Questions tagged [operads]
The operads tag has no usage guidance.
329
questions
11
votes
1
answer
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Conceptual (operadic?) reason for the generalized EHP fiber sequence $J_{q-1}(S^{2n}) \to J S^{2n} \to JS^{2nq}$?
Let $q$ be a prime and $q=p^r$ a power. Then there is a $p$-local fiber sequence from the $q-1$st stage of the James construction on $S^{2n}$, to $J(S^{2n}) = \Omega \Sigma S^{2n}$, to $J(S^{2nq}) = \...
- 61.6k
5
votes
1
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196
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Topological category of topological monoids / operads
The category of topological monoids can be made into a topological category in a naive way. Namely, the space of all continuous homomorphisms between two topological monoids is a closed subspace of ...
- 53
9
votes
2
answers
548
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Obstructions to $E_2$-algebra structure on $E_1$-algebra
Let $A$ be an $E_1$-algebra in chain complexes over $\mathbb Q$.
Is there an easy way to check if $A$ admits the structure of an $E_2$-algebra (or $E_\infty$-algebra)?
- 925
3
votes
0
answers
103
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Generating an enriched multicategory
Let $C$ be an $(M,\otimes,1)$-enriched category. I am looking for a reference for a notion of “generating the morphisms of $C$” (for ordinary categories, but also for multicategories, see below).
My ...
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6
votes
1
answer
324
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Free operad over a monoid object
Let $\mathcal{O}$ be an operad in the monoidal category $M$. Then $\mathcal{O}(1)$ together with the morphisms
$$\mathcal{O}(1)\otimes \mathcal{O}(1)\to \mathcal{O}(1)$$
and the unit $\eta:1\to \...
- 1,623
4
votes
0
answers
187
views
Dyer–Lashof operations for more than 2 inputs
Let $\mathcal{O}$ be a topological operad and $X$ an algebra over it. Let the base ring be $\mathbb{Z}_2$. If $C_*$ denotes the singular chain complex over $\mathbb{Z}_2$, the action of $\mathcal{O}$ ...
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5
votes
1
answer
257
views
Graded commutativity of the $n$th Browder bracket
Let $\mathcal{O}$ be a topological operad and $X$ an algebra over $\mathcal{O}$. Then $H_*(X)$ is an algebra (in the category of $\mathbb{Z}$-graded $R$-modules) over $H_*(\mathcal{O})$. Each $e\in ...
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6
votes
1
answer
194
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Two monoidal structures and copowering
Let $(\mathbf{M},\otimes,1)$ be a closed monoidal category and $(\mathbf{C},\oplus,0)$ an $\mathbf{M}$-enriched monoidal category. Furthermore, assume that we have a copowering $\odot:\mathbf{M}\times\...
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1
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0
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51
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On the notation $C[\lambda]$ where $C$ is a free cooperad in a proof of formality (and other details)
In his paper where details for Tamarkin's proof of formality are given, Hinich considers a Koszul quadratic operad $P$, a graded $P$-algebra $H$, a $P_\infty$-algebra $X$ with $HX=H$ (as $P$-algebras ...
- 1,534
6
votes
1
answer
656
views
Bracket systems (generalization of Poisson brackets)
Related to Why symplectic geometry gives Poisson geometry by coming at it from the other side.
This isn't as fully formalized as it probably should be, but I think enough of the idea is there to ask ...
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1
vote
1
answer
200
views
The table reduction morphism of operads from Barratt-Eccles to Surjection
The Barratt-Eccles operad $E$ in the category of simplicial sets is obtained by applying the nerve functor to the canonical operad $\{\Sigma_n\}_{n>0}$ in groups. Berger-Fresse defined here an ...
- 537
6
votes
1
answer
234
views
"Left Brace Module"
Let $A$ be an algebra over the brace tree operad and $M$ a module over some base ring.
Is there a good notion of a "left brace module" over a brace algebra?
I do not think the definition of a module ...
- 517
3
votes
1
answer
89
views
$H$-space structure on coloured algebras
If $\mathcal{O}$ is a (classical) topological operad with unit $1\in \mathcal{O}(1)$, $\mathcal{O}(0)=\{0\}$ and multiplications $(m;a_1,\dotsc,a_r)\mapsto m(a_1,\dotsc,a_r)$. Let $X$ be an algebra ...
- 1,623
6
votes
1
answer
303
views
Operad structure on Kontsevich's admissible graphs
In his celebrated 97' preprint q-alg/9709040, M. Kontsevich constructs a $L_\infty$-quasi-isomorphism $\mathcal U:\mathcal D_{\rm poly}\to\mathcal T_{\rm poly}$ between the differential graded algebra ...
12
votes
1
answer
585
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 "...
- 1,661
2
votes
0
answers
180
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$E_\infty$-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 ...
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260
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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 ...
- 18.6k
19
votes
4
answers
1k
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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 ...
- 1,635
11
votes
1
answer
508
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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 $H(A)$...
- 39.3k
7
votes
1
answer
397
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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 ...
- 2,719
8
votes
2
answers
238
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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 ...
- 9,052
9
votes
1
answer
630
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"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, ...
- 4,436
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0
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315
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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 ...
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19
votes
2
answers
955
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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
$$F_{Comm_\ast}(z) = \exp(z)-1 \qquad F_{Lie}(z) = \ln(1-z)$$
...
- 61.6k
9
votes
2
answers
386
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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 ...
- 10.1k
2
votes
1
answer
224
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 ...
- 1,372
8
votes
1
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839
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 ...
- 1,372
5
votes
0
answers
266
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 ...
3
votes
0
answers
170
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 $End_V$ ...
5
votes
1
answer
101
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 ...
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177
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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 ...
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17
votes
2
answers
680
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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 ...
- 7,902
3
votes
1
answer
268
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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 ...
- 368
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votes
0
answers
232
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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 ...
- 343
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votes
1
answer
410
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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) $k$-...
- 725
1
vote
0
answers
172
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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 ...
- 87
6
votes
1
answer
291
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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 ...
- 517
11
votes
3
answers
241
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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 ...
- 61.6k
5
votes
2
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484
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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 ...
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2
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409
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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 \}...
- 4,840
2
votes
0
answers
107
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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 ...
- 367
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votes
0
answers
167
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 ...
- 513
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votes
1
answer
426
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 $\...
- 1,221
5
votes
1
answer
364
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 ...
- 1,223
9
votes
2
answers
478
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 ...
- 2,270
10
votes
1
answer
625
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 ...
- 39.3k
4
votes
0
answers
107
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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}...
- 343
1
vote
0
answers
101
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 ...
- 2,014
1
vote
0
answers
75
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 ...
- 2,014
5
votes
0
answers
168
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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},\...
