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Stream: learning: questions

Topic: A categorical proof of the Nielsen–Schreier theorem

Jencel Panic (Aug 04 2026 at 11:19):

I don't know anything about group theory, but I recently read a Wikipedia article about the Nielsen–Schreier theorem.

The theorem states that:

Every subgroup of a free group is itself free.

I don't understand the proofs that are given, but the statements seems somewhat trivial, so I was wondering if there is a simpler categorical way to prove it.

Intuitively, the only way to construct a subgroup of a group is to remove an element from the set of generators (along with all objects that are associated with this generator). And this would give you exactly the free group of the modified set of generators.

An even more interesting question is whether there is a way to prove this using the universal property of free groups.

Jencel Panic (Aug 04 2026 at 11:32):

A subset of a set, can be characterized by a function (the image function) and according to the universal property of free groups, each function between sets induces a unique morphism between the free groups built on them as sets of generators.

Josselin Poiret (Aug 04 2026 at 11:35):

Jencel Panic said:

Intuitively, the only way to construct a subgroup of a group is to remove an element from the set of generators (along with all objects that are associated with this generator). And this would give you exactly the free group of the modified set of generators.

This is not true: take the free group on 2 generators $x$ and $y$, and take the subgroup generated by $xy$. You can't obtain this subgroup by simply removing either $x$ or $y$ as generators.

David Wärn (Aug 04 2026 at 11:58):

You can phrase the "algebraic topology" proof in purely categorical terms, by talking about groupoids (and presheaves / categories of elements) rather than topological spaces (and covering spaces). In this formulation the proof indeed only uses the universal property of free groups.

Tom de Jong (Aug 04 2026 at 12:40):

Since David's brought up this perspective, probably https://doi.org/10.46298/lmcs-18(1:18)2022 by Andrew Swan is worth mentioning.

Oisín FC (Aug 04 2026 at 15:12):

Maybe not quite a categorical statement: but almost. The Nielsen–Schreier theorem has been worked out for some varieties over algebraic operads: https://arxiv.org/abs/2205.05364. I saw a talk on this paper in 2022, and the upshot seemed to be that it is really rather hard to prove these kinds of statements and the proofs seem to rely on very intricate calculations, done by computers! It also seemed like quite a lot of the older literature (at least in ring theory) on the topic had issues, I guess largely thanks to being pre-computer algebra.

Vikraman Choudhury (Aug 04 2026 at 15:15):

I had studied this a while ago, it's in Higgins' 1971 book, see Chapter 14: http://www.tac.mta.ca/tac/reprints/articles/7/tr7.pdf

Oisín FC (Aug 04 2026 at 17:44):

A related question: what are the Nielsen-Schreier monads on Set? There are some trivial cases eg., when the monad is idempotent. We also have the free group monad clearly, I wonder what are the others? I guess this is has probably been studied by someone.

Noam Zeilberger (Aug 05 2026 at 01:41):

An aside, but a little while ago while chatting with @Jonas Frey I idly wondered about how badly the correponding statement for categories failed: when does a category $C$ equipped with a faithful functor $i : C \to D$ into the free category $D = FG$ on a graph $G$ itself fail to be free? The Nielsen-Schreier theorem is not true for monoids, so surely those must provide counterexamples (and indeed Jonas found this MO thread), but are there also more properly categorical counterexamples?

Of course there are.

As target take $D = [3]$, the 3-simplex, which is the free category on the graph $G = 0 \to 1 \to 2 \to 3$.

As source, take $C = \mathbb{I}^2$, the 2-cube, aka the walking commutative square.

$\mathbb{I}^2$ is not a free category, but it admits a pair of faithful functors to $[3]$ by mapping (0,0) to 0 and (1,1) to 3, and (0,1) and (1,0) to either 1 or 2 respectively.

So the naive statement of the Nielsen-Schreier theorem for categories is indeed easily refuted!

