  Category theory is the language of homological algebra, allowing us to state
broadly applicable theorems and results without needing to specify the details
for every instance of analogous objects. However, authors often stray from the
realm of pure abstract category theory in their development of the field,
leveraging the Freyd-Mitchell embedding theorem or similar results, or
otherwise using set-theoretic language to augment a general categorical
discussion. This paper seeks to demonstrate that - while it is not necessary
for most mathematicians' purposes - a development of homological concepts can
be contrived from purely categorical notions. We begin by outlining the
categories we will work within, namely Abelian categories (building off
additive categories). We continue to develop cohomology groups of sequences,
eventually culminating in a development of right derived functors. This paper
is designed to be a minimalist construction, supplying no examples or
motivation beyond what is necessary to develop the ideas presented.
