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
The concept of colimit is that dual to a limit:
a colimit of a diagram in a category is, if it exists, the co-classifying space for morphisms out of that diagram.
The intuitive general idea of a colimit is that it defines an object obtained by sewing together the objects of the diagram, according to the instructions given by the morphisms of the diagram.
Sometimes colimits (or some colimits) are called inductive limits or direct limits; see the discussion of terminology at limit.
A colimit in a category is the same as a limit in the opposite category, .
More in detail, for a functor, its limit is the colimit of .
Here are some important examples of colimits:
A weighted colimit in is a weighted limit in .
The properties of colimits are of course dual to those of limits. It is still worthwhile to make some of them explicit.
Contravariant Hom sends colimits to limits
For a locally small category, for a functor, for and object and writing , we have
Depending on how one introduces limits this holds by definition or is an easy consequence. In fact, this is just rewriting the respect of the covariant Hom of limits (as described there) in in terms of :
Notice that this actually says that is a continuous functor in both variables: in the first it sends limits in and hence equivalently colimits in to limits in .
Proposition – left adjoints commute with colimits
Let be a functor that is left adjoint to some functor . Let be a small category such that admits limits of shape . Then commutes with -shaped colimits in in that
for some diagram, we have
Using the adjunction isomorphism and the above fact that commutes with limits in both arguments, one obtains for every
Since this holds naturally for every , the Yoneda lemma, corollary II on uniqueness of representing objects implies that .