Higher category theory

In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.

An ordinary category has objects and morphisms. A 2-category generalizes this by also including 2-morphisms between the 1-morphisms. Continuing this up to n-morphisms between (n-1)-morphisms gives an n-category.

Just as the category known as Cat, which is the category of small categories and functors is actually a 2-category with natural transformations as its 2-morphisms, the category n-Cat of (small) n-categories is actually an n+1-category.

An n-category is defined by induction on n by:

So a 1-category is just a (locally small) category.

The monoidal structure of Set is the one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite products can be given a monoidal structure. The recursive construction of n-Cat works fine because if a category C has finite products, the category of C-enriched categories has finite products too.

While this concept is too strict for some purposes in for example, homotopy theory, where "weak" structures arise in the form of higher categories, strict cubical higher homotopy groupoids have also arisen as giving a new foundation for algebraic topology on the border between homology and homotopy theory, see the book "Nonabelian algebraic topology" referenced below.

In weak n-categories, the associativity and identity conditions are no longer strict (that is, they are not given by equalities), but rather are satisfied up to an isomorphism of the next level. An example in topology is the composition of paths, where the identity and association conditions hold only up to reparameterization, and hence up to homotopy, which is the 2-isomorphism for this 2-category. These n-isomorphisms must well behave between hom-sets and expressing this is the difficulty in the definition of weak n-categories. Weak 2-categories, also called bicategories, were the first to be defined explicitly. A particularity of these is that a bicategory with one object is exactly a monoidal category, so that bicategories can be said to be "monoidal categories with many objects." Weak 3-categories, also called tricategories, and higher-level generalizations are increasingly harder to define explicitly. Several definitions have been given, and telling when they are equivalent, and in what sense, has become a new object of study in category theory.

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