Definition:Limit of Sets (Category Theory)

Definition
Let $\II$ be a small category.

Let $D : \II \to \mathbf {Set}$ be a diagram in the category of sets $\mathbf {Set}$.

The limit of sets of $D$ is defined as:
 * $\lim D := \set {\family {a_i}_{i \mathop \in \II} : \paren {\forall i \mathop \in \II : a_i \in \map D i} \wedge \paren {\forall i, j \mathop \in \II : \forall f \in \map {\mathrm {Hom}_{\II} } {i, j} : \map {\map D f} {a_i} = a_j} }$

The corresponding projections $\pi_j : \lim D \to \map D j$ are defined by $\map {\pi_j} {\family {a_i}_{i \mathop \in \II}} := a_j$ for all $j \mathop \in \II$.

Notation
$\lim D$ also stands for the categorical limit of $D$. By Limit of Sets is Limit (Category Theory) this notation makes sense for the limit as defined in this article.

Also see

 * Limit of Sets is Limit (Category Theory)