Preservation of limits
Yesterday I talked about functors that create limits. It’s theoretically interesting, but a different condition that’s often more useful in practice is that a functor “preserve” limits.
Given a small category , a functor preserves -limits if whenever is a limiting cone for a functor then is a limiting cone for the functor . A functor is called “continuous” if it preserves all small limits. By the existence theorem, a functor is continuous if and only if it preserves all (small) products and pairwise equalizers.
As stated above, this condition is different than creating limits, but the two are related. If is a category which has -limits and is a functor which creates -limits, then we know also has -limits. I claim that preserves these limits as well. More generally, if is complete and creates all small limits then is complete and is continuous.
Indeed, let be a functor and let be a limiting cone for . Then since creates limits we know there is a unique cone in with , and this is a limiting cone on . But limiting cones are unique up to isomorphism, so this is the limit of and preserves it.
Now it’s useful to have good examples of continuous functors at hand, and we know one great family of such functors: representable functors. It should be clear that two naturally isomorphic functors are either both continuous or both not, so we just need to consider functors of the form .
First let’s note a few things down. A cone in is a family of arrows (into a family of different objects) indexed by , so we can instead think of it as a family of elements in the family of sets . That is, it’s the same thing as a cone , with vertex a set with one element. Also, remember from our proof of the completeness of that if we have a cone then we get a function from to the set of cones on with vertex . That is, . Finally, we can read the definition of a limit as saying that the set of cones is in bijection with the set of arrows
Now we see that
In the first bijection we use the equivalence of cones with vertex and functions from to the set of cones with vertex . In the second we use the fact that a cone in from to a family of hom-sets is equivalent to a cone in . In the third bijection we use the definition of limits to replace a cone from to by an arrow from to .
But now we can use the definition of limits again, now saying that
and since this holds for any set we must have . Therefore preserves the limit.