nLab
discrete morphism

Discrete morphisms

Definition

A morphism f:ABf\colon A\to B in a 2-category KK is called discrete if it is representably faithful and conservative, i.e. if for any object XX the induced functor

K(X,A)K(X,B) K(X,A) \to K(X,B)

is faithful and conservative.

An object AA is called a discrete object if for any XX, the category K(X,A)K(X,A) is (equivalent to) a discrete category, i.e. a set. If KK has a (2-)terminal object 11, this is equivalent to saying that the unique map A1A\to 1 is a discrete morphism.

Caveat

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 f:ABf\colon A\to B is a discrete morphism if is a discrete object in the slice 2-category K/BK/B. In general this does not result in the same notion of “discrete morphism” as the definition we have given. For instance, if BB is the interval category (01)(0\to 1) and AA is the free parallel pair (01)(0 \rightrightarrows 1), then the obvious functor ABA\to B is a discrete object of Cat/BCat/B, but is not faithful.

However, the two definitions do coincide for fibrations, opfibrations, and two-sided fibrations. That is, if f:ABf\colon A\to B is a fibration or an opfibration in BB, then it is faithful and conservative if and only if it is a discrete object of K/BK/B, and similarly if AEBA\leftarrow E \to B is a two-sided fibration, then EA×BE\to A\times B is faithful and conservative if and only if it is a discrete object of K/(A×B)K/(A\times B). 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.

Examples

Discrete categories

A discrete object in the 2-category Cat is, of course, a discrete category.

Discrete groupoids

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.

See also the discussion at discrete space and discrete groupoid.

Factorization systems and discrete reflections

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.

Revised on March 9, 2012 20:17:58 by Urs Schreiber (82.113.106.131)