A functor from a category to a category is called faithful, if for each pair of objects , its function on hom-sets is injective:
More abstractly, we may say that a functor is faithful if it is -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 , -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…
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