Definition:Functor Creating Colimits

From ProofWiki
Jump to navigation Jump to search


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 colimits of type $\mathbf J$ if and only if:

For all $\mathbf J$-diagrams $C: \mathbf J \to \mathbf C$ in $\mathbf C$, given a colimit $\paren {{\varinjlim \,}_j \, FC_j, q_j}$ for $FC: \mathbf J \to \mathbf D$ in $\mathbf D$, the colimit:
$\paren {{\varinjlim \,}_j \, C_j, p_j}$
exists, and furthermore:
$\map F {{\varinjlim \,}_j \, C_j} = {\varinjlim \,}_j \, FC_j$
$F p_j = q_j$
for all objects $j$ of $\mathbf J$.

Also see