What is the notation $M_f \cup_Y M_g$?

Question

Suppose $f:X\longrightarrow Y$ and $g:Y\longrightarrow Z$ are two maps. Also, assume that $M_{f}$ and $M_{g}$ denote mapping cylinders $f$ and $g$, respectively.

What is the notation $M_f \cup_Y M_g$ in the literature?

Answer 1

There are inclusions $i_1 \colon Y \to M_f$ and $i_2 \colon Y \to M_g$ of $Y$ to the top end of $M_f$ and the bottom end of $M_g$, and $M_f \cup_Y M_g$ (which I'd prefer to write as $M_f \sqcup_Y M_g$) is what you get by gluing the copies of $Y$ in $M_f$ and $M_g$ together, the result being a (short) mapping telescope of $X \xrightarrow{f} Y \xrightarrow{g} Z$. It is a pushout: $$\require{AMScd}\begin{CD} Y @>{i_1}>> M_f \\ @V{i_2}VV @VVV \\ M_g @>>> M_f \sqcup_Y M_g \end{CD}$$ and this kind of notation is used for pushouts in general categories. (The maps $i_1,i_2$ are implied in the notation.)

(As for a reference for the notation, I don't have a good one handy but there is an instance in the above link. The notation $X \times_Z Y$ for the dual concept (a pullback of $X \to Z \leftarrow Y$) is seen more frequently in a variety of contexts.)