category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a full and faithful functor.

Formal definitions

Explicitly, let ''C'' and ''D'' be ( locally small)

categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) * Category (Kant) *Categories (Peirce) * ...

and let ''F'' : ''C'' → ''D'' be a functor from ''C'' to ''D''. The functor ''F'' induces a function :

F_\colon\mathrm_(X,Y)\rightarrow\mathrm_(F(X),F(Y))

for every pair of objects ''X'' and ''Y'' in ''C''. The functor ''F'' is said to be *faithful if ''F''_''X'',''Y'' is injectiveJacobson (2009), p. 22 *full if ''F''_''X'',''Y'' is surjectiveMac Lane (1971), p. 14 *fully faithful (= full and faithful) if ''F''_''X'',''Y'' is

bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

for each ''X'' and ''Y'' in ''C''. A mnemonic for remembering the term "full" is that the image of the function fills the codomain; a mnemonic for remembering the term "faithful" is that you can trust (have faith) that

F(X)=F(Y)

implies

X=Y

Properties

A faithful functor need not be injective on objects or morphisms. That is, two objects ''X'' and ''X''′ may map to the same object in ''D'' (which is why the range of a full and faithful functor is not necessarily isomorphic to ''C''), and two morphisms ''f'' : ''X'' → ''Y'' and ''f''′ : ''X''′ → ''Y''′ (with different domains/codomains) may map to the same morphism in ''D''. Likewise, a full functor need not be surjective on objects or morphisms. There may be objects in ''D'' not of the form ''FX'' for some ''X'' in ''C''. Morphisms between such objects clearly cannot come from morphisms in ''C''. A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if ''F'' : ''C'' → ''D'' is a full and faithful functor and

F(X)\cong F(Y)

then

X \cong Y

Examples

* The forgetful functor ''U'' : Grp → Set maps

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

to their underlying set, "forgetting" the group operation. ''U'' is faithful because two

group homomorphisms In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) ...

with the same domains and codomains are equal if they are given by the same functions on the underlying sets. This functor is not full as there are functions between the underlying sets of groups that are not group homomorphisms. A category with a faithful functor to Set is (by definition) a

concrete category In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, ''see Relative concreteness below''). This functor makes it possible to think of the objects of ...

; in general, that forgetful functor is not full. * The inclusion functor Ab → Grp is fully faithful, since Ab (the

category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab ...

) is by definition the full subcategory of Grp induced by the abelian groups.

Generalization to (∞, 1)-categories

The notion of a functor being 'full' or 'faithful' does not translate to the notion of a (∞, 1)-category. In an (∞, 1)-category, the maps between any two objects are given by a space only up to homotopy. Since the notion of injection and surjection are not homotopy invariant notions (consider an interval embedding into the real numbers vs. an interval mapping to a point), we do not have the notion of a functor being "full" or "faithful." However, we can define a functor of quasi-categories to be ''fully faithful'' if for every ''X'' and ''Y'' in ''C,'' the map

F_

is a weak equivalence.

Notes

References