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 $\mathbf{Cone} \left({D}\right)$.

There is believed to be a mistake here, possibly a typo. In particular: A limit for $D$ is an object in $\mathbf C$, isn't it? If yes, then this definition is formally wrong. A terminal object in $\mathbf{Cone} \left({D}\right)$ is a pair of such an object and a morphisms. Concretely, $\paren {\varprojlim_j D_j, \sequence {p_j}_j }$ is the terminal object in $\mathbf{Cone} \left({D}\right)$. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mistake}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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 $5.16$