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Re: in what sense is a limit a functor?
Posted:
Apr 12, 2006 10:30 AM
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Whether this takes the axiom of choice depends on exactly what version
you want, and on what foundation, but basically yes the general result
requires the axiom of choice. So the object part of a lim functor is a
bit of a mystery as you say.
The arrow part, though, is no mystery. So it is not a question of
anafunctors or pseudo-functors. Once you have chosen one limit L for a
diagram F and another L' for a diagram F', then each natural
transformation F-->F' induces a unique cone to F' from L and thus a
unique arrow from L to L' commuting with the cones. This cone
commuting condition obviously gets along with composition of natural
transformations so it takes composite diagram transformations to the
strict composites of the induced arrows between limits.
Pseudo-functors arise in a related stituation using fibered categories,
where you sort of think of cartesian arrows as pullback diagrams yet
they do not give strict functors. But in general cartesian arrows are
actually not defined as pullbacks and even in cases where they are
defined as pullbacks the functoriality problem is not about a
functorial selection of pullbacks. It is about making pasting of
pullback squares into a functor. That is another problem. When you
just want to look at limits of diagrams of a given shape J in a given
category C, you can choose one selected representative of each in a
strictly functorial way if you want to.
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