Contents

category theory

# Contents

## 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.

## Properties

• 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 $F : C \to D$ 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.

## References

• R. Fritsch, D. M. Latch, Homotopy inverses for nerve, Math. Z. 177 (1981), no. 2, 147–179, doi:10.1007/BF01214196.

• Alexandru E. Stanculescu, Constructing model categories with prescribed fibrant objects, Theory and Applications of Categories, Vol. 29, (2014) No. 23, pp 635-653, journal page, arXiv:1208.6005.

Last revised on March 17, 2021 at 09:49:57. See the history of this page for a list of all contributions to it.