nLab
fully faithful morphism

Definition

Let $K$ be a 2-category.

A morphism $f:A\to B$ in $K$ is called (representably) fully-faithful (or sometimes just ff) if for all objects $X\in K$ , the functor

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

is full and faithful.

Remarks

• Fully faithful morphisms in a 2-category may also be called 1-monic, and be said to make their source into a 1-subobject of their target. See subcategory for some discussion.

• Fully faithful morphisms are often the right class of a factorization system. The left class in $\mathrm{Cat}$ consists of essentially surjective functors; in a regular 2-category? it consists of the eso morphisms.

• Just as in a 1-category any equalizer is monic, in a 2-category any inverter or equifier is fully faithful.

Variations

This is not always the “right” notion of fully-faithfulness in a 2-category. In particular, in enriched category theory this definition does not recapture the correct notion of enriched fully-faithfulness. It is possible, however, to characterize $V$-fully-faithful functors 2-categorically; see codiscrete cofibration.

Examples

In the 2-category Cat the full and faithful morphisms are precisely the full and faithful functors.

Related pages

• subcategory
• faithful morphism
• conservative morphism
• pseudomonic morphism
• discrete morphism
• eso morphism