In mathematics, especially category theory, limits and colimits in an ∞-category generalize limits and colimits in a category. Like the counterparts in ordinary category theory, they play fundamental roles in constructions (e.g., Kan extensions) as well as characterizations (e.g., sheaf conditions) in higher category theory.
Definition.
Let formula_1 be a simplicial set and formula_2 an ∞-category (a weak Kan complex). Fix a Grothendieck universe. Then, roughly, a limit of a functor formula_3 amounts to the following isomorphism:
formula_4
functorially in formula_5, where formula_6 denotes the constant functor with value formula_5.
A typical case is when formula_8 is the simplex category or rather its opposite; in the latter case, the functor formula_9 is commonly called a simplicial diagram.
Facts.
The ordinary category of sets has small limits and colimits. Similarly,
Also, many of standard facts about limits and colimits in a category continue to hold for those in an ∞-category.