In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices. K-theory involves the construction of families of K-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the Grothendieck–Riemann–Roch theorem, Bott periodicity, the Atiyah–Singer index theorem, and the Adams operations. In high energy physics, K-theory and in particular twisted K-theory have appeared in Type II string theory where it has been conjectured that they classify D-branes, Ramond–Ramond field strengths and also certain spinors on generalized complex manifolds. In condensed matter physics K-theory has been used to classify topological insulators, superconductors and stable Fermi surfaces. For more details, see K-theory (physics). Grothendieck completion The Grothendieck completion of an abelian monoid into an abelian group is a necessary ingredient for defining K-theory since all definitions start by constructing an abelian monoid from a suitable category and turning it into an abelian group through this universal construction. Given an abelian monoid let be the relation on defined by if there exists a such that Then, the set has the structure of a group where: Equivalence classes in this group should be thought of as formal differences of elements in the abelian monoid. This group is also associated with a monoid homomorphism given by which has a certain universal property. To get a better understanding of this group, consider some equivalence classes of the abelian monoid . Here we will denote the identity element of by so that will be the identity element of First, for any since we can set and apply the equation from the equivalence relation to get This implies hence we have an additive inverse for each element in . This should give us the hint that we should