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11
votes
1answer
377 views

Condition on a Hopf operad for tensor product in the base category to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...
5
votes
1answer
215 views

Explict form of $E_\infty$-morphisms between differential graded commutative algebras

This is a partial duplicate to this MO question, I apologize for that. I'm asking since the answers there still do not allow me to work out an answer to my question, which is a bit more specific. ...
14
votes
0answers
542 views

Splitting of homomorphism from cactus group to permutation group

We all learned in kindergarten that the category of finite-dimensional (type I, say) $U_q(\mathfrak{g})$-modules is braided monoidal for $\mathfrak{g}$ a complex semisimple Lie algebra. This gives an ...
9
votes
0answers
196 views

Does the $(\mathbb Z/2)$-graded isomorphism $E_n \cong E_{n+2}$ have any nice properties?

This question assumes everything is dg. Let's decide to work over the "field" $\mathbb Q[\mu,\mu^{-1}]$ where $\mu$ has homological degree $+2$. Then chain complexes are just $\mathbb Z/2$-graded. ...
9
votes
0answers
173 views

Are pivotal categories the algebras for a cartesian monad?

It seems to be "known" but not written down that the following are more-or-less equivalent: ...
8
votes
0answers
294 views

Galois action on $E_n$-operads

Let $E_n$ be the little $n$-cubes operad which acts on $n$-fold loop spaces (up to group completion, an $E_n$-action is precisely the data needed to perform an $n$-fold delooping). I am looking for ...
8
votes
0answers
253 views

The spheres operad

I have a rather naive question. Consider the space of all maps $$ S^{j_1} \times S^{j_2} \times \cdots \times S^{j_k} \to S^n $$ for all possible natural numbers $n, k, j_1, \cdots , j_k$. This ...
8
votes
0answers
287 views

When can I assure that the representation theory of a PROP is faithful?

Recall that a PROP is a symmetric monoidal category whose object-set is freely $\otimes$-generated by a single object $x\in P$. A representation or algebra for a prop $P$ in a symmetric monoidal ...
8
votes
0answers
272 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 ...
7
votes
0answers
215 views

A model category for E-infty algebras in a non-monoidal model category?

Given a suitable nice symmetric monoidal category $C$, symmetric monoidally enriched, tensored, and cotensored over a symmetric monoidal category $S$, and an operad $\mathcal{O}$ in $S$, we can ...
5
votes
0answers
48 views

Cyclic structure on a balanced (or ribbon) monoidal category

As it is well known, a balanced (and in particular ribbon) monoidal category is an algebra over the framed little 2-discs operad. The latter is homotopy equivalent to the operad of moduli space of ...
4
votes
0answers
184 views

Where is the Courant operad discussed?

Where is the Courant operad discussed? And hopefully defined precisely. By the Courant operad or rather a suitable generalization of operad to accommodate the inner product, the operad whose ...
4
votes
0answers
244 views

Does abstract nonsense of model categories determine the “nonlinear” morphisms of $L_\infty$ algebras?

Recall that a Lie algebra is a module for the operad $\mathrm{Lie}$, which is freely generated by a binary antisymmetric operation $\beta$ modulo an equation that is quadratic in $\beta$. There is a ...
3
votes
0answers
162 views

Straightening for $\infty$-operads

There is this straightening/unstraightening procedure of Jacob Lurie's which takes a symmetric monoidal $\infty$-category (which is the data of a coCartesian morphism of simplicial sets $C^\otimes\to ...
3
votes
0answers
198 views

Boardman Vogt W construction for modules over an operad

The W construction of Boardman and Vogt gives a cofibrant replacement for operads. In http://arxiv.org/abs/math/9907073, Salvatore describes a cofibrant replacement for algebras over an operad. Is ...
2
votes
0answers
70 views

Errata for Getzler-Kapranov “Cyclic operads and Cyclic homology”

Do you know if anyone made an errata for the Getzler-Kapranov paper "Cyclic operads and Cyclic homology"? I was trying to read it, and I have found (at least I think I did) quite a few typos not all ...
2
votes
0answers
141 views

What is a morphism of $B_\infty$ algebra

Let $k$ be a commutative ring and $C$ a $ \mathbb{Z}$ graded $k$ module. A $B_\infty$ structure on $C$ is the datum of a differential and a multiplication on $ BC:= \bigoplus^\infty_{i =0} ...
1
vote
0answers
95 views

Finitely co/continuous monad induced by an operad

It is well known that any operad on a nice monoidal category induces a monad. I was wondering which conditions can be imposed on the operad $T$ (say, in a monoidal closed $\cal V$) so that the ...
1
vote
0answers
100 views

Exact sequence of L-infinity-algebras

We call a sequence of $L_\infty$-algebras (weak) maps $$0\to L\xrightarrow{f} M\xrightarrow{g} N\to 0$$ is exact if it is exact on the the underlying chain complexes level. Thought I don't know ...
1
vote
0answers
82 views

Generating series of free PROs

Let \begin{equation} G := \biguplus_{p \geq 0} \: \biguplus_{q \geq 0} G(p, q) \end{equation} be a bigraded set of generators and $\mathcal{F}(G)$ be the free PRO generated by $G$ (see [1] for a net ...
1
vote
0answers
484 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 ...
0
votes
0answers
349 views

[]-infinity algebra and Projective representation

This is a very vague question. We know that some algebra structures can be viewed as modules of some fantastic stuff, call T. Such examples include: Abelian groups are $\mathbb{Z}$-modules, chain ...