faithful functor

A functor $F \colon \mathcal{C} \to \mathcal{D}$ from a category $\mathcal{C}$ to a category $\mathcal{D}$ is called *faithful*, if for each pair of objects $x, y \in \mathcal{C}$, its function $F_{x,y}$ on hom-sets is injective:

$\array{
\mathcal{C}(x,y)
&\xhookrightarrow{\;\; F_{x,y} \;\;}&
\mathcal{D}(F(x), F(y))
\\
(x \overset{\phi}{\to} y)
&\mapsto&
\big(
F(x)\overset{F(\phi)}{\to} F(y)
\big)
\,.
}$

More abstractly, we may say that a functor is faithful if it is $2$-surjective – or loosely speaking, ‘surjective on equations between given morphisms’.

See also *faithful morphism* for a generalization to an arbitrary 2-category.

And see 0-truncated morphism for generalization to (∞,1)-categories (see there).

This generalization is about extending to morphisms in general (∞,1)-categories the fact that in $\infty{}Grpd$, $0$-truncated morphisms give a reasonable notion of faithful functor.

In particular, the notion of a “faithful morphism in the (∞,1)-category of (∞,1)-categories” does *not* give the right notion of a “faithful functor between (∞,1)-categories”.

A faithful functor reflects epimorphisms and monomorphisms.

(The simple proof is spelled out for instance at *epimorphism*.)

**basic properties of…**