The definition of a category is usually attributed to Mac Lane and Eilenberg (1945). What seems to be less known is that the german mathematician Heinrich Brandt has developed the notion of a groupoid already in 1926 (motivated by questions on quadratic forms). The paper "Über eine Verallgemeinerung des Gruppenbegriffes" introduces, names and studies (connected) groupoids explicitly. The object-free definition is used, so that groupoids look very much like groups except that the product is only defined partially.

Well, first of all I have to say that it is quite amazing that the definition of a category is basically already there in Brandt's paper. Just erase the inverse elements from Axiom III. The paper doesn't really mention the arrow picture stressed by Mac Lane and Eilenberg, although the notions "einander rechts" (sharing the same codomain) and "einander links" (sharing the same domain) are introduced for groupoid elements aka morphisms.

See here for a list of further early publications on groupoids.

Nowadays, groupoids are usually seen as special categories. (Curiously, in homotopy type theory, categories are seen as special $\infty$-groupoids.) But the definition of a groupoid has appeared 20 years before the definition of a category. This leads to many (related) questions:

  • Why did it take 20 years?
  • Does the study of groupoids have influenced the development of the notion of a category?
  • Did Mac Lane and Eilenberg know the work on groupoids?
  • Is Brandt's work rather unknown? (Why?) Or have you heard of it before?
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    $\begingroup$ I imagine groupoids would have turned up in work of Ehresman that is parallel to E and ML. Groupoids seem to naturally turn up if you are looking at inverse semigroups (aka collections of partial symmetries), although I don't know the history here very well. $\endgroup$ – Yemon Choi Mar 13 '15 at 1:02
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    $\begingroup$ Brandt's work is well known to everyone who works in semigroup theory and is discussed in the ~1967 book on semigroups by Clifford and Preston. It is not surprising that groupoids appears before categories since the fundamental groupoid has essentially been there since Poincaré $\endgroup$ – Benjamin Steinberg Mar 13 '15 at 1:38
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    $\begingroup$ Eilenberg and Mac Lane in any event thought the big contribution of their paper was introducing natural transformations as the name of their paper suggests. $\endgroup$ – Benjamin Steinberg Mar 13 '15 at 1:40
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    $\begingroup$ As for the first question, there are really two kinds of categories that category theory studies: "large" categories (e.g., categories of all algebraic structures of some sort) and "small" categories (e.g., the shape of a diagram, which is thought of as a single algebraic structure). My understanding is that when category theory was first invented, it focused on large categories (and in particular, the concept of functors and natural transformations between them). Brandt's theory of groupoids seems to fall squarely on the "small" side of the theory. $\endgroup$ – Eric Wofsey Mar 13 '15 at 4:49
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    $\begingroup$ Thus it seems to me that while they are formally similar, there is a still a wide gulf of intuition between Brandt's theory and the way Eilenberg and Mac Lane thought about category theory. $\endgroup$ – Eric Wofsey Mar 13 '15 at 4:54

The influence of Brandt's groupoid definition on the definition of category by Eilenberg and Mac Lane has been discussed on the category discussion list.

Bill Cockcroft told me in 1964-70 that there was an influence; he had visited Chicago for a year some time earlier. The use of groupoids in algebra was common knowledge in the 1940s, see the 1943 book on rings by Jacobson (N Carolina), and I expect the earlier book by AA Albert (Chicago), though I have not looked at that.

I did ask Eilenberg in 1985 about the influence of groupoids; he denied it and said that if it had they would have put it in as an example! I forgot to ask Mac Lane!

Papers [18,19] on my publication list (pdfs available) also have an extensive bibliography on groupoids.

Paper [147] "Three themes in the work of Charles Ehresmann: Local-to-global; Groupoids; Higher dimensions" gives an impression of Ehresmann's interest in geometric applications of groupoids.

Reidemeister's 1932 book on "Topologie" mentions the fundamental groupoid, and the groupoid determined by a group action. A recent translation to English by John Stillwell is available as arXiv:1402.3906.

The presentation in Galway explains my own interest in groupoids, through irritation that the usual van Kampen theorem did not compute the fundamental group of the circle, THE basic example in topology. I managed to find a solution to that in paper [4], using nonabelian cohomology, but the solution using the fundamental groupoid on a set of base points in [8], inspired by a paper of Philip Higgins, was more useful.

  • $\begingroup$ Yes I read the category mailing list, and this is also a partial motivation for my question. The second paragraph says that there was an influence, the third paragraph says that there was none. Hm. $\endgroup$ – Martin Brandenburg Mar 13 '15 at 23:12
  • $\begingroup$ I have no explanation. The paper listed under General Papers no 8 at people.math.ethz.ch/~knus/publications09.html states that it is believed that the groupoid axioms influenced the work of Eilenberg and Mac Lane. $\endgroup$ – Ronnie Brown Mar 14 '15 at 12:13
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    $\begingroup$ It is not unusual for people to be influenced by a work and then forget about that as they write and rewrite. I have probably done this. $\endgroup$ – Ronnie Brown Mar 14 '15 at 16:52
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    $\begingroup$ Martin writes "groupoids look very much like groups except that the product is only defined partially.". My definition of "higher dimensional algebra" is "the study of algebraic structures with partially defined operations whose domains are determined by geometric conditions", so that we can combine algebra and geometry, opening new worlds. Philip Higgins wrote the first paper on partial algebraic structures, "Algebras with a scheme of operators", Math. Nachr. 27 (1963) 115--132. $\endgroup$ – Ronnie Brown Mar 14 '15 at 21:06
  • $\begingroup$ There is a paper by Michael Barrett `Homotopy ringoids ...'QJM, which takes a related view although it is later that Eilenberg and Mac Lane. This led on to one by Peter Hilton and Walter Ledermann, again avoiding objects as much as possible. The objectless definition did not completely fade. $\endgroup$ – Tim Porter Mar 15 '15 at 16:26

As Yemon suggests in the Comments, perhaps you need to look at Ronnie Brown's paper:


and then, I would add, to look at his book `Topology and groupoids', followed by the other sources on the page:



I would like to address a mathematical, and not historical, point.

Theer is a vast difference between the theories of infinity-groupoids (which may, for example, be realized as Kan complexes) and of infinity-groupoids (which may, for example, be realized as Rezk's complete Segal spaces). The former takes just a few pages (at most a chapter) to set up, the latter a whole book. I doubt that the prior investigation of groupoids has much relevance to the development of category theory, either historically or from the point of view of what the important questions are.

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    $\begingroup$ is one of your "groupoids" supposed to say "categories"? $\endgroup$ – Dylan Wilson Mar 15 '15 at 17:56
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    $\begingroup$ One of the fascinations of mathematics is how structures can be appropriate to help us understand the small and the large. A good example is monoidal categories which crop up for describing the family of braid groups and also of chain complexes. Charles Ehresmann wrote that his aim was to understand the structure of everything; that also includes the structure of structures! $\endgroup$ – Ronnie Brown Mar 16 '15 at 14:46

Not an answer, but a hopefully somewhat relevant comment: The "arrow point of view stressed by Mac Lane and Eilenberg" is worth investigating more closely. it appears that the notation of arrows for maps, exact sequences and so on only took off simultaneously with the establishment of category theory, that is to say between 1947 and 1952. See the nice notes of Beno Eckmann on the subject of "Arrows and Exact Sequences" http://www.indiana.edu/~jfdavis/notes/eckmann.pdf


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