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 be a simplicial set and an ∞-category (a weak Kan complex). Fix a Grothendieck universe. Then, roughly, a limit of a functor amounts to the following isomorphism:
functorially in , where denotes the constant functor with value .
A typical case is when is the simplex category or rather its opposite; in the latter case, the functor is commonly called a simplicial diagram.
Facts The ordinary category of sets has small limits and colimits. Similarly, The ∞-category of ∞-categories and the ∞-category of Kan complexes both have all small limits and colimits. The presheaf category on an ∞-category C has colimits, as a consequence of the above.
Also, many of standard facts about limits and colimits in a category continue to hold for those in an ∞-category. An ∞-category has all small limits if and only if it has coequalizers and small coproducts. If a functor admits a left adjoint, then it commutes with all limits.
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References
Further reading https://ncatlab.org/nlab/show/limits+and+colimits+by+example