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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 $f \mathtt{D}_n$: 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 $SO(n)$ (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, $H_*(\mathtt{D}_2)$ is the operad of Gerstenhaber algebras, whereas $H_*(f\mathtt{D}_2)$ is the operad of BV-algebras -- morally we have a circle action in addition. In general $H_*(f \mathtt{D}_n) = H_*(\mathtt{D}_n) \rtimes H_*(SO(n))$ (see the reference I gave earlier); in particular they have differing homology so they cannot be weakly equivalent.

The first operad $\mathtt{D}_n$ is actually equivalent to $\mathrm{Disk}_n^{fr}$. 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 $f \mathtt{D}_n = \mathtt{D}_n \rtimes SO(n)$, namely I think it is $\mathtt{D}_n \rtimes O(n)$.

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 $\mathtt{D}_n \cong \mathrm{Disk}_n^{fr}$ 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: namely in arity $r$ both spaces are equivalent to the configuration space of $r$ ordered points in $\mathbb{R}^n$. It's finding an equivalence that respects the operad structure that is difficult (but possible).

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).

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 $f \mathtt{D}_n$: 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 $SO(n)$ (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, $H_*(\mathtt{D}_2)$ is the operad of Gerstenhaber algebras, whereas $H_*(f\mathtt{D}_2)$ is the operad of BV-algebras -- morally we have a circle action in addition.

With your (Ginot's?) notation, the first operad $\mathtt{D}_n$ is actually equivalent to $\mathrm{Disk}_n^{fr}$. 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 $f \mathtt{D}_n = \mathtt{D}_n \rtimes SO(n)$, namely I think it is $\mathtt{D}_n \rtimes O(n)$.

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 $\mathtt{D}_n \cong \mathrm{Disk}_n^{fr}$ is equivalent to the operad 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.

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 $f \mathtt{D}_n$: 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 $SO(n)$ (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, $H_*(\mathtt{D}_2)$ is the operad of Gerstenhaber algebras, whereas $H_*(f\mathtt{D}_2)$ is the operad of BV-algebras -- morally we have a circle action in addition. In general $H_*(f \mathtt{D}_n) = H_*(\mathtt{D}_n) \rtimes H_*(SO(n))$ (see the reference I gave earlier); in particular they have differing homology so they cannot be weakly equivalent.

The first operad $\mathtt{D}_n$ is actually equivalent to $\mathrm{Disk}_n^{fr}$. 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 $f \mathtt{D}_n = \mathtt{D}_n \rtimes SO(n)$, namely I think it is $\mathtt{D}_n \rtimes O(n)$.

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 $\mathtt{D}_n \cong \mathrm{Disk}_n^{fr}$ 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: namely in arity $r$ both spaces are equivalent to the configuration space of $r$ ordered points in $\mathbb{R}^n$. It's finding an equivalence that respects the operad structure that is difficult (but possible).

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 $f \mathtt{D}_n$: 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 $SO(n)$ (see "Salvatore, Paola and Wahl, Nathalie, 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, $H_*(\mathtt{D}_2)$ is the operad of Gerstenhaber algebras, whereas $H_*(f\mathtt{D}_2)$ is the operad of BV-algebras -- morally we a circle action in addition.

With your (Ginot's?) notation, the first operad $\mathtt{D}_n$ is actually equivalent to $\mathrm{Disk}_n^{fr}$. 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, $\mathtt{End}_{\mathbb{R}^n}$ in the category of unoriented manifolds and embeddings is equivalent to an operad slightly larger than $f \mathtt{D}_n = \mathtt{D}_n \rtimes SO(n)$, namely I think it is $\mathtt{D}_n \rtimes O(n)$.

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 $\mathtt{D}_n \cong \mathrm{Disk}_n^{fr}$ is equivalent to the operad 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.

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 $f \mathtt{D}_n$: 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 $SO(n)$ (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, $H_*(\mathtt{D}_2)$ is the operad of Gerstenhaber algebras, whereas $H_*(f\mathtt{D}_2)$ is the operad of BV-algebras -- morally we have a circle action in addition.

With your (Ginot's?) notation, the first operad $\mathtt{D}_n$ is actually equivalent to $\mathrm{Disk}_n^{fr}$. 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 $f \mathtt{D}_n = \mathtt{D}_n \rtimes SO(n)$, namely I think it is $\mathtt{D}_n \rtimes O(n)$.

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 $\mathtt{D}_n \cong \mathrm{Disk}_n^{fr}$ is equivalent to the operad 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.