It turns out that there are applications of our functors which make use of the analogous transformations which we like to think of as a change of basis for a fixed root-system — a tilting of the axes relative to the roots which results in a different subset of roots lying in the positive cone. ... For this reason, and because the word 'tilt' inflects easily, we call our functors tilting functors or simply tilts.
Brenner & Butler (1980, p. 103)
In mathematics, specifically representation theory, tilting theory describes a way to relate the module categories of two algebras using so-called tilting modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting module over the first algebra.
Tilting theory was motivated by the introduction of reflection functors by Joseph Bernšteĭn, Israel Gelfand, and V. A. Ponomarev (1973); these functors were used to relate representations of two quivers. These functors were reformulated by Maurice Auslander, María Inés Platzeck, and Idun Reiten (1979), and generalized by Sheila Brenner and Michael C. R. Butler (1980) who introduced tilting functors. Dieter Happel and Claus Michael Ringel (1982) defined tilted algebras and tilting modules as further generalizations of this.