abelian category

abelian category
Name	Abelian category
Field	Category theory, Homological algebra
Introduced	1950s

abelian category An abelian category is an additive category with kernels and cokernels in which every monomorphism and epimorphism is normal, providing a setting for homological algebra and exact sequences. It generalizes the categories of modules and sheaves, underpinning constructions in algebraic geometry, representation theory, and algebraic topology. The concept organizes methods used in the work of mathematicians and institutions such as Alexander Grothendieck, Jean-Pierre Serre, David Hilbert, Emmy Noether, and in programs at Institute for Advanced Study, Princeton University, École Normale Supérieure, Massachusetts Institute of Technology, and University of Cambridge.

Definition

An abelian category is an additive category with a zero object where every morphism has a kernel and a cokernel, and where every monomorphism is the kernel of some morphism and every epimorphism is the cokernel of some morphism. This formalism was crystallized in the mid-20th century in the works of Alexander Grothendieck and Jean-Pierre Serre and has been central in developments at Harvard University, California Institute of Technology, University of California, Berkeley, University of Chicago, Stanford University, Columbia University, University of Oxford, Yale University, and others. The axioms allow one to define exact sequences and derive long exact sequences as used by Henri Cartan, Samuel Eilenberg, Norman Steenrod, Daniel Quillen, and Jean Leray.

Examples

Classical examples include the category of modules over a ring, which figures in the work of Emmy Noether, Richard Dedekind, Ernst Steinitz, and Bartel van der Waerden; categories of abelian groups studied by Niels Henrik Abel influences; and categories of coherent sheaves on schemes central to Grothendieck's school, Alexander Grothendieck and Jean-Pierre Serre. Other notable examples arise in the representation theory of algebras investigated by Issai Schur, William Rowan Hamilton, Hiroshi Matsumura, Alfred Young, Bertram Kostant, and George Mackey; derived categories developed by Grothendieck, Jean-Louis Verdier, and Bernard Malgrange; and categories of perverse sheaves used in the work of Pierre Deligne, Masaki Kashiwara, Mikio Sato, Luca Migliorini, and Mark Goresky. Applications follow in works at Institute des Hautes Études Scientifiques, Max Planck Institute for Mathematics, Courant Institute, Mathematical Sciences Research Institute, Clay Mathematics Institute, and in collaborations leading to theorems like the proof of the Weil conjectures by Pierre Deligne.

Basic properties

In any abelian category one has biproducts which were implicit in the algebraic traditions of Évariste Galois, Carl Friedrich Gauss, and formalized in works at University of Göttingen. Kernels and cokernels interact to give images and coimages, and the canonical map from coimage to image is an isomorphism, a condition reminiscent of structures studied by David Hilbert and Emmy Noether. Exactness properties permit the formulation of short and long exact sequences used by Samuel Eilenberg, Saunders Mac Lane, Hyman Bass, Jean-Louis Verdier, and Jean-Pierre Serre. Additive functors between abelian categories preserve finite biproducts, reflecting themes in the categorical work of Philip Hall, Claude Shannon (information in algebraic topology contexts), and institutions like Royal Society-supported research groups.

Homological constructions

Abelian categories support notions of projective and injective objects, resolutions, and derived functors Ext and Tor developed by Samuel Eilenberg and Norman Steenrod and expanded by Henri Cartan, Jean-Pierre Serre, Grothendieck, and Alexander Grothendieck in the context of spectral sequences and derived categories. Derived functor machinery is central in proofs by Pierre Deligne, Alexander Grothendieck, Jean-Louis Verdier, and in work at Princeton University and IHÉS. Spectral sequences appear in research by Jean Leray, John Milnor, J. H. C. Whitehead, Edwin Spanier, Edward Witten, and Michael Atiyah, and are constructed within abelian categories to compute homology and cohomology groups used in studies at Royal Institution, Institute for Advanced Study, Mathematical Institute, Oxford, and Kavli Institute for Theoretical Physics. The language of Ext and derived categories informs advances by Maxim Kontsevich, Paul Seidel, Alexander Beilinson, Vladimir Drinfeld, and Joseph Bernstein.

Functors and equivalences

Exact functors between abelian categories preserve kernels, cokernels, and exact sequences; left exact and right exact functors give rise to derived functors as shown in the developments by Samuel Eilenberg and Jean-Pierre Serre. Equivalences of abelian categories appear in Gabriel's theorem and in Morita theory explored by Pierre Gabriel, Kiiti Morita, I. M. Gelfand, Dmitri Fuks, and researchers at Moscow State University. Tannakian duality connecting tensor abelian categories to group schemes was formulated by Saavedra Rivano, later refined by Pierre Deligne and applied in contexts studied by Alexander Grothendieck and J. P. Serre at institutions like Institut des Hautes Études Scientifiques.

Applications and significance

Abelian categories provide the ambient setting for coherent sheaf cohomology central to proofs by Alexander Grothendieck and Jean-Pierre Serre, and for modern developments in algebraic geometry, number theory, and mathematical physics pursued at Harvard University, Princeton University, Stanford University, Cambridge University Press-supported research, and by programs at MSRI and ICM presentations. Their structure enables uniform treatments of Ext, Tor, and derived categories used in the Langlands program advanced by Robert Langlands, Pierre Deligne, Edward Frenkel, Ngo Bao Chau, and in string theory contexts by Edward Witten, Cumrun Vafa, Maxim Kontsevich, and Anton Kapustin. The abstraction of abelian categories continues to influence categorical approaches in modern mathematics across collaborations at Clay Mathematics Institute, European Research Council, Simons Foundation, and many universities worldwide.

Category:Category theory