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Concept
Higher category theory
Summary
In mathematics, higher category theory is the part of 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. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as their fundamental .
An ordinary has and morphisms, which are called 1-morphisms in the context of higher category theory. A 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 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:
A 0-category is a ,
An (n + 1)-category is a category over the category n-Cat.
So a 1-category is just a () category.
The structure of Set is the one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite 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 article Nonabelian algebraic topology, referenced in the book below.
Weak n-category
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.
Official source
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