Full and faithful functors

Full and faithful are properties of functors weakly analogous to the functional properties of surjectivity and injectivity, respectively.

Definition

Given a functor $F:𝐂\to 𝐃$, "hom-set mapping" refers to the functor's map $F_{x,y}:𝐂[x\to y]\to 𝐃[Fx\to Fy]$, for $𝐂$-objects $x$ and $y$.

• A functor is called faithful if all of its hom-set mappings are injective.

• A functor is called full if all of its hom-set mappings are surjective.

• If a functor is both faithful and full, it is called fully-faithful.[cite:@riehl2017category][cite:@enwiki:1249350446]

Relation to injectivity and surjectivity

It must be understood that a faithful functor is not necessarily injective on morphisms nor objects, and similarly for full functors.[cite:@riehl2017category] Fullness and faithfulness are "local" conditions that consider the individual homsets in isolation. The difference can be illustrated by an example:[cite:@josh2014answer]

A functor that is injective on morphisms and objects is called an embedding by some authors, though the term isn't widely agreed upon, and even then the word "embedding" is used for other unrelated concepts.[cite:@enwiki:1311108838] Most atrocious is the ambiguity arising from some using the term to mean "injective on objects," and some using the term to mean "fully-faithful."[cite:@enwiki:1297080998]

Let $𝐂$ be a category defined by the following digraph:

(quiver) \begin{tikzcd} A & B \\ C & D \arrow["f", from=1-1, to=1-2] \arrow["g", from=2-1, to=2-2] \end{tikzcd}, ParseError: No such environment: tikzcd at position 7: \begin{̲t̲i̲k̲z̲c̲d̲}̲ A & B \\ C &…

and let $𝐃$ be defined by

(quiver) \begin{tikzcd} P & Q \arrow["h", from=1-1, to=1-2] \end{tikzcd}. ParseError: No such environment: tikzcd at position 7: \begin{̲t̲i̲k̲z̲c̲d̲}̲ P & Q \arrow…

Lastly, define the faithful functor $F:𝐂\to 𝐃$ on objects and morphisms by

The functor $F$ is fully-faithful because each individual homset mapping is injective and surjective, but is clearly non-injective when considering the union of all homsets.