Definition:Limit (Category Theory)
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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.
Also see
- Definition:Colimit: the dual notion
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 5.4$: Definition $5.16$