Could somebody please help me with this?

We know if $C$ is an ordinary additive category, then the addition of maps $f,g:x \rightarrow y$ coincides with the composition

$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x \xrightarrow{\bigtriangleup} x \bigoplus x \xrightarrow{(f,g)} y \bigoplus y \xrightarrow{\bigtriangledown} y$,

which first and last maps are diagonal and codiagonal maps respectively. This map defines an abelian group structure on $\text{Hom}_C(x,y)$.

Now, here is my main question! if $C$ is an additive $\infty$-category, how can we show this operation (on $\text{Map}_C(x,y)$) is commutative and associative up to coherent homotpy? How can we show it is commutative and associative up to just homotopy?

Could somebody please help me with this?

We know if $C$ is an ordinary additive category, then the addition of maps $f,g:x \rightarrow y$ coincides with the composition

$$x \xrightarrow{\bigtriangleup} x \oplus x \xrightarrow{(f,g)} y \oplus y \xrightarrow{\bigtriangledown} y,$$

which first and last maps are diagonal and codiagonal maps respectively. This map defines an abelian group structure on $\text{Hom}_C(x,y)$.

Now, here is my main question! if $C$ is an additive $\infty$-category, how can we show this operation (on $\text{Map}_C(x,y)$) is commutative and associative up to coherent homotopy? How can we show it is commutative and associative up to just homotopy?

Could somebody please help me with this?

We know if $C$ is an ordinary additive category, then the addition of maps $f,g:x \rightarrow y$ coincide with the composition

$x \xrightarrow{\bigtriangleup} x \oplus x \xrightarrow{(f,g)} y \oplus y \xrightarrow{\bigtriangledown} y$,

which first and last maps are diagonal and codiagonal maps respectively. This map defines an abelian group structure on $\text{Hom}_C(x,y)$.

Now, here is my main question! if $C$ is an additive $\infty$-category, how can we show this operation (on $\text{Map}_C(x,y)$) is commutative and associative up to coherent homotpy? How can we actually show it is commutative and associative up to just homotopy?

Could somebody please help me with this?

We know if $C$ is an ordinary additive category, then the addition of maps $f,g:x \rightarrow y$ coincides with the composition

$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x \xrightarrow{\bigtriangleup} x \bigoplus x \xrightarrow{(f,g)} y \bigoplus y \xrightarrow{\bigtriangledown} y$,

which first and last maps are diagonal and codiagonal maps respectively. This map defines an abelian group structure on $\text{Hom}_C(x,y)$.

Now, here is my main question! if $C$ is an additive $\infty$-category, how can we show this operation (on $\text{Map}_C(x,y)$) is commutative and associative up to coherent homotpy? How can we show it is commutative and associative up to just homotopy?

Could somebody please help me with this?

We know if $C$ is an ordinary additive category, then the addition of maps $f,g:x \rightarrow y$ coincides with the composition

$$x \xrightarrow{\bigtriangleup} x \oplus x \xrightarrow{(f,g)} y \oplus y \xrightarrow{\bigtriangledown} y,$$

which first and last maps are diagonal and codiagonal maps respectively. This map defines an abelian group structure on $\text{Hom}_C(x,y)$.

Now, here is my main question! if $C$ is an additive $\infty$-category, how can we show this operation (on $\text{Map}_C(x,y)$) is commutative and associative up to coherent homotopy? How can we show it is commutative and associative up to just homotopy?

Could somebody please help me with this?

We know if $C$ is an ordinary additive category, then the addition of maps $f,g:x \rightarrow y$ coincide with the composition

$x \xrightarrow{\bigtriangleup} x \oplus x \xrightarrow{(f,g)} y \oplus y \xrightarrow{\bigtriangledown} y$,

which first and last maps are diagonal and codiagonal maps respectively. This map defines an abelian group structure on $\text{Hom}_C(x,y)$.

Now, here is my main question! if $C$ is an additive $\infty$-category, how can we show this operation (on $\text{Map}_C(x,y)$) is commutative and associative up to coherent homotpy? How can we actually show it is commutative and associative up to just homotopy?