Tobias Schroeder <tschroed@Mathematik.Uni-Marburg.de> writes:

So I'd be very grateful for answers to one of the following: - 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)?

With a little bit of cheating, you can use domain theory to express the limit as a sequence as a _colimit_ in a partially ordered set. Let D be the partial order consisting of all the closed intervals, including singletons [a,a], ordered by reverse inclusion. We can embed R into D by mapping it to the maximal elements a |---> [a,a], and under a suitable topology on D (the Scott topology), this is a topological embedding--purists may want to throw in R as the smallest element to obtain an honest continuous domain. Let x_i be a Cauchy sequence of real numbers. To say that x_i is a Cauchy sequence is to say that there exist numbers d_i such that (1) For j >= i, the interval [x_i - d_i, x_i + d_i] contains [x_j + d_j, x_j + d_j]. (2) The numbers d_i become arbitrarily small: for every k there is i such that for all j >= i, d_i < 1/k. (Exercise for your students: show that this is equivalent to the usual definition of Cauchy sequence.) In terms of the partial order D, (1) says that the intervals [x_i - d_i, x_i + d_i] form an increasing sequence. Every increasing sequence in D has a supremum, because an intersection of a nested sequence of closed intervals is a closed interval, so let [u,v] = sup_i [x_i - d_i, x_i + d_i] By (2), we get that u = v, and we have obtained the limit of the sequence (x_i) as a supremum. Supremums are the _colimits_ in a partial order. If you prefer limits, you can stand on your head. I do not see how to get by without using the _evidence_ that (x_i) is a Cauchy sequence, i.e., the numbers d_i. This is intuitionistic mathematics creeping in, which is just as well.

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

I am way too young to know the answer to this. Andrej