There are various ways to define homology and cohomology theories in algebra. One standard method is the theory of derived functors. These require two ingredients: a half-exact additive functor between abelian categories and enough projectives or injectives. In the theory of derived functors, we are not concerned with specific resolutions because of comparison lemmas which give us independence from them, although when we actually go to calculate homology, we will need specific resolutions, at least initially.

Another idea in homological algebra is instead to start with a specific resolution or resolution-like “thing” of an object $ X$, and then define a homology theory based upon it by applying a functor with values in an abelian category. Here, the two ingredients are an $ X$-“resolution” in $ \mathcal{C}$ and a functor $ F:\mathcal{C}\to\mathcal{A}$ where $ \mathcal{A}$ is an abelian category. The put quotation marks around “resolution” because $ \mathcal{C}$ may not be abelian, and for that matter the objects in $ \mathcal{C}$ won’t be chain complexes but rather simplicial objects. Hence if $ C\in\mathcal{C}$ then $ F(C)$ won’t be a chain complex but rather a simplicial abelian object, and then we will have to get a chain complex from it. However, all of this is a technicality and will be saved for a future post.

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