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 ${\displaystyle I}$ be a simplicial set and ${\displaystyle C}$ an ∞-category (a weak Kan complex). Fix a Grothendieck universe. Then, roughly, a limit of a functor ${\displaystyle f:I\to C}$ amounts to the following isomorphism:

${\displaystyle \operatorname {Hom} (a_{\textrm {const}},f){\overset {\sim }{\to }}\operatorname {Hom} (a,\varprojlim f)}$

functorially in ${\displaystyle a}$,^[1] where ${\displaystyle a_{\textrm {const}}:I\to C}$ denotes the constant functor with value ${\displaystyle a}$.

A typical case is when ${\displaystyle I=\Delta }$ is the simplex category or rather its opposite; in the latter case, the functor ${\displaystyle f}$ 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.^[2]
• The presheaf category ${\displaystyle {\mathcal {P}}(C)={\textbf {Fct}}(C,{\textbf {Kan}})}$ on an ∞-category C has colimits, as a consequence of the above.^[3]

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.^[4]
• If a functor admits a left adjoint, then it commutes with all limits.^[5]