known that these operads are ...
9
votes
1answer
466 views
“Exactness” of operadic cohomology
There are two somewhat widely known theorems which say
if $A$ is a nonnegatively graded commutative algebra in char $0$, then forgetful map on operadic cohomology $H^*_{Harr}(A, A) \to H^*_{Hoch}(A, ...
8
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0answers
211 views
Did the Goerss-Hopkins manuscript “Multiplicative stable homotopy theory” ever appear?
A citation to "M. J. Hopkins and P. Goerss, Multiplicative stable homotopy theory, unpublished manuscript, 1996" appears in the Hill, Hopkins, Ravenel Annals paper on the Kervaire invariant. It was ...
18
votes
2answers
430 views
Does Koszul duality between $Comm$ and $Lie$ imply the power series identity $\exp(\ln(1-z))-1 = -z$?
To a symmetric sequence $V_\bullet$ of vector spaces, associate the generating function $F_V(z) = \sum_n \frac{\dim(V_n)}{n!} z^n$. Then
$$F_{Comm_\ast}(z) = \exp(z)-1 \qquad F_{Lie}(z) = \ln(1-z)$$
...
7
votes
1answer
195 views
Monoidal structures on modules over derived coalgebras
Given a Hopf-algebra $H$ (over a commutative ring), it is a classical fact that its category of (left) modules is monoidal, even if $H$ is not commutative. Given two left modules $M$ and $N$, we can ...
2
votes
1answer
172 views
Coefficient (or target) category for factorization homology
In the article "Factorization homology of topological manifolds" by Ayala and Francis, a symmetric monoidal $\infty$-category $\mathcal{V}$ is fixed as the target or coefficient category. This ...
8
votes
1answer
439 views
What is the interpretation of the Gerstenhaber bracket?
The homology of an $E_2$-algebra is a Gerstenhaber algebra.
How precisely is the Gerstenhaber structure related to the $E_2$-structure?
Obviously, the Gerstenhaber product is the commutative product ...
4
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0answers
117 views
Higher Braces algebra and operads
1) In [HIGHER OPERATIONS ON HOCHSCHILD COMPLEX], Gerstenhaber and Voronov showed that the Hochschild complex $C_1(\mathcal A)$ of any associative algebra (or e_1 algebra) $\mathcal A$ is naturally ...
2
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0answers
115 views
Generalisation of the notion of operad
Let $\mathscr P$ be an operad in the category of vector spaces. An algebra (of the type encoded by $\mathscr P$) on the vector space $V$ is a morphism of operads $\mu:\mathscr P\to End_V$ with $End_V$ ...
4
votes
1answer
78 views
Equivariant non symmetric operads
The definition of a symmetric $G$-Operad is basically a $G$ object in the category of symmetric operads. As far as I understand there is not a good notion of the non symmetric case. I would like to ...
5
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0answers
105 views
Does real formality descend to rational formality for operads?
A classical theorem in rational homotopy theory says that a space is formal over $\mathbb{Q}$ iff it is formal over any field of characteristic zero. In other words, the algebra $A^*_{PL}(X)$ is ...
15
votes
1answer
441 views
Homotopy theories of operads
I know of three homotopy theories of colored operads.
The (derived) localization category of Berger-Moerdijk's model structure on the category of strict simplicial (or topological) operads, with weak ...
3
votes
1answer
116 views
Is the category of enriched operads (co)complete?
Let $V$ be a symmetric monoidal category
which is complete and cocomplete.
Is the category of small symmetric colored $V$-enriched operads complete and cocomplete?
If $V$ is presentable,
is it ...
8
votes
0answers
157 views
Clarification of Tillmann's construction of the higher genus surface operad
Sorry if this question is inappropriate for overflow. I tried asking on stackexchange yesterday but didn't get any responses, so I thought that this site might be better. Anyway, my question is as ...
7
votes
1answer
165 views
Construction for algebras over little cubes operad
Recently I came across the following construction:
Fix a dimension $k$. Let $C$ denote the space whose points are disjoint rectilinear embeddings $c\colon I^k\to \mathbb R^k$ of the (closed) $k$-...
