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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

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

• subcategory

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

• fully faithful morphism

  subcategory

• conservative morphism

  fully faithful morphism

• pseudomonic morphism

  conservative morphism

• discrete morphism

  pseudomonic morphism

• discrete morphism