nLab full and faithful functor

Definition

A full and faithful functor is a functor which is both full and faithful. That is, a functor $F\colon C \to D$ from a category $C$ to a category $D$ is called full and faithful if for each pair of objects $x, y \in C$, the function

$$F\colon C(x, y) \to D(F(x), F(y))$$

between hom sets is bijective. “Full and faithful” is sometimes shortened to “fully faithful” or “ff.” See also full subcategory.

Remark

It is not sufficient for there simply to exist some isomorphism between $C(x, y)$ and $D(F(x), F(y))$. For instance, consider the category comprising a parallel pair $f, g : x \rightrightarrows y$ and the identity-on-objects endofunctor $F$ sending $f \mapsto f$ and $g \mapsto f$. We have $C(x, y) \cong C(F(x), F(y)) = C(x, y)$, but this functor is not fully faithful. However, it is enough for there to exist some natural isomorphism.

Properties

• Together with bijective-on-objects functors, fully faithful functors form an orthogonal factorization system on $Cat$; see bo-ff factorization system. More invariantly, pair them with essentially surjective functors to get a bicategorial factorization system.

• In particular, fully faithful functors are stable under pullback.

• A fully faithful functor (hence a full subcategory inclusion) reflects all limits and colimits, also all isomorphisms (is a conservative functor). This is evident from inspection of the defining universal property.

• Fully faithful functors that are Injective-on-objects are closed under pushouts in Cat: see full embedding for references.

• Fully faithful functors $F : C \to D$ can be characterized as those functors for which the following square is a pullback, where the vertical maps are source and target, and the horizontal maps are induced by $F$

  $$\array{ C^{[1]} &\to& D^{[1]} \\ \downarrow && \downarrow \\ C \times C &\to& D \times D }$$

• The bijections exhibiting full faithfulness of $F$ form a natural isomorphism, by functoriality of $F$ and of pre- and postcomposition.

• Let $I L \dashv R$ be an adjunction. If $I$ is fully faithful, then $L \dashv R I$. In this case, the two adjunctions induce the same monad. This is Proposition 1.1 of DFH75. (For a converse, see dominant functor.)

Generalizations

• For (∞,1)-categories the corresponding notion of fully faithful functor is described at fully faithful (∞,1)-functor. This is part of a bigger pattern at work here which is indicated at stuff, structure, property and k-surjective functor.

• Inside a 2-category there is a “representable” notion of ff morphism.

• There is also a notion of fully faithful functor in enriched category theory, which in general is stronger than being an ff morphism in the 2-category of enriched categories. But it can be expressed internally in any proarrow equipment.

Related concepts

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

  • homotopy final functor, final (∞,1)-functor

References

• Francis Borceux, Section 1.5 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory [doi:10.1017/CBO9780511525858]

• Aristide Deleanu?, Armin Frei, Peter Hilton, Idempotent triples and completion, Mathematische Zeitschrift 143 (1975) 91-104 [doi:10.1007/BF01173053]