full and faithful functor (changes) in nLab

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Definition

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

F:C(x,y)→D(F(x),F(y))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.

\begin{remark}\label{NeedForNaturality} It is not sufficient for there simply to exist some isomorphism between C(x,y)C(x, y) and D(F(x),F(y))D(F(x), F(y)). For instance, consider the category comprising a parallel pair f,g:x⇉yf, g : x \rightrightarrows y and the identity-on-objects endofunctor FF sending f↦ff \mapsto f and g↦fg \mapsto f. We have C(x,y)≅C(F(x),F(y))=C(x,y)C(x, y) \cong C(F(x), F(y)) = C(x, y), but this functor is not fully faithful. \end{remark}

Properties

• Together with bijective-on-objects functors, fully faithful functors form an orthogonal factorization system on CatCat; 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 are closed under pushouts in Cat. For ordinary categories this was proven by Fritch and Latch; for enriched categories it is proven in Stanculescu, Prop. 3.1, and for (∞,1)-categories it is proven in Simspon, Cor. 16.6.2.

• Fully faithful functors F:C→DF : 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 FF

  C [1] → D [1] ↓ ↓ C×C → D×D \array{ C^{[1]} &\to& D^{[1]} \\ \downarrow && \downarrow \\ C \times C &\to& D \times D }

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

• Let IL⊣RI L \dashv R be an adjunction. If II is fully faithful, then L⊣RIL \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.

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](https://doi.org/10.1017/CBO9780511525858)]

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