Laura Jakobsson

The category of cellular resolutions

Cellular resolutions are a particular type of projective resolutions,
where the modules under consideration are associated with cell complexes and so
these resolutions also have combinatorial and topological information. Cellular
resolutions are 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. Our work has been constructing a category of cellular
resolutions and then looking at the classical category theoretical properties
and methods that have been very fruitful lately in the study of combinatorial
categories. Using these results we have shown that desirable properties
lift from topology to cellular resolutions, like homotopy colimits, and that
the ideas of representation stability can be applied to certain families of
cellular resolutions.