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

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

is full and faithful.

It is said to be co-fully faithful or corepresentably fully faithful if for all objects $X \in K$ , the functor

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

is a full and faithful functor. Note that the shortened name “co-fully faithful” can be misleading, as a functor is corepresentably fully faithful if and only if it is representably fully faithful as a 1-cell in $Cat^{op}$ (rather than $Cat^{co}$.

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 $Cat$ 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. In general, fully faithful morphisms should be defined with respect to a proarrow equipment (or some other context for formal category theory): in particular, this recovers $V$-fully-faithfulness. A 1-cell $f : A \to B$ in a proarrow equipment is fully faithful if the unit of the adjunction $\eta \colon 1 \to f^* f_*$ is invertible.

Examples

In the 2-category Cat the full and faithful morphisms are precisely the full and faithful functors; and the corepresentably fully faithful morphisms are precisely the absolutely dense functors.

Related pages

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