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 connected limit is a limit over a connected category. Similarly, a connected colimit is a colimit over a connected category.
The following are all connected limits:
Construction from pullbacks and equalizers
Let be a finite connected category and a functor. Since is connected, it is inhabited; choose some object . For each object , let be the length of the shortest zigzag from to . Now order the objects of as such that for all we have .
Now we inductively define objects , for , with projections for . We begin with and . Assuming and defined, choose a zigzag from to of minimal length, say
By our choice of the ordering of the objects of , we have for some , and thus we have .
If the final arrow in the zigzag is directed as , then let , let be the composite , and keep the other unchanged. On the other hand, if is directed as , let be the pullback
and define for by composition with .
At the end of this procedure, we have an object with projections for all objects . Now we order the morphisms in as and define, inductively, an object with a morphism . We begin with and . Given and , we let be the equalizer of and , where the ?s denote the indices of the objects that are the source and target of . We then set .
At the end of this procedure, we have an object together with a cone over the diagram , which is easily verified to be a limit of .
Similarly, arbitrary connected limits may be built from wide pullbacks and equalizers.
Let be (finitely) complete, and let be an object of . Then the forgetful functor
preserves and reflects (finite) connected limits.
The wide pullback of the diagram where for all is clearly , as is the equalizer of the pair of identity arrows. Since the product functor preserves arbitrary limits, we see that
i.e., preserves wide pullbacks. The same line of argument shows preserves equalizers, so preserves connected limits.
The functor carries a canonical comonad structure, whose category of coalgebras is . The forgetful functor is comonadic, and thus preserves and reflects any class of limits preserved by the comonad . Thus preserves and reflects all connected limits.
Preservation from wide pullbacks
Recall that a wide pullback is a limit over a diagram whose underlying shape is the poset obtained by freely adjoining a terminal element to a discrete poset, which is certainly connected.
It is not true that if has wide pullbacks then it has connected limits. The saturation of the class of wide pullbacks is the class of connected and “simply connected” limits (limits over categories whose groupoid reflection is trivial).
However, the following is true.
Let be a complete category, and let be locally small. Then a functor preserves connected limits if and only if it preserves wide pullbacks.
The forward direction is clear since wide pullbacks are examples of connected limits. Now suppose preserves wide pullbacks. Then
preserves wide pullbacks for every object of . Put . The underlying functor
reflects and preserves connected limits and in particular wide pullbacks, so that the evident lift
preserves wide pullbacks. It also preserves the terminal object, hence by this proposition it preserves arbitrary limits. Therefore the composite
preserves connected limits, for every object . Since this is the same composite as in (1), and since the representables jointly reflect arbitrary limits, we conclude that preserves connected limits.
In particular, for complete, a functor that preserves wide pullbacks also preserves equalizers.
- Robert Paré, Simply connected limits. Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, CMS