Throughout we fix an integer k ≥ 2. We work with additive structure only up to relations of length k, and we therefore separate the A from its G. Concretely, an is a pair (A, G) where G is an abelian group and ⌀ ≠ A ⊆ G. In most additive–combinatorial applications one has A finite, but the categorical constructions we use are most naturally formulated without finiteness restrictions.
Let (A, G) and
(B, H) be additive
sets. A map f: A → B is called
a if it preserves all additive relations of length k: whenever
a1 + ⋯ + ak = a1′ + ⋯ + ak′ in
G,
with ai, ai′ ∈ A,
one has
f(a1) + ⋯ + f(ak) = f(a1′) + ⋯ + f(ak′) in
H.
We emphasize that the domain and codomain of f are subsets, but the relation to
be preserved is computed in the respective ambient groups. For k ≥ 2 this notion is stable under
composition: if f: A → B and g: B → C preserve
all k-term relations, then so
does g ∘ f. We shall
regard Freiman k-homomorphisms
as the morphisms of our basic category.
It is occasionally convenient to repackage the definition by
introducing the relation set
$$
\mathrm{Rel}_k(A\subseteq G)
:=\left\{\bigl((a_1,\dots,a_k),(a_1',\dots,a_k')\bigr)\in A^k\times A^k:
\sum_{i=1}^k a_i=\sum_{i=1}^k a_i' \text{ in }G\right\}.
$$
Then f: A → B is a
Freiman k-homomorphism
precisely when it sends each relation in Relk(A ⊆ G)
to a relation in Relk(B ⊆ H).
In particular, the definition depends only on the collection of true
k-term relations in the
ambient group, and not on any additional structure of G.
We write FRk∞ for the category whose objects are additive sets (A, G) and whose morphisms are Freiman k-homomorphisms f: A → B. The ambient groups do not appear explicitly in the morphism notation, but the requirement that f preserve relations is always interpreted using the given G and H. When A is finite, we obtain the full subcategory FRk ⊆ FRk∞, which is the regime most familiar from additive combinatorics. We nevertheless prefer FRk∞ as an ambient category because it accommodates filtered colimits, which are required to formalize finitary constructions.
Filtered colimits in FRk∞ may be computed in a concrete way. Given a filtered diagram (Ai, Gi), with structure maps induced by Freiman k-homomorphisms, the colimit group is the filtered colimit of the Gi in Ab, and the distinguished subset is the union of the images of the Ai in that colimit group. The point is that any condition about a k-term relation involves only finitely many elements, hence is detected at some stage of the diagram. This finitary behavior is what ultimately underlies the preservation of filtered colimits by the monad introduced later.
A central role is played by those objects (A, G) in which the ambient group G is, in a precise sense, determined by the k-term additive structure of A. We say that (A, G) has a (or simply that it is ) if for every additive set (B, H) and every Freiman k-homomorphism f: A → B, there exists a unique group homomorphism $\overline f\colon G\to H$ extending f (in the sense that $\overline f(a)=f(a)$ for all a ∈ A). When such a universal property holds, Freiman morphisms out of A are forced to behave as genuine homomorphisms at the ambient-group level.
We denote by UFRk∞ the full subcategory of FRk∞ spanned by universal objects. The inclusion functor will be written ι: UFRk∞ ↪ FRk∞. Later we construct a left adjoint U to ι, so that UFRk∞ becomes a reflective subcategory of FRk∞. For the moment, we only record the guiding principle: universality is designed so that a Freiman k-homomorphism out of A is controlled by an ambient group homomorphism out of G, and this control is unique.
To make universal ambient groups explicit, we shall repeatedly use
the free abelian group on A.
We write ℤ(A) for
the direct sum of copies of ℤ indexed
by A, equipped with its
distinguished basis {ea}a ∈ A.
A k-term relation in G,
$\sum_{i=1}^k a_i=\sum_{i=1}^k
a_i'$,
corresponds to a formal relation vector
$\sum_{i=1}^k e_{a_i}-\sum_{i=1}^k e_{a_i'}\in
\mathbb{Z}^{(A)}$.
We shall denote by Xk(A ⊆ G)
the set of all such vectors and by ⟨Xk(A ⊆ G)⟩
the subgroup they generate. The quotient of ℤ(A) by this relation
subgroup is the ambient group produced by the reflector, and the class
of ea
provides the canonical image of a in that quotient. The essential
point for functoriality is that a Freiman k-homomorphism carries true k-term relations to true k-term relations, hence induces a
homomorphism between the corresponding quotients.
We recall the categorical terminology needed to state our main
constructions. Let 𝒞 be a category and
𝒟 ⊆ 𝒞 a full subcategory with inclusion
ι: 𝒟 → 𝒞. A of 𝒞 onto 𝒟 is
a left adjoint U: 𝒞 → 𝒟 to
ι. Thus there are natural
bijections
Hom𝒟(U(X), Y) ≅ Hom𝒞(X, ι(Y)),
natural in X ∈ 𝒞 and Y ∈ 𝒟. The adjunction provides a
η: id𝒞 ⇒ ιU
and a ε: Uι ⇒ id𝒟.
In the reflective situation the counit is an isomorphism, expressing the
fact that applying the reflector to an object already in 𝒟 does nothing up to canonical
identification.
Any adjunction U ⊣ ι induces a monad T := ιU on 𝒞. A on 𝒞 is an endofunctor T: 𝒞 → 𝒞 equipped with natural transformations η: id𝒞 ⇒ T (unit) and μ: T2 ⇒ T (multiplication) satisfying the associativity and unit axioms. In the case of a reflection, the multiplication is induced by the counit, and one obtains an idempotence phenomenon: T2 is canonically isomorphic to T (equivalently, μ is an isomorphism). We will use this in order to identify the universal objects as the fixed points of the monad.
Given a monad (T, η, μ) on 𝒞, a is a pair (X, α) with X ∈ 𝒞 and a morphism α: T(X) → X such that α ∘ ηX = idX and α ∘ T(α) = α ∘ μX. A morphism of T-algebras (X, α) → (Y, β) is a morphism f: X → Y in 𝒞 with f ∘ α = β ∘ T(f). The resulting category is the 𝒞T. In our setting, 𝒞 = FRk∞, and the concrete description of 𝒞T will be given in terms of split presentations of the ambient group by explicit k-term relations.
Finally, we recall that an endofunctor T on a cocomplete category is called if it preserves filtered colimits. This property will be essential for passing between FRk and FRk∞: a finitary monad on FRk∞ is determined by its behavior on finitely generated data (in particular, finite A together with the finitely many relations relevant to a given computation), and it interacts well with the standard limiting arguments in additive combinatorics. We will verify finitarity for the monad T = ιU by reducing the relevant relation subgroups to finite stages in a filtered system.
We now construct, for each additive set (A, G), a canonical universal ambient group which remembers precisely the k-term additive relations that hold in G on elements of A. The construction is a presentation by generators indexed by A and relations indexed by the true k-term relations in the ambient group.
Let (A, G) be an
object of FRk∞. Consider
the free abelian group
ℤ(A) = ⨁a ∈ Aℤ ea,
with basis {ea}a ∈ A.
