nLab full and faithful functor




A full and faithful functor is a functor which is both full and faithful. That is, a functor F:CDF\colon C \to D from a category CC to a category DD is called full and faithful if for each pair of objects x,yCx, 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.


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:xyf, g : x \rightrightarrows y and the identity-on-objects endofunctor FF sending fff \mapsto f and gfg \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.


  • 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:CDF : 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 ILRI L \dashv R be an adjunction. If II is fully faithful, then LRIL \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.)


basic properties of…


Last revised on December 11, 2023 at 13:42:15. See the history of this page for a list of all contributions to it.