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The result in question is a corollary to proposition 5.4.5.15 in Higher Algebra

Theorem 5.4.5.9. Let $M$ be a manifold and let $C^⊗$ be an $∞$-operad. Composition with the map $$\mathrm{Disk}(M)^⊗ → \mathbb{E}_M^⊗$$ of Remark 5.4.5.8 induces a fully faithful embedding $$θ:\mathrm{Alg}_{\mathbb{E}_M}(C) → \mathrm{Alg}_{\mathrm{Disk}(M)}(C)\,.$$ The essential image of $θ$ is the full subcategory of $\mathrm{Alg}_{\mathrm{Disk}(M)}(C)$ spanned by the locally constant $\mathrm{Disk}(M)^⊗$-algebra objects of $C$.

When $M=\mathbb{R}^n$, the $\infty$-operad $\mathbb{E}_M$ is just the $\mathbb{E}_n$-operad.

Be careful that the book does not use the language of factorization algebras, but if you compare the definitions you'll see that there is a conservative functor from locally constant factorization algebras on $M$ to locally constant $\mathrm{Disk}(M)$-algebras such that the functor in proposition 5.4.5.9 factors as $$\mathrm{Alg}_{\mathbb{E}_M}(C)→\mathrm{Fact}^{lc}_M(C)→\mathrm{Alg}_{\mathrm{Disk}(M)}(C)$$ where the middle term is the $\infty$-category of locally constant factorization algebras and the first arrow is the one given by taking factorization homology. This, plus the existence of the "global sections" functor $\mathrm{Fact}^{lc}_M(C)→\mathrm{Alg}_{\mathbb{E}_M}(C)$ is enough to conclude the result you're after.

The result in question is a corollary to proposition 5.4.5.15 in Higher Algebra

Theorem 5.4.5.9. Let $M$ be a manifold and let $C^⊗$ be an $∞$-operad. Composition with the map $$\mathrm{Disk}(M)^⊗ → \mathbb{E}_M^⊗$$ of Remark 5.4.5.8 induces a fully faithful embedding $$θ:\mathrm{Alg}_{\mathbb{E}_M}(C) → \mathrm{Alg}_{\mathrm{Disk}(M)}(C)\,.$$ The essential image of $θ$ is the full subcategory of $\mathrm{Alg}_{\mathrm{Disk}(M)}(C)$ spanned by the locally constant $\mathrm{Disk}(M)^⊗$-algebra objects of $C$.

When $M=\mathbb{R}^n$, the $\infty$-operad $\mathbb{E}_M$ is just the $\mathbb{E}_n$-operad.

Be careful that the book does not use the language of factorization algebras, but if you compare the definitions you'll see that there is a conservative functor from locally constant factorization algebras on $M$ to locally constant $\mathrm{Disk}(M)$-algebras such that the functor in theorem 5.4.5.9 factors as $$\mathrm{Alg}_{\mathbb{E}_M}(C)→\mathrm{Fact}^{lc}_M(C)→\mathrm{Alg}_{\mathrm{Disk}(M)}(C)$$ where the middle term is the $\infty$-category of locally constant factorization algebras and the first arrow is the one given by taking factorization homology. This, plus the existence of the "global sections" functor $\mathrm{Fact}^{lc}_M(C)→\mathrm{Alg}_{\mathbb{E}_M}(C)$ is enough to conclude the result you're after.

The result in question is a corollary to proposition 5.4.5.15 in Higher Algebra

Proposition 5.4.5.15. Let $M$ be a manifold and let $C^⊗$ be an $∞$-operad. Then composition with map $\mathrm{Disk}(M)^⊗→\mathbb{E}^⊗_M$ of Remark 5.4.5.8 induces a fully faithful embedding $$θ : \mathrm{Alg}^{nu}_{\mathbb{E}_M}(C) → \mathrm{Alg}^{nu}_{\mathrm{Disk}(M)} (C)\,.$$ The essential image of $θ$ is the full subcategory of $\mathrm{Alg}^{nu}_{ \mathrm{Disk}(M)}(C)$ spanned by the locally constant objects.

When $M=\mathbb{R}^n$, the $\infty$-operad $\mathbb{E}_M$ is just the $\mathbb{E}_n$-operad (the small $nu$ on top is just saying that we're talking about "non-unital" algebras).

