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Let $\mathbf C$ be a metacategory.

Let $D: \mathbf J \to \mathbf C$ be a $\mathbf J$-diagram in $\mathbf C$.

Let $\map {\mathbf {Cocone} } D$ be the category of cocones from $D$.

A colimit for $D$ is an initial object in $\map {\mathbf {Cocone} } D$.

It is denoted by $\varinjlim_j D_j$; the associated morphisms $\iota_i: D_i \to \varinjlim_j D_j$ are usually left implicit.

Finite Colimit

Let $\varinjlim_j D_j$ be a colimit for $D$.

Then $\varinjlim_j D_j$ is called a finite colimit if and only if $\mathbf J$ is a finite category.

Also known as

The most important other name for this concept is direct limit.

Some authors speak of colimiting cones.

Also see