nLab complete category

Definition

A category $C$ is complete if it has all small limits: that is, if every small diagram

$$F: D \to C$$

where $D$ is a small category has a limit in $C$.

Sometimes one says that $C$ is small-complete to stress that $D$ must be small; compare finitely complete category. Also compare complete small category, which is different; here we see that any small category that is also small-complete must be thin (at least classically).

Examples

Many familiar categories of mathematical structures are complete: to name just a few examples, Set, Grp, Ab, Vect and Top are complete.

As hinted above, every complete lattice is complete as a category.

A common situation is that of a category of algebras for a monad in a complete category: If there exists a monadic functor $C\to D$ and $D$ is complete, then $C$ is complete (as monadic functors create limits). This includes some obvious examples (such as Grp, Ab, and Vect), as well as some less-obvious examples, such as complete lattices and compact Hausdorff spaces.

The contravariant presheaf category $[C,Set]$ is complete, and the dual Yoneda embedding exhibits it as the free completion of $C$.

Related concepts

• cocomplete category

• bicomplete category

• M-complete category