Any relation of length k in
G,
a1 + ⋯ + ak = a1′ + ⋯ + ak′ (ai, ai′ ∈ A),
determines a formal relation vector in ℤ(A),
$$
\sum_{i=1}^k e_{a_i} \;-\; \sum_{i=1}^k e_{a_i'}.
$$
We let Xk(A ⊆ G) ⊆ ℤ(A)
be the set of all such vectors, and we denote by ⟨Xk(A ⊆ G)⟩
the subgroup they generate. We then define the group
Uk(A ⊆ G) := ℤ(A)/⟨Xk(A ⊆ G)⟩.
We write [x] ∈ Uk(A ⊆ G)
for the class of x ∈ ℤ(A). The
distinguished subset of the reflected object will be the image of A under the map
ηA: A → Uk(A ⊆ G), ηA(a) := [ea].
Thus the promised object is
U(A, G) := (ηA(A), Uk(A ⊆ G)).
By construction, the only relations we impose are those k-term relations that already hold
in G. In particular, Uk(A ⊆ G)
depends on G only through the
predicate ``$\sum_{i=1}^k a_i=\sum_{i=1}^k
a_i'$ in G’’ for tuples
in Ak;
enlarging the ambient group without changing the set of true k-term relations does not change the
quotient.
There is a canonical comparison map back to the original ambient
group. Since any map A → G extends uniquely to a
homomorphism ℤ(A) → G, the
inclusion A ↪ G
induces a unique group homomorphism
p̃A, G: ℤ(A) → G, p̃A, G(ea) = a.
Every element of Xk(A ⊆ G)
lies in ker (p̃A, G),
hence ⟨Xk(A ⊆ G)⟩ ≤ ker (p̃A, G),
so p̃A, G
descends to a homomorphism
pA, G: Uk(A ⊆ G) → G, pA, G([ea]) = a.
Its image is the subgroup ⟨A⟩ ≤ G generated by A, and it is surjective precisely
when A generates G.
We next explain how the assignment (A, G) ↦ U(A, G)
acts on morphisms. Let
f: (A, G) → (B, H)
be a morphism in FRk∞, i.e. a
Freiman k-homomorphism f: A → B. The map
f induces a homomorphism of
free abelian groups
f̃: ℤ(A) → ℤ(B), f̃(ea) = ef(a).
To obtain a map on the quotients we must check that f̃ sends the relation subgroup for
(A, G) into the
relation subgroup for (B, H). Let
$$
r=\sum_{i=1}^k e_{a_i}-\sum_{i=1}^k e_{a_i'}\in X_k(A\subseteq G),
$$
so that $\sum_{i=1}^k a_i=\sum_{i=1}^k
a_i'$ in G. Since f is a Freiman k-homomorphism, we have
$$
\sum_{i=1}^k f(a_i)\;=\;\sum_{i=1}^k f(a_i') \qquad\text{in }H.
$$
Therefore
$$
\widetilde f(r)
=\sum_{i=1}^k e_{f(a_i)}-\sum_{i=1}^k e_{f(a_i')}
\in X_k(B\subseteq H),
$$
and hence f̃(⟨Xk(A ⊆ G)⟩) ⊆ ⟨Xk(B ⊆ H)⟩.
It follows that f̃ descends to
a unique homomorphism
Uk(f): Uk(A ⊆ G) → Uk(B ⊆ H)
such that Uk(f)([ea]) = [ef(a)]
for all a ∈ A. On
distinguished subsets this is exactly the map ηA(A) → ηB(B)
induced by f. We therefore
set
U(f) := Uk(f)|ηA(A): ηA(A) → ηB(B),
viewed as a morphism U(A, G) → U(B, H)
in FRk∞.
The identities and composition laws are inherited from the corresponding
properties of f̃ on free
abelian groups, so U is a
well-defined endofunctor on objects and morphisms of FRk∞, landing
in the full subcategory of universal objects as we verify next.
We claim that (ηA(A), Uk(A ⊆ G))
has a universal ambient group. Concretely, let (B, H) be any additive set
and let
φ: ηA(A) → B
be a Freiman k-homomorphism
(where the k-term relations on
ηA(A)
are computed in the ambient group Uk(A ⊆ G)).
Composing with ηA yields a map
φ ∘ ηA: A → B.
We define a homomorphism
Φ̃: ℤ(A) → H, Φ̃(ea) = φ([ea]) ∈ B ⊆ H,
and we show that Φ̃ kills ⟨Xk(A ⊆ G)⟩.
Indeed, let $r=\sum_{i=1}^k
e_{a_i}-\sum_{i=1}^k e_{a_i'}\in X_k(A\subseteq G)$. Then [r] = 0 in Uk(A ⊆ G),
which is exactly to say that
[ea1] + ⋯ + [eak]= [ea1′] + ⋯ + [eak′] in
Uk(A ⊆ G).
Since φ is Freiman of order
k, applying φ to this k-term relation yields
φ([ea1]) + ⋯ + φ([eak]) = φ([ea1′]) + ⋯ + φ([eak′]) in
H,
i.e. Φ̃(r) = 0. Hence
⟨Xk(A ⊆ G)⟩ ⊆ ker (Φ̃),
and Φ̃ descends to a unique
homomorphism
Φ: Uk(A ⊆ G) → H
such that Φ([ea]) = φ([ea])
for all a ∈ A. By
construction Φ extends φ (on the distinguished subset ηA(A)),
and uniqueness follows because Uk(A ⊆ G)
is generated by the classes [ea]. This
proves that U(A, G) lies in
UFRk∞.
In particular, for each (A, G) we have produced a universal object equipped with the canonical map ηA: (A, G) → ιU(A, G) in FRk∞. The adjunction U ⊣ ι will be obtained by observing that any Freiman k-homomorphism A → B into a universal object (B, H) uniquely extends along ηA to a morphism U(A, G) → (B, H), and conversely any morphism U(A, G) → (B, H) restricts along ηA to a Freiman map A → B. Thus the reflector U is explicit: it freely adjoins an ambient group subject only to the k-term relations already valid in G, and it does so functorially.
We fix an object (A, G) of FRk∞. Our aim is to replace the ambient group G by a canonical one which is generated by formal symbols indexed by A and in which the k-term additive relations witnessed in G are imposed as defining relations. This produces, functorially in (A, G), an object of UFRk∞ equipped with a canonical map from (A, G).
We begin with the free abelian group on the underlying set A,
ℤ(A) = ⨁a ∈ Aℤ ea,
where the direct sum is taken in the usual sense (so elements of ℤ(A) are finitely
supported integer combinations of the basis {ea}a ∈ A).
Each true k-term relation in
the ambient group G,
a1 + ⋯ + ak = a1′ + ⋯ + ak′ (ai, ai′ ∈ A),
gives rise to a vector in ℤ(A),
$$
r(a_\bullet,a'_\bullet)\;:=\;\sum_{i=1}^k e_{a_i}-\sum_{i=1}^k e_{a_i'}.
$$
We collect all such vectors in a set
Xk(A ⊆ G) ⊆ ℤ(A),
and we denote by ⟨Xk(A ⊆ G)⟩
the subgroup generated by them. Since ℤ(A) is abelian, ⟨Xk(A ⊆ G)⟩
is simply the set of all finite integer linear combinations of these
relation vectors.
We define the of A relative to
G by the quotient
Uk(A ⊆ G) := ℤ(A)/⟨Xk(A ⊆ G)⟩.
For x ∈ ℤ(A) we
write [x] ∈ Uk(A ⊆ G)
for its class. The distinguished subset of our reflected object is the
image of A under the map
ηA: A → Uk(A ⊆ G), a ↦ [ea].
