The general notion of a direct limit of a commuting system of embeddings, indexed by pairs in a directed set, has seen heavy use in set theory. It is the same notion as in category theory. I was surprised to find that the general definition does not appear in the book on model theory by Chang and Keisler (MR1059055). Who was the originator of this idea?

2$\begingroup$ Probably it was done first with groups? At first just a sequence of groups. $\endgroup$– Gerald EdgarCommented Apr 19, 2020 at 13:17

2$\begingroup$ It's hard to say when it was first done: it might have been done first in significant particular case, etc. Then maybe formalized but not in a very definite way. Also category theory unified the notion with the dual notions since it allows to treat arrows and their opposite in the same point of view. And of course the categorical definition doesn't supersede any specific case: how some kind of limits can be described in a given category of algebraic structures is also part of the job. $\endgroup$– YCorCommented Apr 20, 2020 at 19:43

$\begingroup$ I think an early example outside groups was Dieudonné and Schwartz's paper on (LF) spaces. Each time it occurs, «limite inductive» is put in (French) quotation marks, as if the term is being used slightly outside its strict meaning. Schwartz's earlier papers describe the topology on compactly supported test functions directly in terms of convergence, rather than as the direct limit of a sequence of Fréchet spaces. $\endgroup$– Robert FurberCommented Apr 20, 2020 at 20:50

$\begingroup$ That what categorists would now call cofiltered limits were first introduced for groups in the 1930s is entirely plausible. The unification of this notion with products, equalisers and pullbacks in category theory was made by Peter Freyd in his thesis in about 1963. $\endgroup$– Paul TaylorCommented Apr 21, 2020 at 15:51

$\begingroup$ @PaulTaylor "Direct limit" is the oldfashioned name for a filtered colimit (indexed by a poset), rather than a cofiltered limit, which were called an "inverse limit", or "projective limit". $\endgroup$– Robert FurberCommented Apr 21, 2020 at 17:24
2 Answers
The definition of a direct limit of groups was given by Pontrjagin in his 1931 paper Über den algebraischen Inhalt topologischer Dualitätssätze
As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwarich in the other answer, the definition can be traced back at least to Lev Pontryagin.
For an early exposition in English, see
MR0007093 (4,84f). Lefschetz, Solomon. Algebraic Topology. American Mathematical Society Colloquium Publications, v. 27. American Mathematical Society, New York, 1942. vi+389 pp.
Specifically, direct limits are defined in Chapter 2, $\S$14 (p. 57).

4$\begingroup$ According to van Est (1999): “Inverse limits of groups had occurred earlier in mathematics (e.g., Brouwer, 1910; van Dantzig, 1930; Herbrand, 1933), and of course examples abound in padics (...) These matters are discussed in the introduction of (Freudenthal, 1937) (...) It is, we think, by this paper that the notions of inverse and direct limit acquired their formal status in mathematics”. $\endgroup$ Commented Apr 19, 2020 at 19:45

2$\begingroup$ Unfortunately what he says is not quite unambiguous. Certainly Pontrjagin is among those quoted by Freudenthal 1937, but you’d have to unravel the direct vs inverse aspect. $\endgroup$ Commented Apr 19, 2020 at 19:58

2$\begingroup$ @Francois I rephrased. Thanks! $\endgroup$ Commented Apr 21, 2020 at 16:24