nLab
discrete morphism

Definition

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

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

is faithful and conservative.

An object $A$ is called a discrete object if for any $X$, the category $K(X,A)$ is (equivalent to) a discrete category, i.e. a set. If $K$ has a (2-)terminal object $1$, this is equivalent to saying that the unique map $A\to 1$ 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 $f:A\to B$ is a discrete morphism if is a discrete object in the slice 2-category $K/B$. In general this does not result in the same notion of “discrete morphism” as the definition we have given. For instance, if $B$ is the interval category $(0\to 1)$ and $A$ is the free parallel pair $(0\rightrightarrows 1)$, then the obvious functor $A\to B$ is a discrete object of $\mathrm{Cat}/B$, but is not faithful.

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