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:

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

functorially in $a$, where $a_{\textrm {const}}:I\to C$ 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 ${\mathcal {P}}(C)={\textbf {Fct}}(C,{\textbf {Kan}})$ 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

• Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200.
• Land, Markus (2021). Introduction to Infinity-Categories. Compact Textbooks in Mathematics. doi:10.1007/978-3-030-61524-6_2. ISBN 978-3-030-61523-9. Zbl 1471.18001.
• Lurie, Jacob (2009), Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, arXiv:math.CT/0608040, ISBN 978-0-691-14049-0, MR 2522659

Further reading

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