I have read it claimed in several places that in a stable $(\infty,1)$-category, the coequalizer of parallel maps $f,g:X\to Y$ can be identified with the cokernel of $f-g$ (i.e. the pushout of the map $X\to 0$ along $f-g$). How is this proven? Is it written down anywhere?



Coequalizers in stable (infinity,1)-categories
It seems to me like the proof for Abelian categories works here too. Consider the following diagram where $d = (id,-id)$, the middle vertical map is the fold (or $+$) map, and the square on the right is a homotopy pushout:


$$ \begin{array}{ccccc}
 X & \xrightarrow{d} & X \oplus X & \xrightarrow{(f,g)} & Y \\
 \downarrow & & \downarrow & & \downarrow \\
 0 & \to & X & \to & C \\
 \end{array}
$$


From the second square being a pushout, $C$ is the coequalizer of $f$ and $g$. If we check the first square is a pushout too, we get that the outer rectangle is a pushout and so $C$ is also $\mathrm{cofib}(f-g)$.


Here's Akhil's proof that the left hand square is a pushout:


$$ \begin{array}{ccccc}
 X & \xrightarrow{\pmatrix{0&1}} & X \oplus X & \xrightarrow{\pmatrix{0 & 1 \\ 1 & -1}} & X \oplus X \\
 \downarrow & & \downarrow \rlap{\scriptstyle\pmatrix{1\\ 0}}& & \downarrow \rlap{\scriptstyle\pmatrix{1\\1}}\\
 0 & \to & X & \xrightarrow{1} & X \\
 \end{array}
$$


The outer rectangle is what we'd like to show is a pushout. The right hand square is a pushout because both horizontal arrows are isomorphisms, and the left hand square is a pushout because it is the (pointwise) direct sum of these two (where all the maps $X \to X$ are the identity):


$$ \begin{array}{cccccc}
 0 & \to & X & & X & \to & X \\
 \downarrow & & \downarrow & & \downarrow & & \downarrow \\
 0 & \to & X & & 0 & \to & 0 \\
 \end{array}
$$


