nLab complete category

Complete categories


Category theory

Limits and colimits

Complete categories


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

F:DC F: D \to C

where DD is a small category has a limit in CC.

Sometimes one says that CC is small-complete to stress that DD 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).


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 CDC\to D and DD is complete, then CC 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][C,Set] is complete, and the dual Yoneda embedding exhibits it as the free completion of CC.

Last revised on May 26, 2020 at 14:47:40. See the history of this page for a list of all contributions to it.