TITLE: "A proof that algebraic topology can never have a non self-contradictory set of abelian groups" - Dr. Sheldon Cooper
QUESTION [30 upvotes]: In the current episode "The Big Bang Theory", Dr. Sheldon Cooper has a booklet titled "A proof that algebraic topology can never have a non self-contradictory set of abelian groups". I'm still an undergrad in mathematics and I have no idea what an algebraic topology is and why it would never have "a non self-contradictory set of abelian groups". My first reaction of course was to read up the wikipedia article, but it's really short and doesn't explain a lot. So instead of reading through tons of articles, I wanted to ask whether it is possible to just very shallowly explain what this is about. 
What is an algebraic topology (I know what a topology is and I'm currently learning algebra)? What is "a non self-contradictory set of abelian groups" (I surely know what abelian groups are)? And how would one prove this (if I had to read up a lot to understand the proof, I guess I'd rather be left without proof)?
I hope this is the right place to ask this question, thanks in advance for answers.

REPLY [4 votes]: It does seem to be nonsense the best I got to making sense out of it was the following way:
There is reference to "an" algebraic topology as opposed to just algebraic topology (it is actually a field of study not an object). This might be interpreted as "a functor from the category of topological spaces to some category of algebraic objects" like the category of groups or abelian groups (like homology groups). This resembles Segal's definition of conformal field theory which is different from the way physicists would define it. 
Then they say it contains no self-contradictory abelian groups. The only thing I can think of there is that this functor would not send two homeomorphic spaces to non-isomorphic abelian groups i.e. the abelian group is an invariant of the space (up to homeomorphism). 
However the definition of a functor already fixes this (which it should since the idea of functor comes from algebraic topology while looking for a way to find this sort of invariants for topological spaces, as far as I know) it has 
$$F(Id_X)= Id_F(X) \\\text{ and }F(f \circ g)=F(f)\circ F(g) $$
so if $f$ is an homeomorphism then just fill in $f$ inverse for $g$ and you see that $F(g)$ is the left and right inverse to $F(f)$. So the algebraic objects are isomorphic. So even though one might want to show this it is just plain silly (even for a five year old) to then only look at abelian groups since it is a purely categorical matter.