We therefore set
U(A, G) := (ηA(A), Uk(A ⊆ G)).
By construction, ηA(A)
generates Uk(A ⊆ G)
as an abelian group, because the classes [ea] generate
ℤ(A) and hence
generate its quotient. In particular, any group homomorphism out of
Uk(A ⊆ G)
is determined by its values on ηA(A).
A useful point of view is that the quotient remembers only the predicate of k-term equality inside G: the subgroup ⟨Xk(A ⊆ G)⟩ is determined by which pairs of k-tuples in Ak have equal sums in G, and nothing else. Thus, if A ⊆ G ⊆ G′ and the sets of true k-term relations among elements of A coincide when computed in G and in G′, then the resulting quotients Uk(A ⊆ G) and Uk(A ⊆ G′) are canonically isomorphic.
The inclusion A ↪ G
defines a unique group homomorphism from the free group,
p̃A, G: ℤ(A) → G, p̃A, G(ea) = a.
By definition of Xk(A ⊆ G),
each relation vector r ∈ Xk(A ⊆ G)
lies in ker (p̃A, G);
hence ⟨Xk(A ⊆ G)⟩ ≤ ker (p̃A, G).
Consequently p̃A, G
descends to a homomorphism
pA, G: Uk(A ⊆ G) → G, pA, G([ea]) = a.
Its image is the subgroup ⟨A⟩ ≤ G generated by A. In particular, when A generates G, the map pA, G
is surjective. We emphasize that we do force pA, G
to be injective: any further relations among A that are consequences of the
imposed k-term relations
remain present in the kernel, and this kernel is precisely what is
needed to encode the Freiman-k
structure of A abstractly.
Let f: (A, G) → (B, H)
be a morphism in FRk∞, i.e. a
Freiman k-homomorphism f: A → B. We first
extend f to a homomorphism
between the free abelian groups:
f̃: ℤ(A) → ℤ(B), f̃(ea) = ef(a).
To descend f̃ to the quotients,
we must verify that f̃ respects
the defining relation subgroups.
Let $r=\sum_{i=1}^k e_{a_i}-\sum_{i=1}^k
e_{a_i'}$ lie in Xk(A ⊆ G).
Then $\sum_{i=1}^k a_i=\sum_{i=1}^k
a_i'$ in G. Since f is Freiman of order k, it preserves this equality of
k-term sums, so
$$
\sum_{i=1}^k f(a_i)\;=\;\sum_{i=1}^k f(a_i')\qquad\text{in }H.
$$
Therefore
$$
\widetilde f(r)\;=\;\sum_{i=1}^k e_{f(a_i)}-\sum_{i=1}^k
e_{f(a_i')}\;\in\;X_k(B\subseteq H),
$$
and hence f̃(⟨Xk(A ⊆ G)⟩) ⊆ ⟨Xk(B ⊆ H)⟩.
It follows that there is a unique induced homomorphism
Uk(f): Uk(A ⊆ G) → Uk(B ⊆ H)
satisfying Uk(f)([ea]) = [ef(a)]
for all a ∈ A.
Restricting to the distinguished subsets yields a morphism in FRk∞,
U(f): ηA(A) → ηB(B), [ea] ↦ [ef(a)],
whose ambient-group component is Uk(f).
Since Uk(f)
is a group homomorphism, it automatically preserves all k-term relations among elements of
ηA(A),
so U(f) is indeed a
Freiman k-homomorphism. The
equalities U(id) = id and
U(g ∘ f) = U(g) ∘ U(f)
follow from the corresponding identities for the maps f̃ on free abelian groups, so U is a well-defined functor FRk∞ → FRk∞.
We now verify that U(A, G) lies in
the full subcategory UFRk∞,
i.e. that Uk(A ⊆ G)
is a universal ambient group for the subset ηA(A).
Let (B, H) be any
object of FRk∞, and
let
φ: ηA(A) → B
be a Freiman k-homomorphism.
Since ℤ(A) is free
on the set A, the assignment
ea ↦ φ([ea]) ∈ B ⊆ H
determines a unique homomorphism
Φ̃: ℤ(A) → H, Φ̃(ea) = φ([ea]).
To show that Φ̃ factors through
the quotient Uk(A ⊆ G),
it suffices to prove that ⟨Xk(A ⊆ G)⟩ ⊆ ker (Φ̃).
Let $r=\sum_{i=1}^k e_{a_i}-\sum_{i=1}^k
e_{a_i'}$ be an element of Xk(A ⊆ G).
By definition of the quotient, [r] = 0 in Uk(A ⊆ G),
equivalently
[ea1] + ⋯ + [eak] = [ea1′] + ⋯ + [eak′] in
Uk(A ⊆ G).
This is a k-term additive
relation among elements of the distinguished subset ηA(A).
Since φ is Freiman of order
k, applying φ yields an equality in H,
φ([ea1]) + ⋯ + φ([eak]) = φ([ea1′]) + ⋯ + φ([eak′]).
But the left-hand side is exactly $\widetilde\Phi\bigl(\sum_{i=1}^k
e_{a_i}\bigr)$ and similarly on the right, so Φ̃(r) = 0. By additivity,
Φ̃ kills the subgroup generated
by such r, and therefore
descends to a unique homomorphism
Φ: Uk(A ⊆ G) → H
with Φ([ea]) = φ([ea])
for all a ∈ A. By
construction, Φ extends φ on ηA(A).
Uniqueness is immediate because Uk(A ⊆ G)
is generated by the elements [ea]. This
establishes the universal ambient group property for U(A, G), and hence
U(A, G) ∈ UFRk∞.
In summary, we have constructed a functor U: FRk∞ → UFRk∞ together with the canonical map ηA: A → ηA(A) on distinguished subsets. In the next step we identify U as the reflector left adjoint to the inclusion ι, by exhibiting the corresponding hom-set bijection and describing the unit and counit explicitly.
Let (A, G) ∈ FRk∞
and let (B, H) ∈ UFRk∞.
We claim that precomposition with the canonical map on distinguished
subsets,
ηA: A → ηA(A) ⊆ Uk(A ⊆ G), a ↦ [ea],
induces a natural bijection
Since ι is the inclusion, the
right-hand side is simply the set of Freiman k-homomorphisms f: A → B.
Given a Freiman k-homomorphism
f: A → B, we
define a map on ηA(A)
by
f♯: ηA(A) → B, f♯([ea]) := f(a).
We must check that f♯ is again a Freiman
k-homomorphism. Thus suppose
that
[ea1] + ⋯ + [eak] = [ea1′] + ⋯ + [eak′] in
Uk(A ⊆ G).
By definition of the quotient Uk(A ⊆ G) = ℤ(A)/⟨Xk(A ⊆ G)⟩,
this equality holds if and only if
$$
\sum_{i=1}^k e_{a_i}-\sum_{i=1}^k e_{a_i'}\in \langle X_k(A\subseteq
G)\rangle,
$$
and in particular it holds whenever $\sum_{i=1}^k a_i=\sum_{i=1}^k a_i'$ in G. But every relation vector in
Xk(A ⊆ G)
arises from such an equality in G, and therefore any equality of
k-term sums among [ea] is
generated by k-term sum
equalities among the corresponding elements of A that already hold in G. Since f is Freiman of order k, it preserves each of those
generating equalities, and hence it preserves their consequences.
