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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 $(0,1)^k\rightarrow (0,1)^k$ of the form $(x_i)\mapsto (a_i i_k+b_i)$ for $x_i\in (0,1)$ and $a_i\geq 0$ for $i=1,...,k$. 2-morphisms are given by isotopies of embeddings, 3-morphisms are isotopies of isotopies etc. Call this operad $\square^k$. Disjoint union equips $\square^k$ with a symmetric monoidal structure.

The $(\infty,1)$-category of $E_k$-algebras with values in a symmetric monoidal $(\infty,1)$-category $(C,\otimes)$ is defined as the $(\infty,1)$-category of symmetric monoidal functors $\text{Fun}^{\otimes}(\square^k,C)$.

The little $k$-disks $(\infty,1)$-operad is similarly defined as the operad of framed embeddings of open(?) disks into one another, with (higher) isotopies as (higher) morphisms. Call this operad $\text{Disk}_k^{fr}$.

The $(\infty,1)$-category of $k$-disk algebras with values in a symmetric monoidal $(\infty,1)$-category $(C,\otimes)$ is again the $(\infty,1)$-category of symmetric monoidal functors $\text{Fun}^{\otimes} ( \text{Disk}_k^{fr},C)$

The question

On the nLab, it is written (see here and here) that the little $k$-cubes operad and the little $k$-disks operad are distinct objects, and the latter is a generalization of the former. The main difference I notice is that the little $k$-disks operad allows one to rotate disks when embedding them, while the little $k$-cubes operad does not. Still, rotations are homotopic to the identity, so it seems (to me) safe to assume that the operads are equivalent as $(\infty,1)$-categories. Am I mistaken?

Also, according to Ginot's notes (page 27, Example 12), the $(\infty,1)$-categories of algebras of $\text{Disk}_k^{fr}$ and $\square^k$ are equivalent, and this leads me to believe that the operads themselves should be equivalent.

I would not be suprised of the above reveals a severe lack of understanding on my part. I am just starting to try to understand these gadgets. Any help will be greatly appreciated.

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    $\begingroup$ I think this is not exactly what you're asking about, but in case it might help, let me say that the little disks and the little cubes (topological) operads are weakly equivalent. Beware, however, that the little disks operad does not allow rotations. That's the framed little disks operad, which is not equivalent to the former. $\endgroup$ – Fernando Muro Sep 25 '14 at 22:20
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    $\begingroup$ I would like to add a comment about your sentence "this leads me to believe that the operads themselves should be equivalent." Be careful, the fact that two categories of algebras are equivalent does not imply in general that the operads themselves are weakly equivalent. The situation is analogous to what happens in representation theory: Morita equivalence does not always imply equivalence. $\endgroup$ – Sinan Yalin Sep 25 '14 at 22:34
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    $\begingroup$ @FernandoMuro Thank you very much. I think this answers my question. $\endgroup$ – Espen Nielsen Sep 25 '14 at 23:36
  • $\begingroup$ @SinanYalin That makes sense. The Morita equivalence example is nice. Thank you very much. $\endgroup$ – Espen Nielsen Sep 25 '14 at 23:37
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    $\begingroup$ But ignoring that, and assuming you just mean that you allow rotations now too, I can tell you that operad is fairly different from the little cubes operad. The unary operations don't match: the space of embeddings of one cube into itself is contractible (there is "only one unary operation"), but the space of embeddings of a disk into itself, where you allow rotations, is homotopic to $SO(k)$. (Your observation that rotations are homotopic to the identity says only that $\pi_0 (SO(k)) = \ast$, which is true but doesn't make $SO(k)$ contractible.) $\endgroup$ – Omar Antolín-Camarena Sep 26 '14 at 0:26
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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 of that do not allow rotations (with some other conditions). In other words, such embeddings $D^n \hookrightarrow D^n$ must preserve the framing.
  • the "framed" version $\mathtt{fD}_n$: here the embeddings are allowed to rotate the disks, and do not necessarily preserve the framing. Basically it is $\mathtt{D}_n$ together with an action of $\mathrm{SO}(n)$ (in fact it's a semi-direct product $\mathtt{D}_n \rtimes \mathrm{SO}(n)$, see P. Salvatore and N. Wahl, Framed discs operads and Batalin-Vilkovisky algebras. Q. J. Math., 2003, 54, 213-231").

These two operads are not weakly equivalent, and their categories of algebras are different. To give you an idea, $H_*(\mathtt{D}_2) = \mathtt{Ger}$ is the operad of Gerstenhaber algebras, whereas $H_*(f\mathtt{D}_2) = \mathtt{BV}$ is the operad of BV-algebras -- morally we have a circle action in addition. More generally, $\mathtt{D}_n(1)$ is contractible, whereas $\mathtt{fD}_n(1) \simeq \mathrm{SO}(n)$ is non-contractible, so the operads cannot be weakly equivalent.

The first operad $\mathtt{D}_n$ is actually equivalent to $\mathrm{Disk}_n^{\mathrm{fr}}$. This makes perfect sense in this context: $\mathtt{D}_n$ is equivalent to the endomorphism operad of $\mathbb{R}^n$ in the category of framed manifolds and embeddings, and you can take the factorization homology of a $\mathtt{D}_n$-algebra only on a framed manifold.

On the other hand, $\mathtt{End}_{\mathbb{R}^n} = \mathrm{Disk}_n$ in the category of unoriented manifolds and embeddings is equivalent to an operad slightly larger than $\mathtt{fD}_n = \mathtt{D}_n \rtimes SO(n)$, I think it is $\mathtt{D}_n \rtimes O(n)$. Its endomorphism operad in the category of oriented manifolds $\mathtt{End}^{\mathrm{or}}_{\mathbb{R}^n}$ is weakly equivalent to $\mathtt{fD}_n$.

Unfortunately, as you can see, the two occurrences of "framed" refer to different things, and are applied in opposite manners. As far as I know, a recent trend in some circles is to do away with the terminology "framed little discs operad" altogether.


With all that being said, it is indeed true that $\mathtt{D}_n \simeq \mathrm{Disk}_n^{fr}$ is equivalent to the operad $\square^n$ of little $n$-cubes (where you don't allow rotations, of course). A few possible references:

  • R. Steiner, A canonical operad pair, Math. Proc. Cambridge Philos. Soc. 86 (1979), 443–449.
  • C. Berger, Opérades cellulaires et espaces de lacets itérés. Ann. Inst. Fourier 46 (1996), 1125–1157.

It's actually not so easy to prove. It's easy to see that they are arity-wise equivalent: in arity $r$, both spaces are equivalent to the configuration space $\operatorname{Conf}_r(\mathbb{R}^n)$ of $r$ ordered points in $\mathbb{R}^n$. It's finding an equivalence that respects the operad structure that is difficult (but possible).

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  • $\begingroup$ Thank you very much for taking the time to write this answer and for the references! I had a suspicion that there was conflicting terminology. This made the matter clear! $\endgroup$ – Espen Nielsen Nov 14 '14 at 17:34

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