Full and faithful functors

Incategory theory,afaithful functoris afunctorthat isinjectiveonhom-sets,and afull functorissurjectiveon hom-sets. A functor that has both properties is called afully faithful functor.

Formal definitions

Explicitly, letCandDbe (locally small)categoriesand letF:C→Dbe a functor fromCtoD.The functorFinduces a function

$F_{X,Y}\colon \mathrm {Hom} _{\mathcal {C}}(X,Y)\rightarrow \mathrm {Hom} _{\mathcal {D}}(F(X),F(Y))$

for every pair of objectsXandYinC.The functorFis said to be

• faithfulifF_X,Yisinjective
• fullifF_X,Yissurjective
• fully faithful(=full and faithful) ifF_X,Yisbijective

for eachXandYinC.

Properties

A faithful functor need not be injective on objects or morphisms. That is, two objectsXandX′ may map to the same object inD(which is why the range of a full and faithful functor is not necessarily isomorphic toC), and two morphismsf:X→Yandf′:X′ →Y′ (with different domains/codomains) may map to the same morphism inD.Likewise, a full functor need not be surjective on objects or morphisms. There may be objects inDnot of the formFXfor someXinC.Morphisms between such objects clearly cannot come from morphisms inC.

A full and faithful functor is necessarily injective on objects up to isomorphism. That is, ifF:C→Dis a full and faithful functor and$F(X)\cong F(Y)$then$X\cong Y$.

Examples

• Theforgetful functorU:Grp→Setmapsgroupsto their underlying set, "forgetting" the group operation.Uis faithful because twogroup homomorphismswith 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 toSetis (by definition) aconcrete category;in general, that forgetful functor is not full.
• The inclusion functorAb→Grpis fully faithful, sinceAb(thecategory of abelian groups) is by definition thefull subcategoryofGrpinduced 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 befully faithfulif for everyXandYinC,the map$F_{X,Y}$is aweak equivalence.

See also

• Full subcategory
• Equivalence of categories

References

• Mac Lane, Saunders(September 1998).Categories for the Working Mathematician(second ed.). Springer.ISBN0-387-98403-8.
• Jacobson, Nathan(2009).Basic algebra.Vol. 2 (2nd ed.). Dover.ISBN978-0-486-47187-7.