Weak completions, bornologies and rigid cohomology

Let V be a complete discrete valuation ring with residue field k of positive characteristic and with fraction field K of characteristic 0. We clarify the analysis behind the Monsky-Washnitzer completion of a commutative V-algebra using completions of bornological V-algebras. This leads us to a functorial chain complex for a finitely generated commutative algebra over the residue field k that computes its rigid cohomology in the sense of Berthelot.