  Cellular resolutions is a well studied topic on the level of single
resolutions and certain specific families of cellular resolutions. One question
coming out of the work on families is to understand the structure of cellular
resolutions more generally. We give a starting point to understanding higher
level structures by defining the category of cellular resolutions. In this
paper we study the properties of this category. The main results are in lifting
homotopy colimits from topology and Morse theory on cellular resolution being
compatible with the category.
