The operads tag has no usage guidance.

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### Does this notion related to species/operads/FI-modules have a name?

Let $B$ be the symmetric monoidal category of finite sets and bijections with disjoint union. Let $C$ be a symmetric monoidal category. Is there a standard name for a lax monoidal functor $F:B \to C$? ...

**3**

votes

**2**answers

342 views

### Definition of E-infinity operad

What is the definition of $E_\infty$-operad in the category of chain complexes over $\mathbb{Z}/p\mathbb{Z}$? J. Smith in http://arxiv.org/abs/math/0004003 define it for complexes over $\mathbb{Z}$ ...

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votes

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68 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 ...

**3**

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**0**answers

124 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 ...

**1**

vote

**1**answer

150 views

### $E_{\infty}$ algebra in characteristic zero

I asked this question on MSE(http://math.stackexchange.com/q/1579026/239218).
Let $A^{\bullet}$ be a cosimplicial commutative algebra over a field $\Bbbk$. Denote with $N(A)^{\bullet}$ the ...

**18**

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**1**answer

278 views

### Positivity of coefficients of the inverse of a certain power series

Consider the unique formal power series $g(z)$ with $g(0)=0$ and $g'(0)=1$ satisfying the equation
$$
g(z)-g(z)^8+g(z)^{15}=z,
$$
that is the inverse of
$$
z-z^8+z^{15}
$$
in the group of formal ...

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votes

**0**answers

135 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 ...

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**1**answer

188 views

### Operads and the Stable Module Category

I am seeking references to places where operads and their algebras have been studied for the stable module category. Colored operads are fine too.
Let $k$ be a field and $R$ a $k$-algebra. The stable ...

**1**

vote

**1**answer

174 views

### Is there an operad that codifies groupoids?

maybe this question is trivial and, then this is the reason I've never seen this written.
The motivation is to define internal $\infty$-groupoids (that are preferably) Kan fibrant and to see if Kan ...

**9**

votes

**1**answer

236 views

### Are $E_n$-operads not formal in characteristic not equal to zero?

This is a short question:
Is it just unproven folklore (yet), or is it definitively known that $E_n$-operads are not formal, if the characteristic of the underlying field is not equal to zero?

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**0**answers

45 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 ...

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votes

**1**answer

108 views

### bijection between S-modules and Schur functors

Given a $\mathbb{S}$-module, we can construct a Schur functor, which is an endo functor in the category of vectorspaces. But given a Schur functor, how can we get back the $\mathbb{S}$-module? In ...

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votes

**1**answer

196 views

### An example for a construction on monads/operads?

Suppose that $C$ is a bicategory. (I only need a monoidal category, i.e. one object bicategory, but I will stick with bicategories, since theory of monads is more commonly stated in that setting). A ...

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**0**answers

446 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 ...

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**1**answer

252 views

### Tensor products over operads and bar constructions

Let $O$ be an operad in spaces, $A$ an $O$-algebra and $R$ an right $O$-module. One can define $R \otimes_O A$ as the coequalizer of the two maps $ROA$ to $RA$. One can also define $B(R,O,A)$ (as in ...

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474 views

### What are algebras for the little n-balls/n-cubes/n-something operads exactly?

As a non expert in the theory of topological operads, I find it pretty hard, to understand what algebras for little balls/cubes/something operads are.
For all the other famous operads I know (like ...

**0**

votes

**1**answer

118 views

### iterated loop spaces and configuration spaces [closed]

In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map
$$
\phi: Hom_T(X,\Omega Y)\to Hom_T(SX,Y)
$$
is defined. And a map
$$
...

**13**

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**1**answer

387 views

### Homotopy transfer in the opposite direction

Let $X\rightleftarrows Y\circlearrowleft$ be a strong deformation retraction of chain complexes (a.k.a. contraction), i.e. $X\rightarrow Y\rightarrow X$ is the identity, $Y\rightarrow Y$ is a homotopy ...

**1**

vote

**0**answers

204 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 ...

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2k views

### Why should have Peter May worked with CGWH instead of CGH in “The Geometry of Iterated Loop Space”?

This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).
The weak Hausdorff rather ...

**8**

votes

**1**answer

293 views

### When is a quasicategory over $N(\Delta)^{op}$ a planar $\infty$-operad?

In Lurie's DAG II, a notion of monoidal $\infty$-category is given that differs from the notion given in his later book Higher Algebra. In the former, the relevant structure is a cocartesian ...

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**1**answer

227 views

### Reference Request: Grouplike Algebras over the little $n$-cubes operad are $n$-fold loop spaces

In Geometry of the iterated loop space, Peter May proved his famous recognition theorem, which is, in a simple form, stated on page 3 as the following.
There exist $\Sigma$-free operads ...

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68 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 ...

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**1**answer

340 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.
...

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**1**answer

185 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 ...

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votes

**1**answer

498 views

### Boardman-Vogt tensor product

Let $\mathbf{sSet}$ be the model category of simplicial sets and $\mathbf{Op}$ the model category of symmetric operads. Equipped with Boardman-Vogt tensor product $ \otimes_{BV}$, the category ...

