# Questions tagged [operads]

The operads tag has no usage guidance.

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### 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 "...

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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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### 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 ...

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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 ...

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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 $H(A)$...

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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 ...

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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 ...

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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, ...

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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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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
$$F_{Comm_\ast}(z) = \exp(z)-1 \qquad F_{Lie}(z) = \ln(1-z)$$
...

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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 ...

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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 ...

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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 ...

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### 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 ...

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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 $End_V$ ...

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### 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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### 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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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 ...

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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 ...

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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 ...

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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) $k$-...

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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 ...

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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 ...

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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 ...

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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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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 \}...

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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 ...

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### 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 ...

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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 $\...

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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 ...

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### 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 ...

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### 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 ...

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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}...

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### 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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### 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 ...

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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},\...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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 ...

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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 ...

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### 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. ...