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0answers
156 views
Modeling scientific theories with category theory (or, how to represent a biological system categorically)
Suppose I wanted to compare Linnaean classification, which arranges species by similarity in ranked taxa, to modern phylogenetic systematics, which appeals to descent with modification and branching ...
5
votes
1answer
201 views
Knot Factorization Homology inputs
Following the paper by Ayala, Francis, and Tanaka: https://arxiv.org/pdf/1409.0848.pdf
If we are talking about knots we are talking about framed 3-manifolds with a framed 1-dimensional sub-manifold ...
10
votes
2answers
138 views
How many operad structures are there on the symmetric sequence of simplices / finitely-supported probability measures?
Consider the symmetric sequence $P_n = \Delta^{n-1}$ of probability measures on finite sets, with coordinatewise $\Sigma_n$-action. There is a natural topological operad structure on $P$ given by ...
5
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2answers
427 views
Can operads (or category theoretic structures more generally) be compared?
I was reading John Baez’s paper on operads and phylogenetics trees where he formalizes a Jukes–Cantor model of phylogenetics. Because biological questions receive different answers depending on the ...
7
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1answer
209 views
Is there a proof of the formality of configuration spaces of Euclidean spaces that do not involve operads?
By "configuration spaces of $\mathbb{R}^n$" I mean ordered configuration spaces:$$\operatorname{Conf}_k(\mathbb{R}^n) = \{ (x_1,\dots,x_k) \in (\mathbb{R}^n)^k \mid x_i \neq x_j, \, \forall i \neq j \}...
2
votes
0answers
97 views
Operads and their applications to define homomorphisms
I have already commenced studying the notion of operads via following Kapranov & Ginzburg's paper (Koszul Duality for Operads) and for instance: Varieties of dialgebras and conformal algebras the ...
3
votes
0answers
123 views
Completion of coalgebras
Is it possible to complete commutative dg associative (conilpotent) coalgebras over $\mathbb{Q}$ in a way so that when we complete the symmetric coalgebra $Sym(V)$ it becomes completed with respect to ...
7
votes
1answer
329 views
When does the enveloping algebra functor lift to the category of bialgebras?
Let $\mathrm{Ass} $ denote the operad, whose algebras are associative unital algebras, considered as a dg-operad.
Denote $\mathrm{Ch} $ the category of chain complexes over a commutative ring $\...
5
votes
1answer
263 views
Homotopy invariant structure: Stasheff versus Segal
To describe homotopy invariant algebraic structures on spaces, there are different approaches.
The Stasheff / Boardman–Vogt / May approach, where operations and equations are replaced by spaces of ...
8
votes
1answer
189 views
One colored infinity operads via symmetric sequences?
The question
One standard approach to the theory of 1-colored (symmetric) operads in classical 1-categorical theory is via monoids in symmetric sequences with respect to the composition product. Has ...
7
votes
1answer
218 views
Tensor products of $\infty$-algebras over operads
Let $A$ and $B$ be $A_\infty$-algebras. It's true, but it's a quite nontrivial fact, that the tensor product $A \otimes B$ can be given the structure of $A_\infty$-algebra, too. What is much easier to ...
3
votes
0answers
76 views
Moduli spaces for the TCFT map $HH(L) \to GW(X)$
Let $L$ be a Lagrangian submanifold of a closed symplectic manifold $X$. What I gather from Costello (see specifically $\S$2.5 there), is that one expects to have a morphism of closed TCFT's
$\tag{1}...
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0answers
73 views
Proof-verification: Existence of an explicit formality morphism from the Barratt-Eccles Koszul dual cooperad
I know asking for proof-verification on MO is a tricky thing. On one hand interesting research level proofs are usually subject of articles and can not be discussed here in detail. On the other hand ...
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0answers
48 views
Morphism from the Koszul associative cooperad into the Koszul Lie cooperad?
Thinking about whether or not there is a natural way to transform $L_\infty$-algebras into $A_\infty$-algebras, I wonder if there is a morphism of cooperads
$\mathcal{A}ss^i\to\mathcal{L}ie^i$
from ...
