# Definition:Limit (Category Theory)

## Definition

Let $\mathbf C$ be a metacategory.

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

Let $\mathbf{Cone} \left({D}\right)$ be the category of cones to $D$.

A limit for $D$ is a terminal object in $\mathbf{Cone} \left({D}\right)$.

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

### Finite Limit

Let $\varprojlim_j D_j$ be a limit for $D$.

Then $\varprojlim_j D_j$ is called a finite limit if and only if $\mathbf J$ is a finite category.

## Also known as

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

Other authors speak of limiting cones, but this is rare.