Noam Zeilberger (Aug 05 2026 at 01:42):

Incidentally, there is a different property that you can place upon a functor $p : C \to D$ into a free category $D = FG$ to ensure that the source category is also free: the unique lifting of factorizations (aka ULF, aka discrete Conduché) property. A ULF functor need not be faithful and a faithful functor need not be ULF, so it is not clear to me whether this basic property of ULF functors has anything to do with the Nielsen-Schreier theorem for groups.

Beppe Metere (Aug 05 2026 at 13:16):

Hi! The Nielsen–Schreier theorem has a broader algebraic generalization. In fact, one defines a class of algebraic varieties, called Nielsen–Schreier varieties, in which every subalgebra of a free algebra is free.

See the classical paper:
https://iopscience.iop.org/article/10.1070/SM1974v022n04ABEH001705

See also:
https://arxiv.org/abs/2205.05364

The latter gives a recent characterization of Nielsen–Schreier varieties of non-associative k-algebras. It was presented by Xabi Garcias-Martinez at CT2025.

Afaik, no categorical characterisation so far, at least in intrinsic terms (i.e. as a set of formal axioms on the given category).

Beppe Metere (Aug 05 2026 at 13:35):

Another relevant recent paper is this: https://link.springer.com/article/10.1007/s10485-025-09828-w
Here Maxime, among other things, characterises projective objects in the category of internal crossed modules within any semi-abelian category, and more generally, analyses the the relation between free and projective objects. One result is recalled there is that internal categories in groups do not give a Schreier variety, because free objects do not coincide with projective objects (this would be a necessary condition), and extend this to semi-abelian context.

Beppe Metere (Aug 05 2026 at 13:42):

One way to face Nielsen-Scherier categorically would be to state the property relative to a monad.
A definition could be: given a monad $T$ on the category $C$, we say that $C$ is $T$-Schreier if free $T$-algebras are closed under subobjects in the Eilenberg-Moore category $C^T$.

Anyone is aware of any investigations in this direction?

Oisín FC (Aug 05 2026 at 15:49):

Beppe Metere said:

Hi! The Nielsen–Schreier theorem has a broader algebraic generalization. In fact, one defines a class of algebraic varieties, called Nielsen–Schreier varieties, in which every subalgebra of a free algebra is free.

See the classical paper:
https://iopscience.iop.org/article/10.1070/SM1974v022n04ABEH001705

See also:
https://arxiv.org/abs/2205.05364

The latter gives a recent characterization of Nielsen–Schreier varieties of non-associative k-algebras. It was presented by Xabi Garcias-Martinez at CT2025.

Afaik, no categorical characterisation so far, at least in intrinsic terms (i.e. as a set of formal axioms on the given category).

This is super interesting!

I mentioned the case of C=Set in this thread yesterday. Upon some thought it probably quite hard in general. It's been quite a while, but as I remember: Vlad and Umirbaev's approach uses this clever homological algebra trick of the "homological freeness criterion" and an unusual form of universal enveloping algebras which, on first glance, seems quite Vect_k-specific. And then they make everything computationally explicit by exploiting Grobener bases for shuffle operads.

I guess one would like to first prove "canonical Nielsen–Schreier theorem" on an analogue of the free group functor F on a monoidal category C (I'm a bit confused here about what these should be because it is groups on Set and associative algebra on Vect_k - I would have thought these latter objects were more like monoids - I guess this is in someway linked to the fact the monoidal product on Vect being used is not Cartesian)

The form of the universal enveloping algebras that they use might have a more general presentation: it's not the usual notion of a left adjoint to a forgetful functor but instead something more combinatorial defined in terms of Joyal's theory of species, see Section 2.7. So the idea would be that given a free algebra TX, one might construct a "universal enveloping F-algebra" F_T X \in Falg via similar procedure. Maybe one can use the presentations of monads in the sense of Power but probably one needs monads that are "built from the monoidal structure" in the same way operads are. Then one proves a theorem (of some kind?) showing that Nielsen-Scherier property of a monad is determined by this universal enveloping algebra functor factoring through free F-algebras.

All this is rather speculative ofc. I also doubt any generalisation of the Grobner bases works, which means all this would not be computationally valuable.