Concretely, applying f♯ gives
f(a1) + ⋯ + f(ak) = f(a1′) + ⋯ + f(ak′) in
H,
which is exactly the Freiman condition for f♯. Thus f♯ is a morphism U(A, G) → (B, H)
in FRk∞.
At this point we use the defining property of (B, H) ∈ UFRk∞:
the map f♯: ηA(A) → B
uniquely extends to a group homomorphism
$$
\overline{f^\sharp}\colon U_k(A\subseteq G)\to H.
$$
This uniqueness is the key ingredient that turns the formal quotient
construction into a reflection: it ensures that a morphism out of U(A, G) is
determined by, and only by, its restriction to the distinguished subset
ηA(A).
We therefore set
Λ(f) := f♯ ∈ UFRk∞(U(A, G), (B, H)).
The assignment f ↦ Λ(f) will be
the inverse of ψ ↦ ψ ∘ ηA
in .
Let ψ: U(A, G) → (B, H)
be a morphism in UFRk∞. Then
ψ ∘ ηA: A → B
is a Freiman map, and applying the construction above yields (ψ ∘ ηA)♯: ηA(A) → B
with
(ψ ∘ ηA)♯([ea]) = (ψ ∘ ηA)(a) = ψ([ea]).
Hence (ψ ∘ ηA)♯ = ψ
as maps on ηA(A),
and therefore as morphisms in FRk∞. This
shows Λ(ψ ∘ ηA) = ψ.
Conversely, let f: A → B be a
Freiman map. Then Λ(f) = f♯
satisfies
(f♯ ∘ ηA)(a) = f♯([ea]) = f(a),
so f♯ ∘ ηA = f.
Thus the two constructions are mutual inverses, establishing the
bijection .
The bijection is natural in both variables. Naturality in (A, G) follows because for
a Freiman map u: (A′, G′) → (A, G)
the diagram
$$
A' \xrightarrow{\eta_{A'}} \eta_{A'}(A') \xrightarrow{U(u)} \eta_A(A)
\qquad\text{equals}\qquad
A' \xrightarrow{u} A \xrightarrow{\eta_A} \eta_A(A),
$$
by construction of U(u) on generators.
Naturality in (B, H)
follows because in UFRk∞ every
morphism v: (B, H) → (C, K)
has a unique ambient-group extension $\overline v\colon H\to K$, and composing
extensions corresponds to extending compositions. In both cases,
uniqueness of the group extension is what forces the hom-set
correspondence to commute with composition.
The unit of the adjunction at (A, G) is the morphism in
FRk∞
η(A, G): (A, G) → ιU(A, G)
whose underlying map on distinguished subsets is precisely a ↦ [ea].
It is characterized by the property that for every (B, H) ∈ UFRk∞
and every Freiman map f: A → B, the
corresponding morphism Λ(f): U(A, G) → (B, H)
satisfies
Λ(f) ∘ η(A, G) = f
as Freiman maps A → B. Equivalently, η(A, G)
is initial among morphisms from (A, G) into universal
objects.
Let (B, H) ∈ UFRk∞.
Consider Uι(B, H) = (ηB(B), Uk(B ⊆ H)).
The identity map idB: B → B
is a Freiman k-homomorphism,
and by the hom-set bijection it corresponds to a unique morphism in
UFRk∞,
ε(B, H): Uι(B, H) → (B, H),
called the counit. Explicitly, it is the Freiman map [eb] ↦ b
on distinguished subsets; its ambient-group extension is the canonical
homomorphism
pB, H: Uk(B ⊆ H) → H, pB, H([eb]) = b.
Because (B, H) is
universal, this extension is uniquely determined by the values on B, and ε(B, H)
is uniquely determined by idB.
The two triangle identities are formal consequences of the hom-set
bijection, but it is instructive to see where uniqueness enters. For
(A, G), we have a
morphism
$$
U(A,G)\xrightarrow{U(\eta_{(A,G)})} U\iota
U(A,G)\xrightarrow{\varepsilon_{U(A,G)}} U(A,G),
$$
and we claim the composite is idU(A, G).
Both maps are morphisms in UFRk∞, hence
are determined by their restrictions to ηA(A).
On generators we compute
εU(A, G)(U(η(A, G))([ea])) = εU(A, G)([e[ea]]) = [ea],
so the composite fixes [ea] for all
a, and therefore is the
identity on Uk(A ⊆ G).
Similarly, for (B, H) ∈ UFRk∞,
the composite
$$
\iota(B,H)\xrightarrow{\eta_{\iota(B,H)}} \iota
U\iota(B,H)\xrightarrow{\iota(\varepsilon_{(B,H)})} \iota(B,H)
$$
restricts on B to b ↦ [eb] ↦ b,
hence equals idB,
and thus is the identity morphism in FRk∞.
We have exhibited U as a
reflector onto UFRk∞, with
unit η given by a ↦ [ea]
and counit ε given by [eb] ↦ b.
The decisive point throughout is that universality converts Freiman maps
into uniquely determined group homomorphisms; the adjunction is
precisely the packaging of this uniqueness into functorial form. Having
identified U ⊣ ι, we
may now pass to the induced monad T = ιU and
describe its unit and multiplication in these explicit terms.
From the adjunction U ⊣ ι we obtain an
endofunctor
T := ι ∘ U: FRk∞ → FRk∞,
together with a unit η: Id ⇒ T and a
multiplication μ: T2 ⇒ T.
On objects, T simply forgets
that U(A, G)
lands in UFRk∞:
thus
T(A, G) = (ηA(A), Uk(A ⊆ G)), ηA(a) = [ea].
On morphisms, if f: (A, G) → (B, H)
is a Freiman k-homomorphism,
then T(f) is induced
by the homomorphism ℤ(A) → ℤ(B)
sending ea ↦ ef(a)
and passing to quotients. In particular, on distinguished subsets we
have T(f)([ea]) = [ef(a)].
For (A, G) ∈ FRk∞,
the unit component
η(A, G): (A, G) → T(A, G) = ιU(A, G)
is the morphism whose underlying map A → ηA(A)
is a ↦ [ea].
This is precisely the map that exhibits T(A, G) as the
universal recipient of Freiman k-maps out of (A, G) into universal
objects: for (B, H) ∈ UFRk∞
and f: A → B
Freiman of order k, there is a
unique morphism Λ(f): T(A, G) → (B, H)
such that Λ(f) ∘ η(A, G) = f.
The multiplication of the monad associated to U ⊣ ι is, by
definition,
μ := ι ε U: ιUιU ⇒ ιU,
where ε: Uι ⇒ IdUFRk∞
is the counit of the adjunction. Evaluating at (A, G), we obtain a
morphism in FRk∞,
μ(A, G) = ι(εU(A, G)): T2(A, G) → T(A, G).
To make this explicit, write U(A, G) = (A1, G1)
where
A1 := ηA(A) ⊆ G1 := Uk(A ⊆ G).
Then
T2(A, G) = T(A1, G1) = (ηA1(A1), Uk(A1 ⊆ G1)),
and the counit εU(A, G): Uι(A1, G1) → (A1, G1)
is characterized by being the identity on the underlying distinguished
subset A1 after
applying the hom-set bijection. Concretely, on distinguished subsets it
is the map
ηA1(A1) → A1, [ex] ↦ x (x ∈ A1),
so that, in particular,
[e[ea]] ↦ [ea] (a ∈ A).
