All Questions
10
questions
11
votes
0
answers
159
views
Generalized $\infty$-operads are an analog of ??? in $\infty$-category theory
In Section 2.3.2 of Higher Algebra, Lurie introduces the notion of generalized $\infty$-operads. This is a functor $p:\mathcal{O}^\otimes \to \mathcal{F}\mathrm{in}_\ast$ of $\infty$-categories, where ...
- 2,154
5
votes
0
answers
323
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
259
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 $\infty$-...
- 2,446
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
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
207
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
3
votes
1
answer
116
views
Pseudo or lax algebras for a 2-monad, reference request
I would like to find explicit definitions of pseudo, or even lax, algebras for a 2-monad, and their lax morphisms, with all the coherence diagrams included.
Alternatively, coherent lax algebras for ...
- 2,129
6
votes
1
answer
352
views
Monadic interpretation of coalgebras over operads
The structure of an algebra $A$ over a operad $O$ is encoded by an operad morphisms from $O$ to $\{Hom(A^{\otimes k},\, A)\}_{k}$. The same structure can be stored using the structure $M_OA\to A$ of ...
- 537
8
votes
1
answer
628
views
Is there a "derived" Free $P$-algebra functor for an operad $P$?
Recall that an operad (in vector spaces, say) $P$ consists of a collection of vector spaces $P(n)$ for $n\geq 0$, such that $P(n)$ is equipped with an action by the symmetric group $S_n$, with maps $P(...
- 53.5k