Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
typical contexts
A morphism in a 2-category is called discrete if it is representably faithful and conservative, i.e. if for any object the induced functor
is faithful and conservative.
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.
A discrete object in the 2-category Cat is, of course, a discrete category. A functor is a discrete morphism if and only if the functor itself is faithful and conservative.
A discrete object in the (2,1)-category Grpd of groupoids is also called a 0-truncated object or 0-groupoid or homotopy 0-type or just 0-type or h-set.
See also the discussion at discrete space and discrete groupoid.
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.
Last revised on December 9, 2023 at 02:55:46. See the history of this page for a list of all contributions to it.