Homological Algebra

There is a subject where Grothendieck’s name is always in the citation
It is a simple and beautiful topic with perplexing notation

Once upon a time there is a colimit named direct limit
Feeling isolated until she met a limit named inverse limit
They met at a category of modules at the edge of a ring

Baffled on why she is a limit of a sequence when she is an initial object
Questioning their uniqueness to her dual whom an inverse limit whilst a terminal object

He convinced her of their uniqueness up to isomorphism
By reciting their universal property of morphism

We are unique
He say,
Yet we are not alone
He assured,

He told her the story about  comohology of a complex
whom definitively is a homology albeit at sixes and sevens as homology is for cocomplex
Yes you are right, they are not even a dual
At the category derived from modules at the edge of a ring

As puzzled as she is
She is happy not being the only one
Also not the most bewildering
As past the morphism
Inside the dark inner part of modules
There are always terminals of the initials
There are always limits of colimits
For the image is the kernel of the cokernel


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