I am a little bit confused about the terminology used in Section 2.4 of this paper. We have a finite pointed CW complex $B$ and a subcomplex $A$. Then "let $C$ be obtained from $B$ by attaching a cone $I\wedge A$ along the subspace $\{1\}\times A=A$". What does this mean explicitly?
I am familiar with mapping cylinders/cones in topology, defined as quotients of a disjoint union and assume it relates to this in some way but it seems unclear.
This appears to be the (standard) mapping cone of the inclusion $A\rightarrow B$. Sections 44 and 45 of the excellent Algebraic Topology notes by Haynes Miller (easy to find online; I am unsure how to attach link as you did) may help. Also fits into adjunction space as pointed out above by @moishekahan. Comments above only appeared as I finished this.