Although fast radio bursts (FRBs) have been an active field in astronomy and cosmology, their origin is still unknown to date. One of the interesting topics is the classification of FRBs, which is closely related to the origin of FRBs. Different physical mechanisms are required by different classes of FRBs. In the literature, they usually could be classified into non-repeating and repeating FRBs. Well motivated by the observations, here we are interested in the possible subclassification of FRBs. By using the first CHIME/FRB catalog, we propose to subclassify non-repeating (type I) FRBs into type Ia and Ib FRBs. The distribution of type Ia FRBs is delayed with respect to the cosmic star formation history (SFH), and hence they are probably associated with old stellar populations, while the distribution of type Ib FRBs tracks SFH, and hence they are probably associated with young stellar populations. Accordingly, the physical criteria for this subclassification of type I FRBs have been clearly determined. We find that there are some tight empirical correlations for type Ia FRBs but not for type Ib FRBs, and vice versa. These make them different in physical properties. Similarly, we suggest that repeating (type II) FRBs could also be subclassified into type IIa and IIb FRBs. A universal subclassification scheme is given at the end. This subclassification of FRBs might help us to reveal quite different physical mechanisms behind them, and improve their applications in astronomy and cosmology.

Recently, fast radio bursts (FRBs) have become a thriving field in astronomy and cosmology. Due to their extragalactic and cosmological origin, they are useful to study the cosmic expansion and the intergalactic medium (IGM). In the literature, the dispersion measure DM of FRB has been considered extensively. It could be used as an indirect proxy of the luminosity distance dL of FRB. The observed DM contains the contributions from the Milky Way (MW), the MW halo, IGM, and the host galaxy. Unfortunately, IGM and the host galaxy of FRB are poorly known to date, and hence the large uncertainties of DMIGM and DM host in DM plague the FRB cosmology. Could we avoid DM in studying cosmology? Could we instead consider the luminosity distance dL directly in the FRB cosmology? We are interested to find a way out for this problem in the present work. From the lessons of calibrating type Ia supernovae (SNIa) or long gamma-ray bursts (GRBs) as standard candles, we consider a universal subclassification scheme for FRBs, and there are some empirical relations for them. In the present work, we propose to calibrate type Ib FRBs as standard candles by using a tight empirical relation without DM. The calibrated type Ib FRBs at high redshifts can be used like SNIa to constrain the cosmological models. We also test the key factors affecting the calibration and the cosmological constraints.