Definition:Functor Creating Limits

Definition

Let $\mathbf C, \mathbf D$ and $\mathbf J$ be metacategories.

Let $F: \mathbf C \to \mathbf D$ be a functor.

Then $F$ is said to create limits of type $\mathbf J$ if and only if:

For all $\mathbf J$-diagrams $C: \mathbf J \to \mathbf C$ in $\mathbf C$, given a limit $\paren {{\varprojlim \,}_j \, FC_j, q_j}$ for $FC: \mathbf J \to \mathbf D$ in $\mathbf D$, the limit:

$\paren {{\varprojlim \,}_j \, C_j, p_j}$

exists, and furthermore:

$\map F {{\varprojlim \,}_j \, C_j} = {\varprojlim \,}_j \, FC_j$

$F p_j = q_j$

for all objects $j$ of $\mathbf J$.

Also see

• Definition:Functor Creating Colimits

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 5.6$: Definition $5.30$