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 
between two (locally small) categories defines a function for each hom set: 
. We now apply our standard definitions to these functions.
We say that a functor is “full” if for every pair of objects 
and 
in 
the function 
is surjective. Similarly, we say that a functor is “faithful” if each function 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 
of a category consists of a subclass of the objects of and a subclass of the morphisms of so that
- For each arrow in both its source and target objects are in .
- For each object in its identity arrow is in .
- For each pair of composable arrows in their composite is in .
These conditions are enough to ensure that is a category in its own right. There is an inclusion functor 
sending each object and morphism of to itself in . Clearly it’s faithful, since any two distinct arrows in are distinct in .
What if this inclusion functor is also full? Then given any two objects and in , every morphism 
in is included in . We say that is a “full subcategory” of . In practice, this means that we’ve restricted which objects of we care about, but we keep all the morphisms between the objects we keep. For example, the category 
of abelian groups is a full subcategory of 
.
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 
to the category 
is faithful, even though it appears to lose a lot of information. Indeed, though we identify both objects from in the image, the function on each hom-set is injective. Similarly, either functor from to is full. One condition on the object map that’s useful is that a functor be “essentially surjective”. A functor 
has this property if each object 
in the target category is isomorphic to an object 
in the image of .
Now, let functors and 
furnish an equivalence of categories. I say that (and, by symmetry, 
) is full, faithful, and essentially surjective. Indeed, since 
we have an isomorphism 
for each object 
, so is essentially surjective. The natural isomorphism 
says that for every arrow 
we have 
. The right side of this equation is a bijection from 
to 
, and the left side is the composite 
. If were not full then the left side could not be always surjective, while if were not faithful then the right side could not be always injective. Since the right side is always a bijection, must be full and must be faithful. By symmetry is essentially surjective and faithful, and is full.
On the other hand, if a functor 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 . Since each object of 
is isomorphic to some object in the image of we pick such an object 
for each and set 
. Then we have to find mappings of the hom-sets of in a coherent way so that the resulting composites 
and 
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.
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June 5, 2007 -
Posted by John Armstrong |
Category theory
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It seems to me that the exposition in the third-from-last paragraph is a bit mixed up. The isomorphism F\circ G\cong1_\mathcal{D} is what makes F essentially surjective.
\eta is reversed from what it usually is, and the positions of C_1 and C_2 need to be swapped in G(F(f))=\eta_{C_1}\circ f\circ\eta_{C_2}^{-1}; so I’d suggest reformulating this to G(F(f))=\eta_{C_2}\circ f\circ\eta_{C_1}^{-1}. Then all goes fine afaics.
Avery, you’re right that I added the phrase about essential surjectivity to the wrong sentence, but I reversed the usual direction of
(and the ensuing alterations to compensate) for a reason: this makes it clearer that an equivalence is actually a type of adjunction. I can flip the directions of these natural isomorphisms because they’re isomorphisms, but once I weaken them to mere natural transformations I no longer have that freedom.
Right, but what about \eta_{C_2}\circ f\circ\eta_{C_1}^{-1} vs \eta_{C_1}\circ f\circ\eta_{C_2}^{-1}; the latter is still looking wrong to me. Too bad categorists don’t use some sort of postfix notation for function-like things.
Yes, that composition was also backwards.. A postfix notation would be useful, but unfortunately everyone else uses this ordering, and if I started out using the other then everyone reading my stuff would be confused when they try to read other material.
oops, \eta_{C_2}^{-1}\circ f\circ\eta_{C_1}, if \eta is from GF to 1_{\mathcal C}.
Incidentally, Avery, try wrapping your LaTeX like this:
::dollar-sign::latex foo::dollar sign::
It comes up:
. And it works on all WordPress-hosted weblogs!
cool. Maybe a `guide for commenters’ would be a useful addition?
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