The main motivation for our definition and our study of exact squares, as exposed through sections 1 to 4 was the study of Paré’s thesis about absolute limits (Paré, 1969) and the characterizations of absolute extensions given in (Guitart, 1977) (partially continuing on the line of some results in Thiébaud (Thiébaud, 1971) and Harting (Harting, 1977)).

We call absolute left extension a diagram in $\Cat$ of the form such that for each $H: Z\to K$ the composite $H\vf$ is a left extension (i.e. $HR\cong \Lan_J HF$).