**14**

votes

**1**answer

466 views

### Free Loop-Space Recognition Principle

It is well-known that one can detect based loopspaces using the machinery of operads. Namely, given a group-like space $X$ with an action of $\mathbb{E}_n$-operad, then it is homotopy equivalent as an ...

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**1**answer

335 views

### Tensor product of dendroidal sets: counter-examples

For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$.
...

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250 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 ...

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votes

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314 views

### Has this construction, which builds a symmetric multicategory from a commutative monoid, been described or studied anywhere, and if so, where?

Whenever $R$ is a commutative ring, write $R[x^{(n)}]$ for the set of all $p \in R[x]$ such that $p$ is a monic polynomial of degree $n$. Then $R[x^{(n)}]$ is not closed under sums, nor does it ...

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239 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 ...

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221 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. ...

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votes

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397 views

### Obstructions for $E_n$-algebras

In Alan Robinson's paper, Classical Obstructions and S-algebras, he provides conditions for a ring spectrum to have an $A_n$ and $\mathbb{E}_\infty$-structure.
Have the obstructions for an object ...

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votes

**2**answers

444 views

### $E_n$-space and n-connected pointed space

Is it true that the homotopy category of group-like $E_n$-spaces is equivalent to the homotopy category of pointed $n$-connected spaces ? If it is true, what should be the statement when ...

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votes

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475 views

### Is the Amitsur-Levitzki identity essentially unique?

Let us consider the matrix algebra. $Mat_n(\mathbb{C})$. The Amitsur-Levitzki identity states that for any matrices $X_1, X_2, ..., X_{2n} \in Mat_n(\mathbb{C})$ the sum $\Sigma_{\sigma \in S_{2n}} ...

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**1**answer

403 views

### $E_{\infty}$ spaces are $A_{\infty}$ spaces

While studying the well-known "Geometry of Iterated Loop Spaces", I found this corollary which is not completely clear to me. (By $\mathcal{M}$ is meant the operad given by $\mathcal{M}(j):=\Sigma_j$, ...

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107 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 ...

**9**

votes

**1**answer

376 views

### $k$-Disk algebras versus $E_k$ algebras

Background:
The little $k$-cubes operad is the $(\infty,1)$-operad defined by embedding disjoint unions of $k$-dimensional open cubes rectilinearly into one another, that is using maps ...

**4**

votes

**1**answer

435 views

### Is there an analog of the Barratt-Eccles construction for group-like E_∞-spaces and E_∞-ring spaces?

The Barratt-Eccles operad is an operad in simplicial sets
that provides a particularly nice model of an E∞-operad;
algebras in spaces over the Barratt-Eccles operad model E∞-spaces,
i.e., homotopy ...

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vote

**0**answers

149 views

### Reference request: modern proof of the recognition principle [closed]

I'm looking for a modern version of P. May's proof of the recognition principle (in "geometry of iterated loop spaces"). Can someone help me?

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**1**answer

342 views

### What is this operad-like structure called?

I'd like to know what's the name (if any) of the following categorical structure, and also references where it has been considered.
Given a category $C$, let $O=\{O(n)\}_{n\geq 0}$ be a sequence of ...

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154 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} ...

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445 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. ...

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**1**answer

514 views

### understanding the definition of $\infty$-operad of module objects

I'm just trying to understand the following definition:
Definition 3.3.3.8 in Higher Algebra by J. Lurie defines the $\infty$-operad of $O$-module objects, and says the following:
Let $O^\otimes$ be ...

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**1**answer

316 views

### What's a good reference for the following obstruction theory yoga?

Fix a colored operad, which I will leave implicit, and a field $\mathbb K$ of characteristic $0$. By algebra in this post I will mean a dg algebra over $\mathbb K$ for the given colored operad. I ...

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vote

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226 views

### comparison between two monadic definitions for an operad

According to May, an operad $\mathcal{C}$ valued in sets is equivalent to a monad in Cat on the endofunctor $C\colon X\mapsto \coprod_i \mathcal{C}(i)\times X^i.$
According to Leinster, an operad is ...

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votes

**0**answers

190 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 ...

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votes

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333 views

### Correspondence between operads and monads requires tensor distribute over coproduct?

In checking the details of the correspondence between operads over a symmetric monoidal category and monads on some associated endofunctor of the category, I cannot make the obvious proof work without ...

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545 views

### Model structure for cooperads and for coalgebras

I am studying the homotopy theory of (algebraic) operads and I came up with several questions I am unable to answer to. I would like to stress that I don't have applications in mind, I just would like ...

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**1**answer

259 views

### A question on the definition of operad

The nlab page says
A (Set-based) operad is a monoid in the monoidal category
$(Psh(ℙ),∘,I)$, where $ℙ$ is the category of $\sqcup_{n\ge 0} S_n$.
The monoidal structure is given by the so ...