Limits And Colimits In An ∞-category

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 $I$ be a simplicial set and $C$ an ∞-category (a weak Kan complex). Fix a Grothendieck universe. Then, roughly, a limit of a functor $f : I \to C$ amounts to the following isomorphism:
:$\operatorname(a_, f) \overset\to \operatorname(a, \varprojlim f)$
functorially in $a$, where $a_ : I \to C$ denotes the constant functor with value $a$.

A typical case is when $I = \Delta$ is the simplex category or rather its opposite; in the latter case, the functor $f$ 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 $\mathcal(C) = \textbf(C, \textbf)$ 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.

Notes

References

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Further reading

*https://ncatlab.org/nlab/show/limits+and+colimits+by+example

Limits (category theory)
Higher category theory

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