Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
A category is complete if it has all small limits: that is, if every 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
Revised on January 19, 2012 18:55:01
by mondic->monadic; hope this is correct?