Its ambient-group extension is the canonical homomorphism
pA1, G1: Uk(A1 ⊆ G1) → G1, pA1, G1([ex]) = x (x ∈ A1),
and μ(A, G)
is this same morphism viewed in FRk∞. Thus
μ(A, G)
is the unique morphism T2(A, G) → T(A, G)
that sends the “second-order generators” [e[ea]]
back to [ea].
Because U is a reflector onto
UFRk∞,
applying U to an already
universal object does not change it, up to the counit isomorphism. In
particular, U(A, G) ∈ UFRk∞
for every (A, G),
hence the counit component
εU(A, G): UιU(A, G) → U(A, G)
is an isomorphism in UFRk∞. After
applying ι, this says
that
μ(A, G): T2(A, G) → T(A, G)
is an isomorphism in FRk∞,
naturally in (A, G).
Equivalently, the monad multiplication μ is a natural isomorphism, and we
may regard T as an idempotent
monad in the usual sense that T2 ≃ T via μ.
It is useful to record the canonical inverse. Since μ(A, G)
is ι(εU(A, G)),
its inverse is ι(εU(A, G)−1).
One may also describe it on distinguished subsets using the unit at
T(A, G): by
the triangle identities, the composite
$$
T(A,G)\xrightarrow{\eta_{T(A,G)}} T^2(A,G)\xrightarrow{\mu_{(A,G)}}
T(A,G)
$$
is the identity, and because μ(A, G)
is an isomorphism this forces ηT(A, G) = μ(A, G)−1.
Concretely, ηT(A, G)
sends [ea] ∈ ηA(A)
to [e[ea]] ∈ ηA1(A1).
By definition, an object (A, G) is a fixed point of
T if η(A, G): (A, G) → T(A, G)
is an isomorphism in FRk∞.
Unwinding the construction, this means precisely that (A, G) already has the
universal ambient-group property: the reflection does nothing.
Equivalently, the canonical projection pA, G: Uk(A ⊆ G) → G
(defined by pA, G([ea]) = a)
is an isomorphism, so that G
is presented by the k-term
relations holding in G among
elements of A. In this sense
T “forgets extraneous ambient
structure” and replaces (A, G) by the minimal
universal object through which all Freiman k-maps out of A into universal objects factor.
We emphasize that fixed points are closed under T in the strongest possible way: if (A, G) is any object, then T(A, G) is always a fixed point, because ηT(A, G) is inverse to μ(A, G) as above. Thus T is a projection (up to canonical isomorphism) onto the full subcategory of universal objects.
A T-algebra structure on (A, G) is a morphism α: T(A, G) → (A, G)
in FRk∞
satisfying the unit and associativity axioms
α ∘ η(A, G) = id(A, G), α ∘ T(α) = α ∘ μ(A, G).
For the present monad, the first axiom already has a strong categorical
meaning: it exhibits (A, G) as a retract of the
universal object T(A, G), with
section η(A, G)
and retraction α. Since T(A, G) is a fixed
point, every algebra is (at least formally) a retract of a fixed point.
This explains why the Eilenberg–Moore category (FRk∞)T
is typically larger than UFRk∞:
universal objects are the fixed points, whereas algebras encode of the
reflection map.
The idempotence of T clarifies the second axiom. Because μ(A, G) is an isomorphism, the equation α ∘ T(α) = α ∘ μ(A, G) may be viewed as coherence of α with the canonical identification T2(A, G) ≅ T(A, G); it ensures that the retraction α is compatible with applying T once more. In particular, when α is itself an isomorphism, the algebra is precisely a fixed point, hence corresponds to an object of UFRk∞. More generally, allowing arbitrary retractions α amounts to passing from fixed points to their retracts, which is the standard mechanism by which the Eilenberg–Moore category of an idempotent monad realizes an idempotent completion (Karoubi envelope) of the reflective subcategory.
In the next step we will analyze how these retractions can be described concretely in terms of the canonical map pA, G: Uk(A ⊆ G) ↠ ⟨A⟩ ≤ G and its sections, and then we will turn to the finitary nature of T, namely its compatibility with filtered colimits in FRk∞.
We now verify that FRk∞ admits
filtered colimits and that the monad T = ιU preserves
them. This is the precise sense in which T is : the relations defining T(A, G) are of
bounded arity (namely k),
hence are already detected on sufficiently small stages of a filtered
diagram.
Let I be a filtered category
and let
D: I → FRk∞, i ↦ (Ai, Gi)
be a filtered diagram. Write the structure maps as Freiman k-homomorphisms
fij: (Ai, Gi) → (Aj, Gj) (i → j in
I).
In particular, each fij is
a function Ai → Aj
preserving k-term additive
relations, and the ambient-group part of the morphism is not part of the
data; nevertheless, since each Ai ⊆ Gi
sits inside an abelian group, we may form the filtered colimit of
ambient groups in Ab,
G := colimi ∈ IGi,
with structure homomorphisms ψi: Gi → G.
We then define a subset A ⊆ G by taking the union
of the images of the distinguished subsets:
A := ⋃i ∈ Iψi(Ai) ⊆ G.
Since I is filtered and each
Ai is
nonempty, the set A is
nonempty as well. The pair (A, G) will be the colimit
object in FRk∞.
To see the universal property, let (B, H) be any object and suppose given a compatible cocone of Freiman k-maps gi: Ai → B. Compatibility means that gj ∘ fij = gi on Ai whenever i → j. For any a ∈ A, choose i and ai ∈ Ai with ψi(ai) = a, and define g(a) := gi(ai). Filteredness and cocone compatibility ensure this is well-defined, and it is immediate that g: A → B is a Freiman k-homomorphism: any k-term relation in G among elements of A may be checked at a sufficiently large stage Gj, where it is preserved by gj, hence by g. Uniqueness is clear since the images of the ψi(Ai) cover A. Thus (A, G) is a filtered colimit of the diagram D in FRk∞.
This computation can be viewed as expressing FRk∞ as the natural Ind-completion of the finite regime FRk: every object is a filtered colimit of its finite subobjects, and filtered colimits are computed by taking direct limits of ambient groups and unions of distinguished subsets.
Fix a filtered diagram D as
above with colimit (A, G). The key observation
is that the subgroup of relations ⟨Xk(A ⊆ G)⟩ ≤ ℤ(A)
is itself the filtered colimit of the corresponding relation subgroups
at finite stages. Concretely, for each i there is an induced homomorphism
of free abelian groups
Φi: ℤ(Ai) → ℤ(A), ea ↦ eψi(a),
and hence an induced subgroup Φi(⟨Xk(Ai ⊆ Gi)⟩) ≤ ℤ(A).
We claim that
The inclusion ``⊇’’ follows from
functoriality of relation vectors: if $\sum_{r=1}^k a_r=\sum_{r=1}^k a'_r$ holds in
Gi, then
applying ψi yields the
corresponding relation in G,
hence the image of any generator of Xk(Ai ⊆ Gi)
lies in Xk(A ⊆ G),
and thus Φi(⟨Xk(Ai ⊆ Gi)⟩) ⊆ ⟨Xk(A ⊆ G)⟩.
For the converse inclusion ``⊆’’, it
suffices to treat generators of Xk(A ⊆ G).
