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.
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.
Weak 3-categories, also called tricategories, and higher-level generalizations are increasingly harder to define explicitly.
These are models of higher categories introduced by Hirschowitz and Simpson in 1998,[3] partly inspired by results of Graeme Segal in 1974.