Many constructions in homotopy theory are special cases of pushouts. Although when being defined sometimes it may be clearer to define the construction as a seperate higher inductive type and then later on prove it is equivalent? to the pushout definition.

Definition

The (homotopy) pushout of a span $A \xleftarrow{f} C \xrightarrow{g} B$ is the higher inductive type$A \sqcup^{C}_{f ; g} B$ (or $A \sqcup^C B$) generated by:

a function $inl : A \to A \sqcup^{C} B$

a function $inr : B \to A \sqcup^{C} B$

for each $c:C$ a path $glue(c) : inl (f (c)) = inr (g( c))$