nLab fully faithful morphism




Let KK be a 2-category.

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

K(X,A)K(X,B)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 XKX \in K , the functor

K(B,X)K(A,X)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 opCat^{op} (rather than Cat coCat^{co}.



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 VV-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 VV-fully-faithfulness. A 1-cell f:ABf : A \to B in a proarrow equipment is fully faithful if the unit of the adjunction η:1f *f *\eta \colon 1 \to f^* f_* is invertible.


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.

Last revised on April 22, 2023 at 16:10:13. See the history of this page for a list of all contributions to it.