pseudofunctor

A pseudofunctor is a concept in category theory, a branch of mathematics that studies the relationships between different mathematical structures. It is a generalization of the notion of a functor, which is a mapping between categories that preserves the categorical structure. While a functor maps objects and morphisms in a way that respects the composition of morphisms and identities, a pseudofunctor relaxes this requirement by allowing for a controlled form of non-associativity.

Formally, a pseudofunctor F between two categories C and D consists of:

1. A mapping of objects F(c) for each object c in C.

2. A mapping of morphisms F(f) for each morphism f in C, such that F(f) is a morphism in D from F(source(f)) to F(target(f)).

3. A family of isomorphisms, called coherence isomorphisms, that ensure the composition of morphisms is preserved up to isomorphism.

The coherence isomorphisms are necessary because, in general, the composition of morphisms in D might not be strictly associative, and the identity morphisms might not be strictly preserved. The pseudofunctor structure ensures that these deviations are controlled and consistent.

Pseudofunctors are particularly useful in the study of higher category theory, where they provide a way to describe functors between higher categories in a way that respects the higher-dimensional structure. They also appear in the context of operads and higher operads, where they help to describe the action of operads on categories.