I know that it has been shown that $E_2$ algebra objects in Categories are simply braided monoidal categories. In particular, Lurie says that an $E_2$-monoidal structure on the infinity-category $N(C)$ (the nerve of a category C) is a braided monoidal structure on C. Is there a generalization (n...

Let $R$ be any ring and let $A\to B\to C\to [1]$ and $A'\to B'\to C'\to [1]$ be distinguished triangles of complexes of $R$-modules. Let $f:A\to A'$, $g:B\to B'$ and $h:C\to C'$ be morphism of complexes such that $(f,g,h)$ is a morphism of distinguished triangles. I wonder whether the following s...