Be careful that the book does not use the language of factorization algebras, but if you compare the definitions you'll see that there is a conservative functor from locally constant factorization algebras on $M$ to locally constant $\mathrm{Disk}(M)$-algebras, that is a left inverse to the functor of proposition 5.4.5.15.

The result in question is a corollary to proposition 5.4.5.15 in Higher Algebra

Theorem 5.4.5.9. Let $M$ be a manifold and let $C^⊗$ be an $∞$-operad. Composition with the map $$\mathrm{Disk}(M)^⊗ → \mathbb{E}_M^⊗$$ of Remark 5.4.5.8 induces a fully faithful embedding $$θ:\mathrm{Alg}_{\mathbb{E}_M}(C) → \mathrm{Alg}_{\mathrm{Disk}(M)}(C)\,.$$ The essential image of $θ$ is the full subcategory of $\mathrm{Alg}_{\mathrm{Disk}(M)}(C)$ spanned by the locally constant $\mathrm{Disk}(M)^⊗$-algebra objects of $C$.

When $M=\mathbb{R}^n$, the $\infty$-operad $\mathbb{E}_M$ is just the $\mathbb{E}_n$-operad.

Be careful that the book does not use the language of factorization algebras, but if you compare the definitions you'll see that there is a conservative functor from locally constant factorization algebras on $M$ to locally constant $\mathrm{Disk}(M)$-algebras such that the functor in proposition 5.4.5.9 factors as $$\mathrm{Alg}_{\mathbb{E}_M}(C)→\mathrm{Fact}^{lc}_M(C)→\mathrm{Alg}_{\mathrm{Disk}(M)}(C)$$ where the middle term is the $\infty$-category of locally constant factorization algebras and the first arrow is the one given by taking factorization homology. This, plus the existence of the "global sections" functor $\mathrm{Fact}^{lc}_M(C)→\mathrm{Alg}_{\mathbb{E}_M}(C)$ is enough to conclude the result you're after.

The result in question is a corollary to proposition 5.4.5.15 in Higher Algebra

Proposition 5.4.5.15. Let $M$ be a manifold and let $C^⊗$ be an $∞$-operad. Then composition with map $\mathrm{Disk}(M)^⊗→\mathbb{E}^⊗_M$ of Remark 5.4.5.8 induces a fully faithful embedding $$θ : \mathrm{Alg}^{nu}_{\mathbb{E}_M}(C) → \mathrm{Alg}^{nu}_{\mathrm{Disk}(M)} (C)\,.$$ The essential image of $θ$ is the full subcategory of $\mathrm{Alg}^{nu}_{ \mathrm{Disk}(M)}(C)$ spanned by the locally constant objects.

When $M=\mathbb{R}^n$, the $\infty$-operad $\mathbb{E}_M$ is just the $\mathbb{E}_n$-operad (the small $nu$ on top is just saying that we're talking about "non-unital" algebras).

Be careful that the book does not use the language of factorization algebras, but if you compare the definitions you'll see immediately that locally constant $\mathrm{Disk}(M)$-algebras are exactly the same thing as locally constant factorization algebras.

The result in question is a corollary to proposition 5.4.5.15 in Higher Algebra

Proposition 5.4.5.15. Let $M$ be a manifold and let $C^⊗$ be an $∞$-operad. Then composition with map $\mathrm{Disk}(M)^⊗→\mathbb{E}^⊗_M$ of Remark 5.4.5.8 induces a fully faithful embedding $$θ : \mathrm{Alg}^{nu}_{\mathbb{E}_M}(C) → \mathrm{Alg}^{nu}_{\mathrm{Disk}(M)} (C)\,.$$ The essential image of $θ$ is the full subcategory of $\mathrm{Alg}^{nu}_{ \mathrm{Disk}(M)}(C)$ spanned by the locally constant objects.

When $M=\mathbb{R}^n$, the $\infty$-operad $\mathbb{E}_M$ is just the $\mathbb{E}_n$-operad (the small $nu$ on top is just saying that we're talking about "non-unital" algebras).

Be careful that the book does not use the language of factorization algebras, but if you compare the definitions you'll see that there is a conservative functor from locally constant factorization algebras on $M$ to locally constant $\mathrm{Disk}(M)$-algebras, that is a left inverse to the functor of proposition 5.4.5.15.