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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?

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    $\begingroup$ Do you know how to prove it in a 1-category? If not, meditate on this diagram (where $\sigma$ is the swap map). $\endgroup$
    – Naïm Favier
    Commented Jul 20 at 15:29
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    $\begingroup$ LaTeX note: \oplus is a binary operations, while \bigoplus is basically a unary operation. $\endgroup$
    – Z. M
    Commented Jul 20 at 15:36
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    $\begingroup$ A reference for this is Remark C.1.5.3 of Lurie's Spectral Algebraic Geometry. The remark only makes use of material from Higher Algebra. $\endgroup$
    – Brian Shin
    Commented Jul 20 at 17:37
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    $\begingroup$ You use the obvious associativity $(x \oplus x) \oplus x \simeq x \oplus (x \oplus x)$, as in this diagram. $\endgroup$
    – Naïm Favier
    Commented Jul 20 at 22:14
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    $\begingroup$ @A.karimi Are you sure that you really are looking for "commutative and associative up to coherent homotopy"? It is somewhat of a challenge to even define what that means, let alone actually demonstrate a particular object satisfies it, so I doubt that you are going to find a proof as simple as the 1-categorical case. $\endgroup$
    – Connor Malin
    Commented Jul 21 at 9:33

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