An object is called a discrete object if for any , the category is (equivalent to) a discrete category, i.e. a set. If has a (2-)terminal object , this is equivalent to saying that the unique map is a discrete morphism.
NB: it is more common to define these concepts in the other order: to first define an object to be discrete, as we have done, and then say that is a discrete morphism if is a discrete object in the slice 2-category . In general this does not result in the same notion of “discrete morphism” as the definition we have given. For instance, if is the interval category and is the free parallel pair , then the obvious functor is a discrete object of , but is not faithful.
However, the two definitions do coincide for fibrations, opfibrations, and two-sided fibrations. That is, if is a fibration or an opfibration in , then it is faithful and conservative if and only if it is a discrete object of , and similarly if is a two-sided fibration, then is faithful and conservative if and only if it is a discrete object of . Since this is usually the case of most interest (giving rise to discrete fibrations and, dually, codiscrete cofibrations), the difference between the two definitions is usually unimportant.
Mike Shulman: I believe that in cases when the two are different, it is the one given above (faithful and conservative) that is often the better one; hence my proposal in writing this page to change terminology slightly. Disagreements are welcome.
Discrete morphisms are often the right class of a factorization system. This factorization system, or one related to it, plays a role in the construction of a proarrow equipment from codiscrete cofibrations.