Grothendieck category

Grothendieck category
Name	Grothendieck category
Field	Category theory
Introduced	1957
Introduced by	Alexander Grothendieck

Grothendieck category is an additive, cocomplete abelian category satisfying exactness conditions and possessing a generator, introduced in the work of Alexander Grothendieck. It generalizes categories of modules over rings and categories of sheaves on topological and algebraic objects, linking ideas from Émile Cartan, Jean-Pierre Serre, and Claude Chevalley to modern treatments related to Pierre Deligne, Michel Demazure, and Alexander Beilinson. Grothendieck categories form a foundational setting for homological algebra used by Jean-Louis Verdier, Henri Cartan, and Jean-Pierre Serre in the development of derived categories and toposes.

Definition and basic properties

A Grothendieck category is an abelian category that is complete and cocomplete with exact filtered colimits and that admits a generator, as formalized by Alexander Grothendieck and written up in seminars influenced by Jean Dieudonné and Jean-Pierre Serre. Core properties include the existence of arbitrary direct sums (coproducts) akin to those in Paul Erdős-era module theory over Emil Artin's rings, the AB5 condition (exactness of filtered colimits) used by Lorenzo Ramero and Henri Cartan in sheaf cohomology contexts, and a generator object enabling Yoneda-style embeddings into functor categories as exploited by Saunders Mac Lane and Samuel Eilenberg. These properties imply the existence of injective envelopes and Baer-type results in the spirit of Irving Kaplansky and yield well-behaved derived functors as in the work of Jean-Louis Verdier.

Examples and non-examples

Standard examples include the category of right modules over a ring R, a setup originating in the algebra of Emil Artin and extended by Nathan Jacobson, and the category of sheaves of abelian groups on a topological space as studied by Henri Cartan and Jean-Pierre Serre. Categories of quasi-coherent sheaves on schemes introduced by Alexander Grothendieck and developed by Jean-Pierre Serre, Pierre Deligne, and Michael Artin are principal examples used in algebraic geometry. Functor categories of the form Additive Functors from a small preadditive category to abelian groups, following constructions by Saunders Mac Lane and Samuel Eilenberg, also give Grothendieck categories. Non-examples include certain abelian categories lacking a generator as in some constructions by Jean Dieudonné and pathological abelian categories without exact filtered colimits studied in counterexamples by Bernhard Neumann and Israel Gelfand.

Exactness and generators

The AB5 condition (exactness of filtered colimits) and the existence of a generator are central, following axiomatizations influenced by Alexander Grothendieck and formalized in later expositions by Pierre Gabriel and Jean-Pierre Serre. Generators provide faithful exact functors to module categories, paralleling Gabriel's embedding theorem and echoing structural insights of Emmy Noether and Emil Artin. Exactness properties guarantee injective cogenerators and allow use of Baer-type criteria for injectivity as in the work of Irving Kaplansky and Hans Zassenhaus. These features support Grothendieck’s use of spectral sequences and derived functors later exploited by Jean-Louis Verdier and Pierre Deligne.

Localization and quotient categories

Localization theory in Grothendieck categories generalizes Gabriel localization of module categories and the localization techniques used by Alexander Grothendieck in algebraic geometry and by Jean-Pierre Serre in sheaf theory. Gabriel–Popescu theorem and Gabriel localization connect Grothendieck categories to module categories over rings and to localizing Serre subcategories studied by Pierre Gabriel and André Joyal. Quotient categories formed by localizing at a Serre subcategory preserve the Grothendieck structure under conditions analogous to those in the theory of schemes developed by Alexander Grothendieck and later employed by Alexander Beilinson and Joseph Bernstein in representation-theoretic contexts. This machinery parallels localization techniques in the work of Ivo Herzog and Henning Krause on representation theory of algebras.

Homological algebra in Grothendieck categories

Grothendieck categories support a robust homological algebra: existence of enough injectives follows from Grothendieck’s original arguments and from methodologies used by Jean-Pierre Serre and Jean-Louis Verdier for derived categories. Derived functors, Ext and Tor analogues, and spectral sequences can be constructed following frameworks by Alexander Grothendieck, Jean-Louis Verdier, and Pierre Deligne. The development of derived categories by Alexander Grothendieck’s students and collaborators such as Jean-Louis Verdier and applications in perverse sheaves by Alexander Beilinson and Pierre Deligne rely on these homological foundations. Techniques from model category theory championed by Daniel Quillen and triangulated category theory by Jean-Louis Verdier also interact fruitfully with Grothendieck categories.

Relationships with other abelian categories

Grothendieck categories sit above module categories, extending classical module-theoretic results of Emil Artin and Nathan Jacobson and connecting to locally presentable categories studied by Jiří Adámek and Jiří Rosický. The Gabriel–Popescu theorem relates Grothendieck categories to reflective localizations of module categories over rings with enough idempotents, a theme pursued by Pierre Gabriel and Georges Bergman. Connections to topos theory initiated by Alexander Grothendieck and advanced by William Lawvere and André Joyal illuminate relationships with sheaf categories and geometric morphisms. Comparisons with length categories, noetherian abelian categories, and Artinian categories draw on classification themes from Emmy Noether and Emil Artin.

Category:Category theory Category:Homological algebra Category:Algebraic geometry