Let
$$
x \;=\; \sum_{r=1}^k e_{a_r}\;-\;\sum_{r=1}^k e_{a'_r}\ \in\
X_k(A\subseteq G),
$$
so that $\sum_{r=1}^k a_r=\sum_{r=1}^k
a'_r$ in G. Choose
indices ir, ir′ ∈ I
and representatives ãr ∈ Air,
ãr′ ∈ Air′
mapping to ar, ar′
in A. By filteredness there
exists j ∈ I
receiving morphisms from all ir and ir′,
so that all these elements have images in Gj; denote these
images by br, br′ ∈ Aj ⊆ Gj.
The equality ∑ar = ∑ar′
in the colimit group G implies
that the elements ∑ψj(br)
and ∑ψj(br′)
are equal in G. A basic
property of filtered colimits in Ab is
that if two elements of some stage Gj become equal
in the colimit, then they become equal at a further stage: thus there
exists a morphism j → ℓ in I such that the images of ∑br and ∑br′
coincide in Gℓ.
Equivalently, ∑cr = ∑cr′
holds in Gℓ for the
images cr, cr′ ∈ Aℓ.
It follows that
$$
x \;=\; \Phi_\ell\!\left(\sum_{r=1}^k e_{c_r}-\sum_{r=1}^k
e_{c'_r}\right)
$$
lies in Φℓ(Xk(Aℓ ⊆ Gℓ)),
and hence in the union on the right-hand side of . Since the right-hand
side is a subgroup, it contains the subgroup generated by all such x, proving .
We now compare Uk(A ⊆ G) = ℤ(A)/⟨Xk(A ⊆ G)⟩
with the filtered colimit of the groups Uk(Ai ⊆ Gi).
First, since ℤ(−) is left
adjoint to the forgetful functor Ab → Set, it preserves all colimits; in
particular,
ℤ(A) ≅ colimi ∈ Iℤ(Ai)
via the maps Φi. Second,
filtered colimits in Ab are exact,
hence preserve cokernels. Using , we may write
⟨Xk(A ⊆ G)⟩ ≅ colimi ∈ I⟨Xk(Ai ⊆ Gi)⟩
as a filtered colimit of subgroups, compatibly embedded into the colimit
ℤ(A). Exactness
then gives
Uk(A ⊆ G) = coker (⟨Xk(A ⊆ G)⟩ ↪ ℤ(A)) ≅ colimi ∈ Icoker (⟨Xk(Ai ⊆ Gi)⟩ ↪ ℤ(Ai)) = colimi ∈ IUk(Ai ⊆ Gi).
Moreover, the distinguished subset ηA(A) ⊆ Uk(A ⊆ G)
is the union of the images of ηAi(Ai)
under the colimit maps, because each [ea] depends on
a single element a, hence
appears at some stage.
Applying the previous discussion to T(A, G) = (ηA(A), Uk(A ⊆ G)),
we obtain a canonical identification
T(A, G) ≅ colimi ∈ IT(Ai, Gi)
in FRk∞,
natural in the diagram D.
Hence T preserves filtered
colimits.
Finally, since every object (A, G) is a filtered colimit of its finite subobjects (A0, ⟨A0⟩) with A0 ⊆ A finite, the above implies that T is determined by its restriction to FRk: computing T(A, G) reduces to computing Uk(A0 ⊆ ⟨A0⟩) for finite A0, and then passing to the filtered colimit. This is exactly the finitary behavior required in the sequel, where we pass from universal objects (fixed points) to general T-algebras by splitting the canonical projection pA, G on progressively larger finite pieces.
We now make explicit the Eilenberg–Moore category (FRk∞)T
for the monad T = ιU. Since
T(A, G) = (ηA(A), Uk(A ⊆ G))
is obtained by freely adjoining exactly the k-term relations already valid in
G, a T-algebra structure on (A, G) should be understood
as a choice of coherent ``evaluation’’ map from this universal group
back to the ambient group. The point is that the monad axioms force this
evaluation to admit a group-theoretic splitting, and it is precisely
this splitting which is recorded in the presentation category Presk.
We define Presk as
follows. An object of Presk is a triple
(A ⊆ G, p, s)
where G is an abelian group
generated by A, the map
p = pA, G: Uk(A ⊆ G) ↠ G
is the canonical surjective homomorphism determined by p([ea]) = a
for all a ∈ A,
and
s: G → Uk(A ⊆ G)
is a group homomorphism section of p, i.e. p ∘ s = idG.
We emphasize that p is not
arbitrary: it is the unique homomorphism extending the identity-on-A map A → G, a ↦ a, and the condition
that A generate G simply serves to exclude
irrelevant ambient direct summands which play no role in k-term relations on A.
A morphism
(A ⊆ G, p, s) → (B ⊆ H, q, t)
in Presk is a
Freiman k-homomorphism f: A → B such that
its unique group extension f̄: G → H (which
exists because A generates
G and f respects the defining relations
encoded by p) makes the
evident comparison with the splittings commute, namely
where Uk(f): Uk(A ⊆ G) → Uk(B ⊆ H)
is the homomorphism induced by f as in Lemma~1. Either equality in
implies the other upon composing with p and using p ∘ s = id, q ∘ t = id.
Let ((A, G), α) be a
T-algebra, i.e. an object
(A, G) equipped with
a morphism
α: T(A, G) = (ηA(A), Uk(A ⊆ G)) → (A, G)
in FRk∞
satisfying the usual unit and associativity axioms. Because the source
object T(A, G) is
universal (it lies in UFRk∞ by
construction), the Freiman map α extends uniquely to a group
homomorphism on ambient groups; we denote this extension by
ᾱ: Uk(A ⊆ G) → G.
The unit axiom α ∘ η(A, G) = id(A, G)
forces α([ea]) = a
on the distinguished subset, hence ᾱ coincides with the canonical
projection pA, G.
Thus every T-algebra
canonically determines the surjection p: Uk(A ⊆ G) ↠ ⟨A⟩ ≤ G,
and after replacing G by ⟨A⟩ (which does not change Xk(A ⊆ G)
and hence does not change T(A, G)), we may
and do assume A generates
G.
The genuinely additional content of the T-algebra axioms is the existence of
a section s. Concretely,
consider T2(A, G) = T(T(A, G)).
Since T(A, G) is already
universal, Lemma~3 identifies the multiplication
μ(A, G): T2(A, G) → T(A, G)
with the comparison isomorphism induced by the canonical isomorphism of
ambient groups
Uk(ηA(A) ⊆ Uk(A ⊆ G)) ≅ Uk(A ⊆ G).
Applying T to α yields a morphism Tα: T2(A, G) → T(A, G),
and the associativity axiom
α ∘ Tα = α ∘ μ(A, G)
translates, after passing to ambient groups via universality at each
free stage, into the statement that the canonical surjection p admits a homomorphic right
inverse. We therefore extract a homomorphism
s = sα: G → Uk(A ⊆ G)
with p ∘ s = idG.
Informally, s provides a
coherent choice of representing each element of G by a formal ℤ-linear combination of basis elements [ea], compatible
with the k-term relations
coming from G itself. This
produces a functor
ℰ: (FRk∞)T → Presk, ((A, G), α) ↦ (A ⊆ G, pA, G, sα),
and on morphisms ℰ sends a morphism of
T-algebras f: (A, G) → (B, H)
to the same underlying Freiman map f: A → B; the
algebra-morphism condition is precisely the commutativity .
