In order to investigate the subtype relationships and the properties of categorical data types which usually contain both construction operations and observation operations,this paper adopts the bialgebras to bring a bout aunified mathematical description of the construction operations and observations operations of categorical data types,and analyzes categorical definitions and their semantics,which are both used to describe the algebraic subtypes of inductive data types and the coalgebraic subtypes of coinductive data types,as well as the algebraic and coalgebraic subtypes in bialgebraic structures.Thus,the researches on the algebraic and coalgebraic subtypes are extended to the bialgebras.