Homological systems and bocses

Homological Computations for Term Rewriting Systems

An important problem in universal algebra consists in finding presentations of algebraic theories by generators and relations, which are as small as possible. Exhibiting lower bounds on the number of those generators and relations for a given theory is a difficult task because it a priori requires considering all possible sets of generators for a theory and no general method exists. In this art...

Categories and Homological Algebra

The aim of these Notes is to introduce the reader to the language of categories with emphazis on homological algebra. We treat with some details basic homological algebra, that is, categories of complexes in additive and abelian categories and construct with some care the derived functors. We also introduce the reader to the more sophisticated concepts of triangulated and derived categories. Ou...

Homological Algebra and Data

These lectures are a quick primer on the basics of applied algebraic topology with emphasis on applications to data. In particular, the perspectives of (elementary) homological algebra, in the form of complexes and co/homological invariants are sketched. Beginning with simplicial and cell complexes as a means of enriching graphs to higher-order structures, we define simple algebraic topological...

Homological Algebra

Homology and Homological Algebra,

= convex hull of {a0, . . . , an}. The points a0, . . . , an are geometrically independent if a1 − a0, . . . , an − a0 is a linearly independent set over R. Note that this is independent of the order of a0, . . . , an. In this case, we say that the simplex a0 . . . an is n -dimensional, or an n -simplex. Given a point Pn i=1 λiai belonging to an n-simplex, we say it has barycentric coordinates ...