Conversely, given an object (A ⊆ G, p, s)
of Presk, we define
a T-algebra structure on (A, G) as follows. The
structure map
α = αs: T(A, G) = (ηA(A), Uk(A ⊆ G)) → (A, G)
is the Freiman k-map ηA(A) → A
sending [ea] ↦ a.
This is well-defined in FRk∞ because
any k-term relation among the
[ea] in
Uk(A ⊆ G)
is mapped by p to the
corresponding relation among the a in G. The unit axiom is immediate from
the definition.
The associativity axiom is encoded by the chosen splitting s. Indeed, under the identification of T2(A, G) with the free universal object on ηA(A) ⊆ Uk(A ⊆ G), the two composites α ∘ Tα and α ∘ μ coincide exactly because the section s makes evaluation along p stable under the canonical comparison isomorphisms defining μ. Equivalently, the data of s ensures that evaluating a formal combination in two stages (first in the free universal group and then in G) agrees with evaluating it directly in G. Thus (A, G) becomes a T-algebra, and a morphism in Presk automatically induces a morphism of T-algebras because is exactly the compatibility required with the induced maps on free universal groups.
We obtain a functor
𝒫: Presk → (FRk∞)T, (A ⊆ G, p, s) ↦ ((A, G), αs).
The constructions ℰ and 𝒫 are quasi-inverse. On the Presk side, starting from
(A ⊆ G, p, s),
forming αs, and then
extracting the resulting section recovers s because the extraction process is
characterized by the requirement p ∘ s = idG
together with the coherence built into the algebra axioms. On the (FRk∞)T
side, starting from ((A, G), α),
extracting sα, and
rebuilding αsα
recovers α since α is determined on the distinguished
subset and the remaining coherence is exactly what sα encodes.
Hence we have an equivalence of categories
(FRk∞)T ≃ Presk.
Under this equivalence, the full subcategory UFRk∞ corresponds to the full subcategory of Presk on those objects for which p is an isomorphism (equivalently, for which the T-algebra structure map is an isomorphism, i.e. the fixed points of T). Indeed, by Lemma~3, (A, G) is universal if and only if pA, G: Uk(A ⊆ G) → G is an isomorphism, in which case the only possible section is s = p−1, and the presentation becomes tautological.
We record a few computations which serve two purposes. First, they
confirm that the reflector
U(A, G) = (ηA(A), Uk(A ⊆ G))
behaves as expected in standard additive-combinatorial situations.
Second, they illustrate the additional rigidity enforced by passing from
FRk∞ to
its Eilenberg–Moore category: a T-algebra structure is not merely a
choice of ambient group, but a choice of splitting s of the canonical surjection p, and this splitting can fail to
exist unless enough k-term
relations are visible inside A.
Suppose A ⊆ G is a
subgroup (or more generally A = G, or A contains 0 and generates a subgroup H = ⟨A⟩ which we take as
the ambient group). Then every additive relation in G among elements of A is already a relation in the group
generated by A, and in this
case the presentation implicit in Uk(A ⊆ G)
is tautological: the canonical map
pA, G: Uk(A ⊆ G) → G, pA, G([ea]) = a,
is an isomorphism. Indeed, any group homomorphism out of ℤ(A) is determined by the
images of the basis elements ea, and the
subgroup ⟨Xk(A ⊆ G)⟩
is precisely the set of k-term
relations that must vanish to make the assignment ea ↦ a
respect the equalities already valid in G. Consequently (A, G) is universal, T(A, G) ≅ (A, G),
and the only possible T-algebra structure is the identity
(equivalently, in Presk we have s = p−1). This
is the conceptual reason that, on genuine subgroups, Freiman k-homomorphisms reduce to ordinary
homomorphisms.
Let A = {0, 1, …, n} ⊆ ℤ and
take k = 2. In G = ℤ we have the elementary 2-term
relations
(i + 1) + (j − 1) = i + j (1 ≤ j ≤ n, 0 ≤ i ≤ n − 1),
and hence, in ℤ(A),
the corresponding relation vectors
ei + 1 + ej − 1 − ei − ej ∈ X2(A ⊆ ℤ).
Modulo ⟨X2⟩, these
imply that the successive differences [em + 1] − [em]
are all equal. Writing
u := [e1] − [e0] ∈ U2(A ⊆ ℤ),
we obtain inductively
[em] = [e0] + mu (0 ≤ m ≤ n).
Thus the subgroup of U2(A ⊆ ℤ)
generated by ηA(A) = {[em]}
is generated by [e0] and u, and the only relations among the
[em] are
those forced by the linear parametrization [em] = [e0] + mu.
Pushing forward along p sends
[e0] ↦ 0 and u ↦ 1, so p identifies the u-direction with ℤ. In particular, after discarding the
irrelevant free summand generated by [e0] (which maps to 0 and does not affect the additive structure
on the translate {1, …, n}),
we recover the expected universal ambient group:
U2(A ⊆ ℤ) ≅ ℤ (up to
the harmless choice of basepoint).
This matches the informal principle that a one-dimensional arithmetic
progression has no hidden 2-term
additive structure beyond that of ℤ
itself.
A related sanity check is that U ignores ambient direct summands
that do not participate in k-term relations on A. For instance, embed the same
progression A = {0, 1, …, n} into ℤ2 via m ↦ (m, 0). The set X2(A ⊆ ℤ2)
is identical to X2(A ⊆ ℤ), since
equality of sums in ℤ2
reduces to equality in the first coordinate on this subset. Hence the
quotients U2(A ⊆ ℤ2)
and U2(A ⊆ ℤ)
coincide. This exemplifies the general phenomenon: Uk(A ⊆ G)
depends on G only through the
k-term relations that hold on
A, and not on ambient
structure invisible to A.
The role of torsion is subtle because Uk only imposes
relations of length k that are
already witnessed inside A.
Consider k = 2 and G = ℤ/4ℤ.
(1) Let A = {0, 1} ⊆ G. There are
no nontrivial relations of the form a1 + a2 = a1′ + a2′
involving only 0 and 1 in ℤ/4ℤ
(since 1 + 1 = 2 but 2 ∉ A). Thus
U2(A ⊆ ℤ/4ℤ) ≅ ℤ(A) ≅ ℤe0 ⊕ ℤe1,
and the canonical projection
p: U2(A ⊆ G) ↠ G, p(e0) = 0, p(e1) = 1,
is surjective. However, p
admits homomorphic section s: G → U2(A ⊆ G):
any such section would have to send 1 ∈ ℤ/4ℤ to an element of order 4 in ℤ2, but ℤ2 is torsion-free. Hence (A ⊆ G) does not underlie
any object of Pres2, and
equivalently (A, G)
admits no T-algebra structure.
This illustrates that the Eilenberg–Moore category is genuinely smaller
than FRk∞: the
existence of a section encodes a nontrivial coherence requirement.
(2) Enlarge to A′ = {0, 1, 2} ⊆ ℤ/4ℤ.
Now the equality 1 + 1 = 2 + 0 holds
inside A′, so X2(A′ ⊆ G)
contains 2e1 − e2 − e0.
Moreover, 2 + 2 = 0 + 0 gives 2e2 − 2e0.
In the quotient U2(A′ ⊆ G)
we can write
[e2] = 2[e1] − [e0], 4([e1] − [e0]) = 0,
so U2(A′ ⊆ G)
contains an element of order 4, namely
[e1] − [e0].
