Ab-categories and Preadditive Categories (Pt. II)
Point of Post: This is a continuation of this post.
Given two -categories and there is an obvious way in which functors can interact with the abelian structures of and respectively. Namely, for each objects in we know that produces a (set) map –thus if we want to not only provide a categorical correspondence between and but one which respects the -category structure of both, it seems logical to consider functors for which these induces maps are group homomorphisms. To make a definition out of it, if and are -categories and is a functor such that the induced set map is a group homomorphism for each objects in then we call an additive functor.
Let’s look at some examples of additive functors:
We claim that covariant Hom functor is an additive functor. Indeed, this amounts to showing that if are two -maps then where, as usual, . But, this is simple since for each morphism one has that
In fact, this is a simple consequence of the following general fact:
Theorem: Let be an -category and an object of . Then, the contravariant Hom functor is an additive functor .
Proof: We have, by assumption, that is an abelian group for each an object of , and furthermore that for each (where, as usual, denotes the image of under ). Thus, it remains to be shown that if is a -morhphism then is a group homomorphism, and that for all morphisms . In fact, each of these facts follows from one side of the distributivity axioms. For example, is a group homomorphism since applying right distributivity gives us
so that, indeed, . To prove that for all morphism we merely note that left distributivity gives us
for all , so that follows.
For our next example let be some fixed unital ring. Then, an additive functor is nothing more than a unital left – module . Indeed, suppose first that is such a functor and let be the abelian group which is the image under of the unique object of . Note then that since our functor gives us a map such that (by definition of a functor) and (by definition of an additive functor) , and such that . Thus, we see that an additive functor can be thought of as nothing more than an abelian group and a unital ring homomorphism (where, as usual, is the endomorphism ring)–but from first principles this is nothing more than a left (or right) -module. This examples often leads people to define a –module for an -category to be an additive functor .
For our last example, we show that the direct limit functor is additive. Indeed, this amounts to showing that if are morhpisms then . But, we recall that if is a limit cone for and is a limit cone for then is the unique -map such that . But, we note then that
from where the desired equality follows from the aforementioned uniqueness.
One of the interesting facts about additive functors is that they necessarily take zero objects to themselves. To prove this we first note the following general facts:
Theorem: Let be an -category and three objects in . Then, if then .
Proof: Indeed, and so the rest follows by cancellation. The other direction follows similarly.
Remark: Of course, this is a generalization of the fact that .
Theorem: Let be an -category. Then, an object in is a zero object if and only if is trivial.
Proof: Evidently if is a zero object then is trivial. Conversely, suppose that was trivial and let be an arbitrary object of . Conversely, to show that is a zero object, we will show that given any object there exists unique morphisms and . But, since we know there exists such morphisms (by assumption that is an -category) it suffices to prove their uniqueness. To do this we note that since is trivial we must have that and so, by the previous theorem, for any morphisms we have that and similarly if then . The conclusion follows.
With this we can confidently state and prove:
Theorem: Let and be -categories and an additive functor. If is a zero object for then is a zero object for .
Proof: By the previous theorem it suffices to show that is trivial, or that . But, since is a group homomorphism we have that and since is a functor we have that . Recalling that finished the argument.
Corollary: Let and be -categories and an additive functor, then if is a zero map, then is a zero map.
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