A persistence module assigns a vector space to each value of a filtration parameter and a linear map to each increase of that parameter. Equivalently it is a functor from a preordered set to vector spaces, a sheaf on that set in the Alexandrov topology, or a graded module over the monoid ring built from the parameter monoid. Taking those three viewpoints seriously lets the standard machinery of homological algebra be run on persistence modules: two different tensor products (a graded one and a sheaf-theoretic one), the matching internal Hom functors and their adjunctions, and the derived functors Tor and Ext obtained from projective and injective resolutions. The interval modules that make up a barcode can then be classified by which ones are projective, injective or flat, and Kuenneth and universal-coefficient theorems relate the persistent homology of a product filtration to the persistent homology of its factors, with the additive filtration on the product corresponding to the graded tensor product and the maximum filtration to the sheaf tensor product. The category of persistence modules is itself a Grothendieck category and is closed symmetric monoidal, so it is enriched over itself.
A persistence module is a diagram of algebraic objects indexed by a preordered set: in topological data analysis, the vector spaces produced by applying homology to a filtration of a space, indexed b…