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 functor is final (often called cofinal), if we can restrict diagrams on to diagrams on along without changing their colimit.
Dually, a functor is initial (sometimes called co-cofinal) if pulling back diagrams along it does not change the limits of these diagrams.
Beware that this property is pretty much unrelated to that of a functor being an initial object or terminal object in the functor category . The terminology comes instead from the fact that an object is initial (resp. terminal) just when the corresponding functor is initial (resp. final).
Let be a functor
The following conditions are equivalent.
For all functors the natural function between colimits
is a bijection.
For all categories and all functors the natural morphism between colimits
is a isomorphism.
For all functors the natural function between limits
is a bijection.
For all categories and all functors the natural morphism
is an isomorphism.
If is final then is connected precisely if is.
If and are final, then so is their composite .
If and the composite are final, then so is .
If is a full and faithful functor and the composite is final, then both functors seperately are final.
The generalization of the notion of final functor from category theory to (∞,1)-higher category theory is described at
The characterization of final functors is also a special case of the characterization of exact squares.
If has a terminal object then the functor that picks that terminal object is final: for every the comma category is equivalent to . The converse is also true: if a functor is final, then its image is a terminal object.
In this case the statement about preservation of colimits states that the colimit over a category with a terminal object is the value of the diagram at that object. Which is also readily checked directly.
Let be a pair of adjoint functors.To see that is final, we may for instance check that for all the comma category is non-empty and connected:
It is non-empty because it contains the adjunction unit . Similarly, for
two objects, they are connected by a zig-zag going through the unit, by the universal factorization property of adjunctions
The inclusion of any category into its idempotent completion is final.
See at idempotent completion in the section on Finality.
The inclusion of the cospan diagram into its cocone
Section 2.5 of
Section 2.11 of
Notice that this says “final functor” for the version under which limits are invariant.
Section IX.3 of