32
votes
Accepted
How many natural operations on subsets are there?
Here is a classification: Natural operations $\tau : P(-)^n \to P(-)$ correspond to maps $M : P([n]) \to P([n])$ that are deflationary, i.e. $M a \subseteq a$ for each $a$.
A deflationary map $M$ on $...
- 21.3k
22
votes
Accepted
Why are operads sometimes better than algebraic theories?
First - yes, for symmetric set-operads this functor is "injective", though it is not fully faithful. It is faithful on general maps and fully faithful on isomorphisms. Its image can easily ...
- 42.4k
18
votes
Accepted
Does Koszul duality between $Comm$ and $Lie$ imply the power series identity $\exp(\ln(1-z))-1 = -z$?
Let me flesh out the answer a little. The general statement is given by Theorem 7.5.1 in the book Algebraic Operads by Loday and Vallette.
First a definition. Let $P = P(E,R)$ be a quadratic operad, ...
- 5,040
15
votes
Accepted
Homotopy theories of operads
The answer is yes: see the paper of Chu-Haugseng-Heuts, "Two models for the homotopy theory of ∞-operads", arXiv:1606.03826.
In brief, already Cisinski and Moerdijk ("Dendroidal sets and simplicial ...
- 40.2k
13
votes
Accepted
What is the interpretation of the Gerstenhaber bracket?
I think the most transparent interpretation is by identifying $E_2$ algebras with brace algebras (which was proved by McClure and Smith).
Namely, a brace algebra satisfies the relation
$$ab - (-1)^{|...
- 3,357
13
votes
Does Koszul duality between $Comm$ and $Lie$ imply the power series identity $\exp(\ln(1-z))-1 = -z$?
Yes. (Mathoverflow won't let me make this my total answer, so ...)
Koszul duality says that a certain chain complex of graded vector spaces is acyclic. Thus the alternating sum of the Poincare ...
- 11.1k
13
votes
Accepted
"Exactness" of operadic cohomology
The second result you mention is significantly easier than the first. Indeed from the PBW theorem we know that $L$ is a direct summand of $UL$ as an $L$-module, and the second result follows. But in ...
- 40.2k
13
votes
Accepted
Tensor product of a DGA and an $A_\infty$ algebra
In fact the tensor product of two $A_\infty$ algebras can be made into an $A_\infty$ algebra in an explicit way: there are two constructions, one by Saneblidze-Umble and one by Loday. See the paper ...
- 40.2k
12
votes
Are there prominent examples of operads in schemes?
Mikhail Kapranov's invited lecture at the 1998 ICM was about exactly this:
Mikhail Kapranov, Operads and algebraic geometry. Documenta Mathematica Extra Volume ICM II (1998), 277-286. (PDF of the ...
- 27.7k
12
votes
Homotopy Gerstenhaber algebras: description via operads vs derivations
The equivalence between these two notions is nontrivial, since it amounts to a choice of formality isomorphism for the operad of little disks.
Let $D_2$ be the little disks operad. The easy part ...
- 3,928
11
votes
Are there prominent examples of operads in schemes?
The homology of $M_{0,n}$ is an operad in a natural way that is compatible with AG structure (like weights). The construction uses the operad structure on the compactification plus Gysin maps-- it is ...
- 3,928
11
votes
How many natural operations on subsets are there?
In a comment Peter LeFanu Lumsdaine has conjectured a classification, which shows that there are $16$ (!) binary subset operations. They can be combined from $1 \cdot 2 \cdot 2 \cdot 4$ independent ...
10
votes
Are there prominent examples of operads in schemes?
I recommend reading the paper Mixed Hodge structures and formality of symmetric monoidal functors by Cirici and Horel. They give several examples, and the paper is a great motivation for why one would ...
- 5,040
9
votes
Accepted
Is there a proof of the formality of configuration spaces of Euclidean spaces that do not involve operads?
When $n$ is even these spaces are complex algebraic varieties, so the cohomology comes with a mixed Hodge structure. Moreover, this mixed Hodge structure is pure: the cohomology ring is generated in ...
- 40.2k
9
votes
Accepted
Two monoidal structures and copowering
No. Consider the case where $(M,\otimes,1)$ is $(\mathbf{Set},\times,1)$, so the enrichment is vacuous, and $(C,\oplus,0)$ is $(\mathbf{Set},+,0)$, with copowering $\odot$ given by $\times$.
Then ...
