fully faithful morphism in nLab

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Contents

Definition

Let KK be a 2-category.

A morphism f:A→Bf:A\to B in KK is called (representably) fully-faithful (or sometimes just ff) if for all objects X∈KX \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 X∈KX \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}.

• 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 CatCat 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 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:A→Bf : A \to B in a proarrow equipment is fully faithful if the unit of the adjunction η:1→f *f *\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.

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