modules, since they provide more structure for studying the shape of data. Namely, multiparameter modules can have greater density sensitivity and robustness to outliers than single-parameter modules, making them a potentially useful tool for data analysis. One downside of multiparameter persistence is its inherent complexity. This makes performing computations related to multiparameter persistence modules difficult. In the worst case, the computational complexity of multidimensional persistent homology is exponential. Examples Homology Modules When using homology with coefficients in a field, a homology group has the structure of a vector space. Therefore, given a filtration of spaces , by applying the homology functor at each index we obtain a persistence module for each called the (th-dimensional) homology module of . The vector spaces of the homology module can be defined index-wise as for all , and the linear maps are induced by the inclusion maps of . Homology modules are the most ubiquitous examples of persistence modules, as they encode information about the number and scale of topological features of an object (usually derived from building a filtration on a point cloud) in a purely algebraic structure, thus making understanding the shape of the data amenable to algebraic techniques, imported from well-developed areas of mathematics such as commutative algebra and representation theory. Interval Modules A primary concern in the study of persistence modules is whether modules can be decomposed into "simpler pieces", roughly speaking. In particular, it is algebraically and computationally convenient if a persistence module can be expressed as a direct sum of smaller modules known as interval modules. Let be a nonempty subset of a poset . Then is an interval in if For every if then For every there is a sequence of elements such that , , and are comparable for all . Now given an interval we can define a persistence module index-wise as follows: ; . The module is called an interval module. Free Modules Let . Then we can define a persistence module with respect to where the spaces are given by , and the maps defined via . Then is known as a free (persistence) module. One can also define a free module in terms of decomposition into interval modules. For each define the interval , sometimes called a "free interval." Then a persistence module is a free module if there exists a multiset such that . In other words, a module is a