For a fixed small category D and homotopical category M, the homotopy colimit functor should be a derived functor of colim: MD → M. Assuming that the homotopical category M is saturated, which it always is in practice, the homotopy type of a homotopy colimit will not depend on which flavor of derived functor is used. Because colimits are left adjoints, we might hope that colim has a left derived functor and, dually, that lim: MD → M has a right derived functor, defining homotopy limits.

For special types of diagrams (e.g., if D is a Reedy category), there are simple modfications that produce the right answer. For instance, homotopy pullbacks can be computed by replacing the maps by fibrations between fibrant objects. Or if the homotopical category M is a model category of a particular sort (cofibrantly generated for colimits or combinatorial for limits), then there exist suitable model structures (the projective and injective, respectively) on MD for which Quillen's small object argument can be used to produce deformations for the colimit and limit functors.

These well-documented solutions are likely familiar to those acquainted with model category theory, but they are either quite specialized (those depending on the particular diagram shape) or computationally difficult (those involving the small object argument). By contrast, the theory of derived functors developed in Chapter 2 does not require the presence of model structures on the diagram categories.