the number of points on an algebraic curve or variety of higher dimension. Grothendieck's discovery of the ℓ-adic étale cohomology, the first example of a Weil cohomology theory, opened the way for a proof of the Weil conjectures, ultimately completed in the 1970s by his student Pierre Deligne. Grothendieck's large-scale approach has been called a "visionary program". The ℓ-adic cohomology then became a fundamental tool for number theorists, with applications to the Langlands program. Grothendieck's conjectural theory of motives was intended to be the "ℓ-adic" theory but without the choice of "ℓ", a prime number. It did not provide the intended route to the Weil conjectures, but has been behind modern developments in algebraic K-theory, motivic homotopy theory, and motivic integration. This theory, Daniel Quillen's work, and Grothendieck's theory of Chern classes, are considered the background to the theory of algebraic cobordism, another algebraic analogue of topological ideas. Category theory Grothendieck's emphasis on the role of universal properties across varied mathematical structures brought category theory into the mainstream as an organizing principle for mathematics in general. Among its uses, category theory creates a common language for describing similar structures and techniques seen in many different mathematical systems. His notion of abelian category is now the basic object of study in homological algebra. The emergence of a separate mathematical discipline of category theory has been attributed to Grothendieck's influence, although unintentional. In popular culture The novel Colonel Lágrimas (Colonel Tears in English, available by Restless Books) by Puerto Rican–Costa Rican writer Carlos Fonseca is a semibiographic novel about Grothendieck. The band Stone Hill All Stars have a song named after Alexander Grothendieck. In the novel When We Cease to Understand the World, Benjamin Labatut dedicates one chapter to the story of Grothendieck. In the novel The Passenger and its sequel Stella Maris by Cormac McCarthy one of the main characters is a student of Grothendieck. Publications See also ∞-groupoid λ-ring AB5 category Abelian category Accessible category Algebraic geometry Algebraic stack Barsotti–Tate group Chern class Descent (mathematics) Dévissage Dunford–Pettis property Excellent ring Formally smooth map Fundamental group scheme K-theory Hilbert scheme Homotopy hypothesis List of things named after Alexander Grothendieck Nakai conjecture Nuclear operator Nuclear space Parafactorial local ring Projective tensor product Quasi-finite morphism Quot scheme Scheme (mathematics) Section conjecture Semistable abelian variety Sheaf cohomology Stack (mathematics) Standard conjectures on algebraic cycles Sketch of a program Tannakian formalism Theorem of absolute purity