nLab faithful morphism

Definition

A morphism $f\colon A\to B$ in a 2-category $K$ is said to be (representably) faithful if for all objects $X$, the induced functor

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

is faithful. In Cat, this is equivalent to $f$ being faithful in the usual sense.

Remarks

• Faithful morphisms may also be called 2-monic and said to make their source into a 2-subobject of their target. See subcategory for discussion.

• Faithful morphisms often form the right class of a factorization system. In Cat, the left class consists of essentially surjective and full functors.

• Of course, any fully faithful morphism is also faithful, and in particular any inverter or equifier is faithful. Moreover, any inserter is also faithful, though not generally fully-faithful.

Related pages

• the analogous concept in (∞,1)-categories is that of 0-truncated morphisms (see there).

• subcategory

• fully faithful morphism

• conservative morphism

• pseudomonic morphism

• discrete morphism