The Unapologetic Mathematician

Mathematics for the interested outsider

Full and Faithful Functors

We could try to adapt the definitions of epics and monics to functors, but it turns out they’re not really the most useful. Really what we’re most concerned with in transforming one category into another is their morphisms. The more useful notions are best phrased specifically with regard to the hom-sets of the categories involved.

Remember that a functor F:\mathcal{C}\rightarrow\mathcal{D} between two (locally small) categories defines a function for each hom set: F_{X,Y}:\hom_\mathcal{C}(X,Y)\rightarrow\hom_\mathcal{D}(F(X),F(Y)). We now apply our standard definitions to these functions.

We say that a functor is “full” if for every pair of objects X and Y in \mathcal{C} the function F_{X,Y} is surjective. Similarly, we say that a functor is “faithful” if each function F_{X,Y} is injective. When we apply these definitions to monoids (considered as categories with one object) we recover monomorphisms and epimorphisms of monoids, so we know we’re on the right track.

Now a subcategory \mathcal{S} of a category \mathcal{C} consists of a subclass of the objects of \mathcal{C} and a subclass of the morphisms of \mathcal{C} so that

  • For each arrow in \mathcal{S} both its source and target objects are in \mathcal{S}.
  • For each object in \mathcal{S} its identity arrow is in \mathcal{S}.
  • For each pair of composable arrows in \mathcal{S} their composite is in \mathcal{S}.

These conditions are enough to ensure that \mathcal{S} is a category in its own right. There is an inclusion functor I:\mathcal{S}\rightarrow\mathcal{C} sending each object and morphism of \mathcal{S} to itself in \mathcal{C}. Clearly it’s faithful, since any two distinct arrows in \mathcal{S} are distinct in \mathcal{C}.

What if this inclusion functor is also full? Then given any two objects X and Y in \mathcal{S}, every morphism f:X\rightarrow Y in \mathcal{C} is included in \mathcal{S}. We say that \mathcal{S} is a “full subcategory” of \mathcal{C}. In practice, this means that we’ve restricted which objects of \mathcal{C} we care about, but we keep all the morphisms between the objects we keep. For example, the category \mathbf{Ab} of abelian groups is a full subcategory of \mathbf{Grp}.

Note that we have not given any conditions on the map of objects induced by a functor. In fact, the (unique) functor from the category \mathbf{2} to the category \mathbf{1} is faithful, even though it appears to lose a lot of information. Indeed, though we identify both objects from \mathbf{2} in the image, the function on each hom-set is injective. Similarly, either functor from \mathbf{1} to \mathbf{2} is full. One condition on the object map that’s useful is that a functor be “essentially surjective”. A functor F has this property if each object D in the target category is isomorphic to an object F(C) in the image of F.

Now, let functors F:\mathcal{C}\rightarrow\mathcal{D} and G:\mathcal{D}\rightarrow\mathcal{C} furnish an equivalence of categories. I say that F (and, by symmetry, G) is full, faithful, and essentially surjective. Indeed, since F\circ G\cong1_\mathcal{D} we have an isomorphism F(G(D))\cong D for each object D\in\mathcal{D}, so F is essentially surjective. The natural isomorphism \eta:G\circ F\rightarrow 1 says that for every arrow f:C_1\rightarrow C_2 we have G(F(f))=\eta_{C_2}^{-1}\circ f\circ\eta_{C_1}. The right side of this equation is a bijection from \hom_\mathcal{C}(C_1,C_2) to \hom_\mathcal{C}(G(F(C_1)),G(F(C_2))), and the left side is the composite G_{F(C_1),F(C_2)}\circ F_{C_1,C_2}. If G were not full then the left side could not be always surjective, while if F were not faithful then the right side could not be always injective. Since the right side is always a bijection, G must be full and F must be faithful. By symmetry G is essentially surjective and faithful, and F is full.

On the other hand, if a functor F:\mathcal{C}\rightarrow\mathcal{D} is full, faithful, and essentially surjective, then it is one side of an equivalence of categories. The proof is a bit more involved, but basically it proceeds by building a functor G. Since each object of \mathcal{D} is isomorphic to some object in the image of F we pick such an object F(C)\cong D for each D and set G(D)=C. Then we have to find mappings of the hom-sets of \mathcal{D} in a coherent way so that the resulting composites F\circ G and G\circ F are naturally isomorphic to the identity functors. I won’t fill in all the details here, but suffice it to say it can be done.

This, then, characterizes equivalences of categories the way injectivity and surjectivity of functions characterize isomorphisms of sets.

June 5, 2007 Posted by | Category theory | 10 Comments