Categories of Functors

Many constructions on various mathematical objects depend not just on the elements of those objects but also on the morphisms between them. Such constructions can thus be effectively formulated in the corresponding category of objects. A “topos” is a category in which a number of the most basic such constructions (product, pullback, exponential, characteristic function,…) are always possible. With these constructions available, many other properties can be efficiently developed. Superficially quite different categories, arising in geometry, topology, algebraic geometry, group representations, and set theory, all turn out to satisfy the axioms defining such a topos.