Limits and colimits

Definition 1.7.1 (Limit/colimit). Let \(I\) and \(C\) be \(\infty \)-categories and let \(F\colon I \to C\) be a functor.

(1)

A cone on \(F\) is an object \(X\) in \(C\) together with a natural transformation \(\epsilon \colon \const _X \to F\) of functors \(I \to C\).

(2)

A cone \((X,\epsilon )\) is called a limit cone if for every other object \(Y\) of \(C\) the composite \[ \Hom _C(Y,X) \xrightarrow {\const } \Nat (\const _Y, \const _X) \xrightarrow {\epsilon \circ -} \Nat (\const _Y,F) \] is an equivalence.

In this case, the object \(X\) is called the limit of \(F\). It is unique, and will be denoted either by \(\lim _I F\) or \(\lim _{i \in I} F(i)\).

Dually, one defines a cocone to be a natural transformation \(\eta \colon F \to \const _X\). It is a colimit cocone if for every \(Y\) the composite \[ \Hom _C(X,Y) \xrightarrow {\const } \Nat (\const _X, \const _Y) \xrightarrow {- \circ \eta } \Nat (F,\const _Y) \] is an equivalence. The object \(X\) is then called the colimit of \(F\).