The tag has no usage guidance.

learn more… | top users | synonyms

6
votes
1answer
306 views

Semidirect products and PROPs

$\newcommand{\p}{\mathcal{P}}$Let G be a group, and let $G_n$ denote the wreath product $G^n \rtimes \Sigma_n$. There seems to be a notion of a PROP $\p$ where the role of the symmetric groups ...
6
votes
1answer
562 views

Classifying spaces of E_1 - spaces

Hello, I try to understand aspects of homotopy coherence, in particular "recognition principle" of May. About the following I did not think a lot, but I decided to ask here anyway, so to save ...
4
votes
1answer
984 views

A-infinity structure of E-infinity algebras

This is perhaps somewhat related to this question. Fix a field $k$ of characteristic $p>0$. Suppose that $A$ is an $E_\infty$-algebra over $k$. Then $A$ also has an $A_\infty$-algebra structure, ...
16
votes
1answer
690 views

Higher homotopy algebraic structure on the homology of an operad

Given a DGA $A$, then by standard techniques such as homological perturbation theory, the ring structure on the homology $H(A)$ extends to a minimal $A_\infty$-algebra structure such that $H(A)$ is ...
8
votes
0answers
279 views

Question about $A_\infty$ maps

Given $A_\infty$-spaces $X$ and $Y$, Boardman and Vogt defined an $A_\infty$-map from $X$ to $Y$ to be a map $f: X \to Y$ of underlying based spaces and an $A_\infty$-structure on the reduced mapping ...
18
votes
1answer
3k views

What's up with Wick's theorem?

Sorry about the dumb title. I'd like to understand Wick's theorem. More specifically, I have seen it pop up in several different contexts and I am really puzzled by the different statements of it ...
1
vote
0answers
531 views

Recognition principle

Hello, The classical statement of the recognition principle (after Boardman, Vogt, Milgram and May) that I know is: Let $X$ be a (group-like) topological space acted on by the little $n$-discs ...
6
votes
1answer
490 views

How does Berger-Moerdijk's relative Boardman-Vogt work?

In "The Boardman-Vogt resolution of operads in monoidal model categories," the authors construct factorizations of sufficiently nice operad maps $P\to Q$ into a cofibration followed by a weak ...
4
votes
1answer
423 views

Repairing the Lie operad in characterstic 2?

Recall: We present an operad (with $S_n$-action) in $R-Mod$ for any commutative ring $R$ not of characteristic 2 generated by a single element in degree $2$ satisfying the following identities: ...
15
votes
2answers
3k views

Kontsevich's conjectures on the Grothendieck-Teichmüller group?

Reading Kontsevich's "Operads and Motives in Deformation Quantization", I was wondering about the state of the many conjectures concerning the Grothendieck-Teichmüller group in chapter 4. (Also, where ...
3
votes
1answer
636 views

Automorphisms of the rooted tree operad

This follows Ryan Budney's comment to the question asked here. What is the automorphism group of the rooted tree operad? (By the rooted tree operad, I just mean the operad with object rooted ...
1
vote
2answers
711 views

What are operad automorphisms?

What is the general concept of an *operad automorphism*$?$ Is there a "standard" definition? [added after comment] If an operad automorphism is an invertible operad endomorphism, how then is operad ...
16
votes
1answer
1k views

Little disks operad and $Gal (\bar {Q}/Q)$

My question is simple: How do the little disks operad and $Gal (\bar {Q}/Q)$ relate? I realize that a huge amount of heavy-machinery can be brought into an answer to this, but I'm struggling ...
12
votes
1answer
975 views

koszul duality and algebras over operads

Given a pair of Koszul dual algebras, say $S^*(V)$ and $\bigwedge^*(V^*)$ for some vector space $V$, one obtains a triangulated equivalence between their bounded derived categories of ...
6
votes
1answer
465 views

Do you recognise this variant of the cubes operad?

In the subject of operads one comes across many quirky variants of more or less the same operad. The cubes operad has many incarnations with various interesting properties. In a recent paper I came ...
26
votes
5answers
2k views

What are natural questions to ask about an operad?

I just discovered that something I've been working with has the structure of an operad. So I'm wondering what natural basic questions does one ask about operads? For example, if I knew I had the ...
2
votes
1answer
377 views

Extending a property of commutative algebras to C infinity algebras

If A is a commutative algebra and B is an X- algebra, then the tensor product $A \otimes B$ is an X-algebra (so for example, $Com \otimes Lie$ is a Lie algebra). This is seen using the language of ...
5
votes
2answers
653 views

Operad terminology - Operads with and without O(0).

In the Markl, Schneider and Stasheff text, topological operads are an indexed collection of spaces $O(n)$ for $n \in \{1,2,3,\cdots\}$ satisfying some axioms. In May's text, the index set is allowed ...
29
votes
3answers
2k views

Algebras over the little disks operad

Hello, The so-called "recognition principle" of Boardman-Vogt and May leaves me unsatisfied. My problem is the following: The "recognition principle" says that every "group-like" algebra over the ...
13
votes
1answer
627 views

Are G_infinity algebras B_infinity? Vice versa?

What is the relationship between $G_\infty$ (homotopy Gerstenhaber) and $B_\infty$ algebras? In Getzler & Jones "Operads, homotopy algebra, and iterated integrals for double loop spaces" (a paper ...
9
votes
2answers
643 views

Right actions of operads and monads

Given an operad $A$, there is an associated monad $M_A$ given by $M_A(X) = \coprod_n A(n) \otimes X^{\otimes n}$ such that being an $A$-algebra and being an algebra over the monad $M_A$ is the same ...
8
votes
2answers
581 views

Is the cohomology of a topological operad a cooperad?

For cohomology with coefficients in a field $F$ the map $H^\cdot(X;F) \otimes H^\cdot(Y;F) \to H^\cdot(X \times Y;F)$ of the Kunneth theorem is an isomorphism of algebras over $F$. I am correct in ...
6
votes
1answer
350 views

Transmutation versus operads

A while ago, I was reading Majid's book Foundations of quantum group theory, and Section 9.4 has a rather fascinating description of a Tannaka-Krein reconstruction result for quantum groups. In ...
9
votes
2answers
307 views

When do PROP-morphisms induce adjunctions?

If (C,tensor,1) is a symmetric monoidal category and f:A-->B is a morphism of PROPs (or monoidal cats = colored PROPs), one gets a forgetful functor f^*:B-Alg(C)-->A-Alg(C) (where ...
6
votes
1answer
448 views

Pontryagin product from an operad

For a topological group G, we have a Pontryagin product in homology by multiplying representative cycles. This gives the homology the structure of an associative graded algebra. Am I correct in ...
15
votes
4answers
2k views

Deligne's conjecture (the little discs operad one)

Deligne's conjecture states that the Hochschild cochain complex of an A-infinity algebra is an algebra over the operad of chains on the topological little discs operad. Of course the conjecture has ...