Beck–Chevalley condition

In category theory, the Beck–Chevalley condition is a coherence condition relating adjoint functors associated with a commutative square of categories. It states that, under suitable assumptions, two canonical ways of transporting information around the square coincide up to a natural isomorphism.

The condition appears throughout category theory, especially in the theory of fibrations, toposes, indexed categories, descent theory, categorical logic, and algebraic geometry. It is named after Jonathan Mock Beck and Claude Chevalley.

See also

• Category theory
• Adjoint functor
• Grothendieck fibration
• Topos
• Descent theory
• Indexed category
• 2-category

Bibliography

• Mac Lane, Saunders; Moerdijk, Ieke (1994). Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Universitext. Springer-Verlag. ISBN 978-0-387-97710-2. MR 1300636.
• Jacobs, Bart (1999). Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics. Vol. 141. North Holland, Elsevier. ISBN 0-444-50170-3. A comprehensive monograph written by a computer scientist; it covers both first-order and higher-order logics, and also polymorphic and dependent types. The focus is on fibred category as universal tool in categorical logic, which is necessary in dealing with polymorphic and dependent types.
• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.