# Why are free objects "free"?

What is the origin/motivation for the adjective "free" in the term "free object"?

Does it refer to them coming "for free" (as being constructed from a set in a straight-forward manner) or does it refer some type of freedom they enjoy?

• "free of any relations" (other than those implied by the identities that define the class) Oct 29 '17 at 6:52
• Freely admitting extensions of maps from the generating set to homomorphism, depending on context. Oct 29 '17 at 6:56
• I have searched a bit for similar past questions. I found this one: Why are free modules called “free”? Maybe somebody will have better luck. Oct 29 '17 at 8:16
• I don't understand the downvotes either. The question isn't about the definition, it's about the history of the term. Does everyone know the history of the term, because I sure don't. Oct 29 '17 at 9:36
• It's also possible that "free" was first introduced in another language such as French (libre) or German (frei). In French, the meaning of "libre" is the same as free in the sense of freedom, but never has the meaning of free in "for free". Also I don't think that people used to think of free objects to be constructed in a straightforward way (explicit constructions, such as bases of free Lie algebras, are not trivial at all).
– YCor
Oct 29 '17 at 12:47

Free objects were first defined* by MacLane in Duality for Groups. That paper gives "free" a curious political context, I quote from page 486:

Call the dual (in this sense) of a free (nonabelian) group a fascist group. R . Baer has shown me a proof of the elegant theorem: every fascist group consists only of the identity element.

You can find a critical discussion of this joke on "freedom versus fascism" at nForum and at MSE.

*As pointed out by Lee Mosher and Peter LeFanu Lumsdaine, free groups were introduced much earlier. MacLane's 1950 paper gave the first definition of a free object in the context of an arrow composition from category theory.

• This is a footnote to "In this sense, free abelian groups are dual to infinitely divisible abelian groups". I can't deduce what definition MacLane has in mind for a f$*****$t group.
– YCor
Oct 29 '17 at 14:22
• The universal mapping definition of "free" already appears in Samuel's 1948 "On Universal Mappings and Free Topological Groups". Oct 29 '17 at 19:20
• This is an amusing later wordplay on the terminology, and a useful citation on the history of the category-theoretic unification of universal properties — but it tells us absolutely nothing about the origin or the original motivation of the term free, since (as the other answer shows) the terminology predates Mac Lane by almost thirty years Oct 29 '17 at 21:33
• The other answer provides an occurrence of "free" 30 years before MacLane's 1950's paper.
– YCor
Mar 18 '18 at 20:49

Jakob Nielsen's 1921 paper in Danish, with the title "Om Regning med ikke-kommutative Factorer og den Anvendelse i Gruppetoerien", is where he proved that subgroups of free groups are free. I don't know about the original Danish, but the English translation by Anne W. Neumann is available in Nielsen's collected works, with translated title "On calculation with Non-commutative Factors and its Application to Group Theory", and it contains the following sentences

In our case we will call it the 'free group $G_n$' generated by the generators $a_1,...,a_n$ and will denote it $[a_1,...,a_n]$.

...

In $G_n$, two distinct products of the $a_i$, irreducible in the $a_i$, always describe two different elements. It is for this reason that $G_n$ is called a free group.

One finds free groups described but not named in earlier works, such as Dehn's 1912 paper "Über unendliche discontinuierliche Gruppen", with English translation by John Stilwell titled "On infinite discontinuous groups" in Dehn's collected works:

Consider, say, the group which is given by two generators $S_1$ and $S_2$ without defining relations...

There is a telling passage later in this same paper. Dehn's focus in this paper is to understand groups in which there is a presentation with a single relator, and every generator appears exactly twice in that relator (he proves that this characterizes surface groups). Along the way he considers variations on this property, such as groups in which "each generator appears in the relation either twice or not at all", and then he introduces some terminology:

... we arrive at the following normal form of the group: among the generators are $w$ generators which appear in no relation ("free generators")...

• A nice early example in its original published language is Nielsen’s 1924 paper Die Isomorphismengrupper der freien Gruppen (‘The isomorphism group of the free group’), from Matematische Annalen 91. Oct 29 '17 at 21:37
• Nielsen's 1921 paper is on JStor, and there the phrase is "frie Gruppe." The Danish "fri" indeed translates to "free", but not in the monetary sense, confirming @YCor's suggestion. Oct 30 '17 at 13:32