5
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0answers
100 views
Question about terminology, and reference request related to the braid operad
Let $\Delta_n$ stand for the Garside element of the braid group $B_n$. It turns out that the family of all Garside elements have the following ``operadic'' property:
$$
\Delta_n\left[ \Delta_{k_1},\...
14
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0answers
434 views
Is this an $E_\infty$-algebra?
I have a particular kind of algebraic structure that's come up in my work. It's basically a chain complex equipped with a multiplication which is commutative and associative up to homotopy in a ...
3
votes
1answer
102 views
A differential graded Lie algebra with the Hochschild differential
Let $(V,\cdot)$ be an associative algebra and $W$ be a vector space endowed with a bimodule structure $\triangleright:V\otimes W\to W$ and $\triangleleft:W\otimes V\to W$ such that the following ...
2
votes
0answers
69 views
Algebras for (Koszul) Hopf operads
If necessary, we can restrict the following to the case where we consider only Hopf (co)operads in the category of chain complexes over fields of characteristic zero.
In case of ordinary operads, ...
2
votes
1answer
97 views
Pseudo or lax algebras for a 2-monad, reference request
I would like to find explicit definitions of pseudo, or even lax, algebras for a 2-monad, and their lax morphisms, with all the coherence diagrams included.
Alternatively, coherent lax algebras for ...
2
votes
1answer
115 views
Is there a Hopf structure on the dg-endomorphism operad?
This is a short question:
In the symmetric monoidal category of chain complexes (over a field if necessary), does the endomorphism operad carries a Hopf structure, i.o.w. can it be considered as a ...
1
vote
0answers
93 views
Obstruction theory on $A_{\infty}, C_{\infty}$-algebras
Let $\mathcal{P}_{\infty}$ be $A_{\infty}$ or $C_{\infty}$. Let $A=A^{1}\oplus A^{2}$ be a graded vector space concentrated in degree 1 and 2. Let $m_{n}\: : \:{A^{1}}^{\otimes n}\to A^{2}$ be a ...
5
votes
1answer
225 views
Monadic interpretation of coalgebras over operads
The structure of an algebra $A$ over a operad $O$ is encoded by an operad morphisms from $O$ to $\{Hom(A^{\otimes k},\, A)\}_{k}$. The same structure can be stored using the structure $M_OA\to A$ of ...
5
votes
0answers
112 views
Classification of formality morphisms for chains and Drinfel'd associators
In his 1997 preprint q-alg/9709040, M. Kontsevich proved constructively the existence of a $L_\infty$-quasi-isomorphism between the differential graded algebra structure on the deformation complex of ...
4
votes
1answer
161 views
3-Gerstenhaber algebra structure on the cohomology of deformation complexes?
In a seminal paper "On the Deformation of Rings and Algebras", M. Gerstenhaber showed that the deformation complex of any associative algebra (known as the Hochschild complex) is naturally endowed ...
11
votes
1answer
233 views
What is the relation between 2-Gerstenhaber, CohFT, and Gerstenhaber geometrically?
Background. As we know from Fred Cohen's Thesis, taking homology of the little 2-discs operad $\mathcal{D}_2$ with coefficients in a field of characteristic zero produces the Gerstenhaber operad $\...
2
votes
0answers
106 views
Monad induced by actegory
It seems to be folklore that if we have an actegory, i.e. a monoidal functor from a monoidal category $C$ to an endofunctor category $Cat(D,D)$, we can obtain from it a monad on $D$. This appears for ...
3
votes
0answers
70 views
What is the correct generalization of “sigma-free” to props?
This is a question about props, a generalization of operads (used to model operations with several inputs and several outputs).
By forgetting the composition structure of an operad one obtains a so ...
3
votes
1answer
162 views
Reference on Operads
I was reading 'Category for Scientists' by David Spivak and I'd like some references on 'Operads' with that kind of approach, using only "basic Category Theory", nothing too advanced.
I appreciate it ...
2
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0answers
84 views
Does the totality of $E_n$-operads in a given category has any interesting structure?
Suppose we are given a fixed ambient symmetric monoidal model category (I'm mostly interested in chain complexes over char zero fields). Then we have the notion of an $E_n$-operad in that category. ...
