**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.

framedlittle disks operad, which is not equivalent to the former. $\endgroup$ – Fernando Muro Sep 25 '14 at 22:20