categories: Re: Limits

[Peter Freyd <pjf@saul.cis.upenn.edu>]

Tobias Schroeder asks:

- Can the limit of a sequence of real numbers be expressed
  as a categorical limit (of course it can if the sequence is
  monotone, but what if it is not)?

A good question. I have no answer, only a similar (and ancient)
question: is there a setting in which adjoint operators on Hilbert
spaces can be seen to be examples of adjoint functors between
categories?

As for his second question:

- Why have people chosen the term "limit" in category theory?
  (And, by the way, who has defined it first?)

In the beginning, the only diagrams that had limits were "nets", that
is, diagrams based on directed posets. I believe it was Norman
Steenrod in his dissertation who first used the term. Before his
dissertation the Cech cohomology of a space was defined only as the
numberical invarients that arose as a limit of a directed set of such
invariants. It was Steenrod who perceived that Cech cohomology could
be defined as an abelian group. For that he needed to invent the
notion of a limit of a directed diagram of groups.

In the 50s the fact that one didn't need the diagram to be directed
was considered startling.

At least two of us tried to avoid the word "limit" in this more
general setting. Jim Lambek was pushing "inf" and "sup", a suggestion
I wish I had heard. Not having heard it, I was pushing "left root" and
"right root" (one was, after all, supplying a root to a generalized
tree. sort of).

All to no avail. So now we have "finite limits" and "finitely
continuous".