Chain Complexes (Pt. II)
Point of Post: This is a continuation of this post.
I’d like to pause to point out a useful fact. Although not at all apparent from the definition of , is actually just the image of under a full embedding . Indeed, for each object in define to be the element of with and for . Define then a functor which, on objects, is and on morphisms just sends to the chain map with at and the zero map elsewhere.
Ok, back to buisness. So, right now we have that is preadditive whenever is. So, the next step is to show that is additive whenever is. So, unsurprisingly, to show that has direct sums we only need to do things piecewise. Namely, suppose that we have two objects and in . We claim that if are the ingredients to the direct sum of the individual objects then the object along with the chain maps give a direct sum of and in . Of course, this is quite easy to check, and I leave it to you. Thus, we may conclude that is an additive category if is.
Now, we’re on to discussing how to make preabelian when . Not surprisingly, considering how things have been going so far, we can get by just by defining everything piecewise. That said, unlike the other constructions where everything just kind of “worked out” there is a bit of subtlety here.Namely, suppose that we have a chain map . Define then to be the object of with and with boundary maps equal to….what? There is no obvious candidate to get from to . They key though is this. A priori we only know have the following diagram
But, since is a chain map we get that and so we get a unique arrow making the following diagram commute
Let’s check real quick that these actually compose to . Namely, to check that . But writing out the diagram one can check that and since is a monomorphism we may conclude that as desired.
Now before we go on to actually construct the kernel of we’d like to note something. Not only were the maps between the objects in our new construction less obvious in their definition than they were before, they are also less obvious as to what these look like in tame abelian categories, like . I mean, everyone knows what the direct sum of maps in looks like (just in case ) but it’s not at all obvious the form that takes in . Luckily enough this is all simple. Recalling that in our map is just the inclusion one can check that our are nothing more than the restriction of the maps to .
Ok, so back to the kernel of the matter (bad, I know). So, know we at least have an element of with and with boundary map our constructed –we denote this element of as . We can then construct a “chain map” –I put scare quotes around the chain map just because it’s not a priori obvious that it satisfies the necessary commutativity of a chain map. But, by some kind of miracle (:wink:) the have been constructed precisely so that this is true. Namely, we constructed to be the morphism which came from the fact that annihilated , but then we know that this is made to make a commutative triangle and so (if one writes out the diagram) give –tada!
The last thing we need to check is that is an honest to god categorical kernel of , but considering all the work we’ve already done with this construction I believe the interested reader would have absolutely no trouble doing it themselves. Thus, with all of this we can finally state that if is preabelian then so is .
So, our quest is coming to a close since all we now have left to do is verify that is whenever is. But, I claim this is simple–well, if you’re willing to have a little faith. After doing forty-five kabillion dealings with things in just being the piecewise versions of things in I don’t think you’ll find it hard to believe that if we are given a morphism then and and the natural map is nothing more than the chain map and since a chain map is an isomorphism if and only if each coordinate map is an isomorphism everything basically just falls out.
So, after all that work we can definitely say:
Theorem: Let be an -category, preadditive, additive, preabelian, or abelian. Then, so is .
 Weibel, Charles A. An Introduction to Homological Algebra. Cambridge [England: Cambridge UP, 1994. Print.
 Schapira, Pierre. “Categories and Homological Algebra.” Web. <http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf>.
 Rotman, Joseph. An Introduction to Homological Algebra. Dordrecht: Springer, 2008. Print.
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