Pseudofunctors

Pseudofunctors are a concept in category theory, a branch of mathematics that studies the abstract structures and relationships between mathematical objects. They were introduced by Grothendieck as a generalization of functors, which are mappings between categories that preserve the structure of the categories.

A pseudofunctor is a functor-like structure that does not necessarily preserve composition strictly, but only up to a specified isomorphism. This means that the composition of pseudofunctors is not necessarily associative, but it is associative up to a natural isomorphism. This relaxation of the functorial properties allows for a broader range of examples and applications.

Pseudofunctors are particularly useful in the study of higher category theory, where they provide a way to describe the behavior of functors between higher categories. They also appear in the context of topological quantum field theories, where they are used to describe the behavior of quantum systems under changes of topology.

In summary, pseudofunctors are a generalization of functors that allow for a more flexible and inclusive framework for studying the relationships between mathematical objects. They have found applications in various areas of mathematics and theoretical physics.