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.


  • 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.


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Last revised on March 17, 2021 at 13:49:57. See the history of this page for a list of all contributions to it.