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14
votes
0answers
596 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
190 views

Grothendieck Construction, Categories of Operators and Opposites

Given a symmetric monoidal category $C$, we can construct its endomorphism operad (or multicategory) $End(C)$ whose objects are the objects of $C$, and for which the multimorphisms from $\{c_1,\ldots,...
9
votes
0answers
230 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
300 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 ...
9
votes
0answers
184 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
212 views

When does an $E_\infty$ algebra come from a commutative differential graded algebra?

Suppose that $K$ is an $E_\infty$-algebra on a space $X$ (more generally, any ringed topos; also, feel free to assume that $X$ is a point). That is, $K$ is a cochain complex of sheaves on $X$, endowed ...
8
votes
0answers
338 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
319 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
286 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
261 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 ...
7
votes
0answers
362 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 ...
6
votes
0answers
146 views

When are the categories of algebras over props (co)complete?

Suppose P is a (colored) prop in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable? It seems that one can relatively easy ...
5
votes
0answers
55 views

Intuition for resolutions of operads

Is there an intuitive way to understand resolutions of operads for someone not well-versed in model categories? In particular, I'd like to have some idea, even if not a complete understanding, of how ...
5
votes
0answers
74 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
73 views

Invert quasi-isomorphisms of symmetric cooperads

The theory of symmetric operads in chain complexes (say over a good enough field) is in some sense nice, because we have a well defined homotopy theory. In particular we have a notion of infinity-...
4
votes
0answers
191 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 ...
3
votes
0answers
124 views

Can ring spectra be thought of as some sort of operad in $Top$?

It is a result of May's work on operads that the homotopy category (or $\infty$-category, if you prefer) of connective spectra is equivalent to a full subcategory of the category of representations of ...
3
votes
0answers
170 views

Complete Segal operads and dendroidal sets

There is a Quillen equivalence between the model category presenting Lurie's $\infty$-operads (which are inner fibrations $\mathcal{C}\to\mathrm{N}(\mathbf{F})$ satisfying certain conditions) and the ...
3
votes
0answers
130 views

Finiteness for 2-dimensional contractible complexes

While thinking about graph-complex and related operadic stuff, I found a quite interesting (at least for me) question. However, I'm a novice in the algebraic topology, so I'm unable to resolve it by ...
3
votes
0answers
256 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
221 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
227 views

E-infinity operads explicit examples

I was looking for particular and explicit examples of $E_\infty$-operads. I know the $E_\infty$-operad defined by Smith in http://arxiv.org/abs/math/0004003, and the Barratt-Eccles operad, but it is ...
2
votes
0answers
65 views

Composition law in the colored operad whose algebras are (symmetric) operads

In Berger-Moerdijk, Resolution of coloured operads and rectification of homotopy algebras (1.5.6) there is a description of the operad, whose algebras are non-colored symmetric operads. I have some ...
2
votes
0answers
158 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} C[1]^{\...
1
vote
0answers
74 views

suspension operad

There are many candidates of suspension operad in literature. Among them, $\Lambda= {\rm End}_{s\mathbb{K}}$ and $\Lambda'=\{s^{1-n}\mathbb{K}\otimes sgn_n\}_{n\geq 0}$ are typical ones. The oeprad ...
1
vote
0answers
39 views

Is it sufficient for all “critical monomials” to be confluent for an operadic rewriting law to induce a distributive law?

See [LV] Loday–Vallette, Algebraic Operads, Section 8.6 for all the definitions. Let $\mathtt{P} = \mathbb{F}(V)/(R)$ and $\mathtt{Q} = \mathbb{F}(W)/(S)$ be two quadratic operads presented by ...
1
vote
0answers
114 views

Structures on the free path space object

Let $X$ be a connected topological space. Then its based loops space $\Omega X$ is an examples of $A_{\infty}$ space. Let $PX$ be the free paths space. Which type of homotopical structures are ...
1
vote
0answers
282 views

Does the Boardman-Vogt tensor product of operads commute with their W-construction

I have absolutely no idea whether this is true or not but it could well be useful for me in the future if it is. If we have topological operads $\mathcal{P}$ and $\mathcal{Q}$ and we let $W$ denote ...
1
vote
0answers
112 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
113 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
86 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
557 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
462 views

maps from labelled configuration space to section space / iterated loop space

In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, 2014, page 3, Section 3: for a $m$-manifold $M$, consider the disc bundle $D(M)$ in the tangent bundle $T(M)$...
0
votes
0answers
366 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 ...