  We introduce notions of lax semiadditive and lax additive
$(\infty,2)$-categories, categorifying the classical notions of semiadditive
and additive 1-categories. To establish a well-behaved axiomatic framework, we
develop a calculus of lax matrices and use it to prove that in locally
cocomplete $(\infty,2)$-categories lax limits and lax colimits agree and are
absolute. In the lax additive setting, we categorify fundamental constructions
from homological algebra such as mapping complexes and mapping cones and
establish their basic properties.
