# nLab complete category

category theory

## Applications

#### Limits and colimits

limits and colimits

# Complete categories

## Definition

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

Revised on January 19, 2012 18:55:01 by mondic->monadic; hope this is correct? (131.111.145.118)