- 21.3k
9
votes
Accepted
Free operad over a monoid object
Let me mention that this is related to this earlier question of mine (which is unanswered :-( ) and more generally to semi-direct products of operads by bialgebras. You construction is the semi-direct ...
- 5,040
9
votes
Conceptual (operadic?) reason for the generalized EHP fiber sequence $J_{q-1}(S^{2n}) \to J S^{2n} \to JS^{2nq}$?
In my view, the fact of that this is a fibration sequence is something to be cherished, and I wouldn't think that it generalizes without complication.
Regarding your comments that the James-Hopf ...
- 11.1k
9
votes
Accepted
How to characterize $E_n$ morphisms from $\mathrm{Mod}(A)$ to $\mathrm{Mod}(B)$?
By Corollary HA.4.8.5.20, the functor from $\mathbb{E}_{n+1}$-algebras to $\mathbb{E}_n$-monoidal categories and colimit-preserving, $\mathbb{E}_n$-monoidal functors is fully faithful. (Notice that ...
- 13.5k
9
votes
Accepted
Non(skew)commutative Lie algebras?
Such objects are known as Leibniz algebras.
A Leibniz algebra is a module $M$ together with a bilinear pairing $$[-,-]\colon M⊗M→M$$ that satisfies the Leibniz identity: $$[a,[b,c]]=[[a,b],c]+[b,[a,c]]...
- 37.8k
9
votes
How many natural operations on subsets are there?
In the comments, Tobias Fritz suggested to look at the subfunctor $P' \subseteq P$ of non-empty subsets. Here the classification is much easier: Every operation $(P')^n \to P'$ is of the form $(U_1,\...
8
votes
Suspension operad
$\Lambda$ is correct. $\Lambda'$ is not an operad because the $\circ_i$ maps are not equivariant with respect to the symmetric group actions. If $\sigma$ is the nontrivial element of $S_2$, then for ...
- 4,588
8
votes
Accepted
Can operads (or category theoretic structures more generally) be compared?
Yes, operads can be compared.
There are lots of kinds of operad (enriched in various categories, symmetric or plain or defined with respect to a monad, one-colored or many-colored, and don't even get ...
- 63.9k
8
votes
Accepted
Functoriality of infinite loop space machines?
Here is a nice gentle old-fashioned answer. Symmetric monoidal categories are functorially equivalent as symmetric monoidal categories to permutative (symmetric strict monoidal) categories, and ...
- 30.4k
8
votes
Accepted
Is there a filtered splitting of product labelling spaces?
The answer to your first question is no. And this can be seen by homology considerations. Note that this equivalence induces an isomorphism of Hopf algebras
$$ H_*(C(\mathbb R; X \vee Y \vee (X\...
- 11.1k
8
votes
Accepted
Enriched coends which preserve equivalences
One sufficient condition is that either $B$ or both $A$ and $A'$ are cofibrant in the projective model structure.
If $C$ is a Reedy category, possibly in the generalized sense defined by Berger and ...
- 10.9k
8
votes
Accepted
How now to study operads in homotopy theory?
One of the most comprehensive references today is certainly:
B. Fresse, Homotopy of Operads and Grothendieck–Teichmüller Groups, Mathematical Surveys and Monographs 217. https://bookstore.ams.org/...
- 5,040
8
votes
Accepted
Combinatorial type construction of the free operad
There are in fact infinitely many nonplanar rooted tree structures having a given nonempty set of leaves $S$, because for example such a tree can look like a linear stalk of any finite height topped ...
- 53.3k
7
votes
Does every equivalence of operads in the category of small categories have a weak inverse?
No, not even when you restrict to groupoids. In fact there's a counterexample in the book of Fresse that you cite (and it's explicitly said in §I.5.2.2 that the arity-wise inverse don't always ...
- 5,040
7
votes
What homotopy classes can attaching an $E_n$-cell kill?
[I write $A//\alpha$ for the pushout computed in $E_n-A$-algebras, and $E/\alpha$ for the pushout computed in $A$-modules.]
Here is an attempt (for $k>0$), though I should make it clear that I ...
- 1,961
7
votes
Accepted
Dioperads vs polycategories
In Martin Markl's article "Operads and PROPs," just after Def. 64, the dioperad-polycategory connection is briefly mentioned. Markl attributed this observation to Leinster.
In my book with Mark W. ...
- 466
Only top scored, non community-wiki answers of a minimum length are eligible