# Tag Info

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

I'm not quite certain what Peter May had in mind 40 years ago, but probably he had in mind the fact that pushouts are a lot better behaved in CGWH than in CGH. Specifically, CGWH is closed under ...
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• 48.3k

### In what sense are operads "better" than PROPs?

One thing you can do with an operad that you cannot do with a prop is write down a monad such that algebras over the monad correspond to algebras over the operad. For example, Hopf algebras have a ...
• 110k
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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 ...
• 37.9k
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### Obstructions for $E_n$-algebras

Let me expand a little on what Qiaochu and Craig mentioned. If you want an obstruction theory for building an uber-gadget, you'll need (i) an algebraic approximation to such gadgets, and (ii) a way ...
• 12.8k
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### 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)^{|...
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### 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 ...
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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 ...
• 37.9k
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### What are algebras for the little n-balls/n-cubes/n-something operads exactly?

An $E_n$ algebra (an algebra over the little $n$-cubes operad, etc.) is intuitively an object with $n$ compatible monoid structures. All of the subtlety in this theory lies in making "compatible" ...
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### 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 ...
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Operads $\mathcal{C}$ can be defined in any symmetric monoidal category, and then $E_{\infty}$ operads are specified in accordance with the (or a) notion of equivalence relevant to that category. In ...
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### 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 ...
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### 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 ...
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### $k$-Disk algebras versus $E_k$ algebras

There is an unfortunate clash of terminologies here. Traditionally, the little discs operad comes in two variants: the "usual" $\mathtt{D}_n$: the space of arity $r$ operations consists of embeddings ...
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### Reference Request: Grouplike Algebras over the little $n$-cubes operad are $n$-fold loop spaces

May gave a proof in the case $n = \infty$ in this followup paper to "Geometry of iterated loop spaces," which relies on knowing that a certain map (from a free algebra on $X$ to the free infinite loop ...
• 48.3k
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### Boardman-Vogt tensor product

The answer to 2) is yes for cofibrant operads, see http://arxiv.org/abs/1102.1311 by Fiedorowicz and Vogt. The answer to 1) is no; the Boardman-Vogt tensor product does not interact well with ...
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### 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 ...
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### 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 ...
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### Errata for Getzler-Kapranov "Cyclic operads and Cyclic homology"

I am using this version of the paper, and it seems like the published version looks exactly like that. Here is what I have found. First, a warning: Getzler and Kapranov use the term "quadratic operad"...
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### $E_{\infty}$ spaces are $A_{\infty}$ spaces

If $X$ and $Y$ are $G$-spaces for any group $G$, then the projection $\pi\colon X\times Y\to Y$ is a $G$-map, trivially: $\pi (gx,gy) = gy$. The map $\pi_2$ of your question is a very special case; ...
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### Are $E_n$-operads not formal in characteristic not equal to zero?

Ok, Sean. I'll write in terms of homology operations. Let $\mathcal C$ be any $\Sigma$-free operad, in spaces or in chain complexes, makes no real difference to the answer. For definiteness, take ...
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### Is there an operad that codifies groupoids?

The question (under the relaxed notion of operad in the original post) will be settled negatively if we show that the category $\mathrm{Grpd}$ (internal groupoids in $\mathrm{Set}$) isn't monadic over ...
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### 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 ...