A category is complete if it has all small limits: that is, if every small diagram
where is a small category has a limit in .
Sometimes one says that is small-complete to stress that 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 and is complete, then 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 is complete, and the dual Yoneda embedding exhibits it as the free completion of .
Last revised on May 26, 2020 at 14:47:40. See the history of this page for a list of all contributions to it.