Strict Higher Categories
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). 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.
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.
Read more about this topic: Higher Category Theory
Famous quotes containing the words strict, higher and/or categories:
“Yet if strict criticism should till frown on our method, let candor and good humor forgive what is done to the best of our judgment, for the sake of perspicuity in the story and the delight and entertainment of our candid reader.”
—Sarah Fielding (1710–1768)
“Every man wants a woman to appeal to his better side, his nobler instincts and his higher nature—and another woman to help him forget them.”
—Helen Rowland (1875–1950)
“The analogy between the mind and a computer fails for many reasons. The brain is constructed by principles that assure diversity and degeneracy. Unlike a computer, it has no replicative memory. It is historical and value driven. It forms categories by internal criteria and by constraints acting at many scales, not by means of a syntactically constructed program. The world with which the brain interacts is not unequivocally made up of classical categories.”
—Gerald M. Edelman (b. 1928)