Homological Algebra for Persistence Modules

Homological Algebra for Banach Modules?

Injectivity is an important concept in algebra, homotopy theory and elsewhere. We study the ‘injectivity consequence’ of morphisms of a category: a morphism h is a consequence of a set H of morphisms if every object injective w.r.t. all members of H is also injective w.r.t. h. We formulate a very simple logic which is always sound, i.e., whenever a proof of h from assumptions in H exists, then ...

Persistence modules: Algebra and algorithms

Abstract. Persistent homology was shown by Zomorodian and Carlsson [35] to be homology of graded chain complexes with coefficients in the graded ring k[t]. As such, the behavior of persistence modules — graded modules over k[t] — is an important part in the analysis and computation of persistent homology. In this paper we present a number of facts about persistence modules; ranging from the wel...

Homological Algebra in the Category of Γ-modules

We study homological algebra in the abelian category Γ̃, whose objects are functors from finite pointed sets to vector spaces over Fp. The full calculation of Tor ∗ -groups between functors of degree not exceeding p is presented. We compare our calculations with known results on homology of symmetric groups, Steenrod algebra and functor homology computations in the abelian category F of functors...

Homological Algebra

Homological Algebra for Schwartz Algebras

Let G be a reductive group over a non-Archimedean local field. For two tempered smooth representations, it makes no difference for the Ext-groups whether we work in the category of tempered smooth representations of G or of all smooth representations of G. Similar results hold for certain discrete groups. We explain the basic ideas from functional analysis and geometric group theory that are ne...