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:

$\mathrm{Hom}(a_{\text{const}},f)\stackrel{\sim }{\to }\mathrm{Hom}(a,\underleftarrow{\lim }f)$

functorially in $a$, where $a_{\text{const}}:I\to C$ denotes the constant functor with value $a$.

A typical case is when $I=\mathrm{\Delta }$ 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 $𝒫(C)=\mathbf{\text{𝐅𝐜𝐭}}(C,\mathbf{\text{𝐊𝐚𝐧}})$ 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.

References

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Further reading

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

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