We introduce and study the new concepts of cosilting complexes, cosilting modules and AIRcotilting
modules. We prove that the three concepts AIR-cotilting modules, cosilting modules
and quasi-cotilting modules coincide with each other, in contrast with the dual fact that AIRtilting
modules, silting modules and quasi-tilting modules are different. Further, we show that
there are bijections between the following four classes (1) equivalent classes of AIR-cotilting
(resp., cosilting, quasi-cotilting) modules, (2) equivalent classes of 2-term cosilting complexes,
p3q torsion-free cover classes and p4q torsion-free special precover classes. We also extend a
classical result of Auslander and Reiten on the correspondence between certain contravariantly
finite subcategories and cotilting modules to the case of cosilting complexes.
This is a joint work with Peiyu Zhang.