Coarse property C and decomposition complexity

The coarse category was established by Roe [20] to distill the salient features of the large-scale approach to metric spaces and groups that was started by Gromov [13]. In this paper, we use the language of coarse spaces to define coarse versions of asymptotic property C [6] and decomposition complexity [16]. We prove that coarse property C implies coarse property A; we also show that these coarse versions share many of the features of their metric analogs such as preservation by products or unions.