Higher category theory

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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.

[edit] Strict higher categories

Main article: n-category

N-categories are defined inductively using the enriched category theory: 0-categories are sets and (n+1)-categories are categories enriched over the monoidal category of n-categories (with the monoidal structure given by finite products).^[1] This construction is well defined, as shown in the article on n-categories. This concept introduces higher arrows, higher compositions and higher identities, which must well behave together. For example, the category of small categories is in fact a 2-category, with natural transformations as second degree arrows. However this concept is too strict for some purposes (for example, homotopy theory), where "weak" structures arise in the form of higher categories.^[2]

[edit] Weak higher categories

Main article: Weak n-category

In weak n-categories, the associativity and identity conditions are not strict anymore (that is, they are not given by equalities) but they are satisfied up to an isomorphism of the next level. An example in topology is the composition of paths which is associative only up to homotopy. These isomorphisms must well behave between them 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, which makes say that bicategories are "monoidal categories with many objects". Weak 3-categories, also called tricategories, are harder to define explicitly, and so on. 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.