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Definition (1), found in [CG, 6.1.4]. We call $F$ a factorization algebra if for each Weiss cover $\mathcal{U}$ of an open set $U\subset M$, the map $$ T\to FU $$ is a quasi-isomorphism, where $T$ denotes the totalization of the normalized chain complex in $\cal C$ (regarded as a cochain complex in $\cal C$ concentrated in negative degrees) associated with the simplicial object $\{\bigoplus_{U_{0},\dots,U_{n}\in\mathcal{U},\ U_i \mathrm{all\ distinct}}F\left(U_{0}\cap\cdots\cap U_{n}\right)\}_{n\geq0}$.

Definition (1), found in [CG, 6.1.4]. We call $F$ a factorization algebra if for each Weiss cover $\mathcal{U}$ of an open set $U\subset M$, the map $$ T\to FU $$ is a quasi-isomorphism, where $T$ denotes the totalization of the normalized chain complex in $\cal C$ (regarded as a cochain complex in $\cal C$ concentrated in negative degrees) associated with the simplicial object $\{\bigoplus_{U_{0},\dots,U_{n}\in\mathcal{U}}F\left(U_{0}\cap\cdots\cap U_{n}\right)\}_{n\geq0}$.

Definition (1), found in [CG, 6.1.4]. We call $F$ a factorization algebra if for each Weiss cover $\mathcal{U}$ of an open set $U\subset M$, the map $$ T\to FU $$ is a quasi-isomorphism, where $T$ denotes the totalization of the normalized chain complex in $\cal C$ (regarded as a cochain complex in $\cal C$ concentrated in negative degrees) associated with the simplicial object $\{\bigoplus_{U_{0},\dots,U_{n}\in\cal U}F\left(U_{0}\cap\cdots\cap U_{n}\right)\}_{n\geq0}$.

Definition (3), found in [CFM, Definition 2.2]. We call $F$ a factorization algebra if for each open set $U\subset M$ and each Weiss cover $\mathcal{U}$ of $M$, the map $$ \operatorname{hocolim}_{\alpha\subset\mathcal{U}\text{ finite}}F\left(\bigcap_{V\in\alpha}V\right)\to FU $$ is a quasi-isomorphism, and moreover for each pairwise disjoint open sets $U_{1},\dots,U_{k}\subset M$, the map $m_{U_{1},\dots,U_{k};\bigcup_{i}U_{i}}$ is a quasi-isomorphism.

Definition (1), found in [CG, 6.1.4]. We call $F$ a factorization algebra if for each Weiss cover $\mathcal{U}$ of an open set $U\subset M$, the map $$ T\to FU $$ is a quasi-isomorphism, where $T$ denotes the totalization of the normalized chain complex in $\cal C$ (regarded as a cochain complex in $\cal C$ concentrated in negative degrees) associated with the simplicial object $\{\bigoplus_{U_{0},\dots,U_{n}\in\mathcal{U},\ U_i \mathrm{all\ distinct}}F\left(U_{0}\cap\cdots\cap U_{n}\right)\}_{n\geq0}$.

Definition (3), found in [CFM, Definition 2.2]. We call $F$ a factorization algebra if for each open set $U\subset M$ and each Weiss cover $\mathcal{U}$ of $U$, the map $$ \operatorname{hocolim}_{\alpha\subset\mathcal{U}\text{ finite}}F\left(\bigcap_{V\in\alpha}V\right)\to FU $$ is a quasi-isomorphism, and moreover for each pairwise disjoint open sets $U_{1},\dots,U_{k}\subset M$, the map $m_{U_{1},\dots,U_{k};\bigcup_{i}U_{i}}$ is a quasi-isomorphism.

I am unfamiliar with the computation of homotopy colimits of cochain complexes, but looking this MO question, I suspect that the object $T$ in Definitions (1) and (2) are homotopy colimits of the corresponding simplicial object. So they are similar in the sense that they both use simplicial objects, but they differ in their choices of covers.

I am unfamiliar with the computation of homotopy colimits of cochain complexes, but looking at this MO question, I suspect that the object $T$ in Definitions (1) and (2) are homotopy colimits of the corresponding simplicial object. So they are similar in the sense that they both use simplicial objects, but they differ in their choices of covers.