All Questions
Tagged with higher-category-theory operads
26
questions
6
votes
0
answers
140
views
A "lax Boardman-Vogt tensor product," or what object represents duoidal categories?
Let me preface this by saying I'm not sure what the fundamental examples should be, and perhaps that's part of my question.
The Boardman-Vogt tensor product of $\infty$-operads $\mathcal{O}$ and $\...
- 221
3
votes
2
answers
238
views
Is the free algebra functor over an $\infty$-operad symmetric monoidal?
Suppose $F: \mathcal{O}^\otimes \to \mathcal{P}^\otimes$ is a map of $\infty$-operads, and $\mathcal{C}$ is a symmetric monoidal $\infty$-category that admits small colimits, such that the tensor ...
- 894
5
votes
0
answers
322
views
What is an $\infty\text{-}E_{\infty}$ morphism?
My question is essentially what the title says, but here is some background that I have gathered from skimming through the literature. Please feel free to correct me if my understanding is wrong at ...
- 235
7
votes
1
answer
258
views
$(\infty,n)$-operads?
I wonder whether there is (or should be) a theory of colored $(\infty,n)$-operads or multicategories?
We know that multicategories are generalizations of categories, and nonsymmetric colored $\infty$-...
- 2,416
4
votes
0
answers
113
views
For which operads $O$ does $\operatorname{coAlg}_O(C) = C$ whenever $C$ is cartesian monoidal?
Let $O$ be an operad, and let $D$ be a symmetric monoidal category. Then there is a forgetful functor $\operatorname{Alg}_O(D) \to D$. This functor is an equivalence in either of the following cases:
...
- 62.6k
3
votes
0
answers
133
views
Transporting $\mathbb E_n$-monoidal structures between categories
Suppose given an $\mathbb E_n$-monoidal presentable $\infty$-category $\mathcal C$ (wrt. the Lurie tensor product $\otimes$), and $\mathcal D$ a presentable $\infty$-category. Suppose given a pair of ...
- 435
3
votes
0
answers
133
views
Riemann-Hilbert-type correspondence for locally constant factorization algebras
This is related to a previous post, but a bit softer and should probably stand on its own.
In Appendix A of "Higher Algebra", Lurie shows that for a reasonably good topological space, there ...
- 894
6
votes
1
answer
386
views
$\mathbb{E}_M$ as colimit of little cubes operads
In Lurie's "Higher Algebra", Remark 5.4.5.2 towards the end, there is the following statement: "It follows that $\mathbb{E}_M$ can be identified with the colimit of a diagram of $\infty$...
- 894
3
votes
0
answers
181
views
Augmented algebras over $\infty$-operads via the envelope
Let $\mathcal{O}^\otimes$ be an $\infty$-operad and $\mathcal{C}^\otimes$ a symmetric monoidal $\infty$-category, both in the sense of Lurie's Higher Algebra.
By augmented $\mathcal{O}^\otimes$-...
- 1,072
5
votes
0
answers
136
views
Definition of $E_{n}$-operad in dgCat
In "Derived Algebraic Geometry and Deformation Quantization" Toën defines in 5.1.2 an $E_{n}$-monoidal A-linear dg-category as an $E_{n}$-monoid in the symmetric monoidal $\infty$-category $...
- 1,447
2
votes
0
answers
176
views
What is an invertible operad?
Let $\mathcal V$ be a nice symmetric monoidal ($\infty$-)category, and consider the ($\infty$-)category $Op(\mathcal V)$ of $\mathcal V$-enriched (symmetric) operads, symmetric monoidal under the ...
- 62.6k
7
votes
0
answers
204
views
Higher categorical / operadic approach to homotopy associative, homotopy commutative, $H_\infty$ ring spectra?
Let $\mathcal C$ be a symmetric monoidal $\infty$-category, and let $O$ be an operad (for example, $O$ could be an $A_m$ or $E_n$ operad or a tensor product thereof, and $\mathcal C$ could be spaces ...
- 62.6k
5
votes
0
answers
103
views
Is there a n-category structure on algebras for $e_n$-like operads?
I'm fishing in troubled waters here and therefore the question is vague and meant to be as general as possible. In particular "$e_n$-like operad" can be an algebraic or topological $e_n$ operad, as ...
- 2,034
5
votes
0
answers
197
views
The notion of $\infty$-Cooperads for which Bar-Cobar duality is an equivalence
In the paper Bar-Cobar Duality by Michael Ching, he proves that the category of operads in spectra is equivalent via the Bar-Cobar adjunction to some model category of co-operads defined in the paper. ...
- 7,679
9
votes
2
answers
390
views
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 ...
- 10.3k