Consequently p now split: we
may define s(1) = [e1] − [e0],
and extend by homomorphism. In other words, once A is sufficiently 2-relation-complete to witness the torsion of
G, the universal group U2(A ⊆ G)
acquires the torsion needed for a section to exist.
Let k = 2, A = {0, 1, 4} ⊆ ℤ, B = {0, 2, 4} ⊆ ℤ, and define φ: A → B by φ(0) = 0, φ(1) = 2, φ(4) = 4. Since A supports no nontrivial 2-term relations internal to A, every map A → B is automatically a
Freiman 2-homomorphism; in particular
φ is a morphism in FR2, and it is bijective as a set
map. Nevertheless φ is not an
isomorphism in FR2: the
inverse map ψ: B → A fails to
be Freiman because B satisfies
the nontrivial relation 0 + 4 = 2 + 2,
whereas its image under ψ
would require 0 + 4 = 1 + 1, which is
false in ℤ.
The reflector U makes the
asymmetry visible at the level of ambient groups. For A, as above, there are no nontrivial
relations, hence
U2(A ⊆ ℤ) ≅ ℤ(A) ≅ ℤe0 ⊕ ℤe1 ⊕ ℤe4.
For B, the relation 0 + 4 = 2 + 2 is internal, so
U2(B ⊆ ℤ) ≅ ℤ(B)/⟨e0 + e4 − 2e2⟩,
which already forces a linear dependence between the generators. The
induced homomorphism
U2(φ): U2(A ⊆ ℤ) → U2(B ⊆ ℤ)
is surjective, but it cannot be injective because the target satisfies
an additional relation not present in the source. Thus the failure of
φ to be invertible in FR2 is reflected by the failure of
U2(φ) to
be an isomorphism in Ab.
From the Eilenberg–Moore viewpoint, the same example explains why T-algebra morphisms are stricter than mere Freiman maps. A choice of T-algebra structure on (A, ℤ) is a choice of section sA: ℤ → U2(A ⊆ ℤ), and similarly for B. Compatibility condition forces U2(φ) ∘ sA to coincide with sB ∘ φ̄ on ℤ. The additional relation in U2(B ⊆ ℤ) constrains sB, and in general there is no reason a section sA can be chosen so that its image lands in the corresponding constrained cosets. In this precise sense, the T-algebra structure ``detects’’ the missing relation coherence that a bare Freiman bijection ignores: bijectivity on A does not suffice, because coherence must also hold on the ambient group via the chosen splittings.
These examples support the intended interpretation of Uk(A ⊆ G)
as the universal ambient group generated by A subject only to the k-term relations already valid on
A inside G. In the subgroup and progression
cases, U recovers the expected
minimal ambient group (up to inessential basepoint choices). In torsion
situations, the existence of a T-algebra structure is genuinely
conditional: a section exists only when Uk has acquired
the torsion dictated by the visible k-term relations. Finally, bijective
Freiman maps may fail to be isomorphisms, and the passage to U and to T-algebras identifies exactly where
the defect lies: not at the level of sets, but at the level of
ambient-group coherence encoded by the splittings.
The preceding results place Freiman k-homomorphisms into a familiar
categorical framework: ι: UFRk∞ ↪ FRk∞
is reflective with reflector U, and the resulting monad T = ιU is finitary
and idempotent up to canonical isomorphism. Conceptually, this says that
passing to the universal ambient group'' is a localization process: the unit \[ \eta_X\colon X\longrightarrow TX \] is initial among maps from \(X\) into universal objects, and the fixed points are precisely the universal objects. Two immediate consequences are worth emphasizing. First, any invariant or construction on \(\mathsf{UFR}_k^\infty\) can be transported back to \(\mathsf{FR}_k^\infty\) functorially by precomposing with \(U\) (equivalently, by applying \(T\)). Second, because \(T\) is finitary, computations reduce to finitely generated (indeed finitely supported) data: every relation in \(X_k\) involves only finitely many elements, so \(U_k(A\subseteq G)\) can be approximated by running through finite subconfigurations of \(A\). This is the categorical form of the common heuristic that Freiman structure islocal
in A’’.
Given a monad T on a category
𝒞, the standard bar construction
produces an augmented simplicial object
$$
\cdots \Longrightarrow T^3X \Longrightarrow T^2X \Longrightarrow TX
\xrightarrow{\ \epsilon_X\ } X,
$$
with face maps induced by the multiplication μ: T2 ⇒ T
and degeneracies induced by the unit η: id ⇒ T. In our setting
𝒞 = FRk∞.
Since T is idempotent up to
canonical isomorphism, the bar construction simplifies dramatically: the
multiplication μX: T2X → TX
is an isomorphism, so all higher iterates Tn + 1X
are canonically identified with TX. The resulting
simplicial object is therefore best viewed as the ech nerve of the unit
ηX: X → TX,
i.e. the simplicial object whose n-simplices encode (n + 1)-fold compatibilities of a
putative lift of X to the
reflective subcategory.
This perspective suggests a practical ``resolution by universal objects’’: even when X is not universal and does not admit a T-algebra structure, the augmented simplicial diagram built from TX is universal by construction and functorial in X. In particular, if one is interested in applying additive invariants (in the homological sense) to Freiman data, the bar construction provides a canonical replacement of X by a simplicial object in which every level is controlled by universal ambient groups and hence by honest group homomorphisms.
Although FRk∞ is not
abelian, many natural observables factor through abelian groups. Two
basic examples are:
(A, G) ↦ Uk(A ⊆ G) ∈ Ab, (A, G) ↦ ⟨A⟩ ≤ G ∈ Ab.
Composing such functors with the bar resolution yields simplicial
abelian groups, hence chain complexes by the Dold–Kan correspondence.
Their homology groups provide derived invariants of the original object
X ∈ FRk∞,
functorial in X. Concretely,
the augmentation TX → X is the
universal comparison map from a universal ambient group to the given
ambient group data, so the resulting homology measures the failure of
X to lie in the reflective
subcategory, in a way that is stable under filtered colimits (by
finitarity of T).
There is also an intrinsic candidate for a ``defect module’’. For
X = (A, G)
write H = ⟨A⟩ ≤ G and
consider the canonical surjection (Lemma~2)
pA, G: Uk(A ⊆ G) ↠ H.
Its kernel KA, G := ker (pA, G)
depends only on the k-term
relations internal to A and
measures precisely the additional relations that are forced in the
universal group but vanish in H. The assignment X ↦ KA, G
is functorial for morphisms compatible with the canonical maps p, hence becomes most naturally a
functor on the Eilenberg–Moore side, i.e. on Presk. The bar resolution
offers a way to extend this functorially back to FRk∞ without
choosing a splitting.
Under the equivalence (FRk∞)T ≃ Presk,
a T-algebra structure on (A, G) is exactly a choice
of section s: H → Uk(A ⊆ G)
of pA, G.
Since all groups are abelian, the existence of such a section is
equivalent to the splitting of the short exact sequence
$$
0\longrightarrow K_{A,G}\longrightarrow U_k(A\subseteq
G)\xrightarrow{\,p_{A,G}\,} H\longrightarrow 0.
$$
Thus the obstruction to endowing (A, G) with a T-algebra structure is the extension
class of this sequence in Extℤ1(H, KA, G).
This makes precise the informal statement that a T-algebra structure is ``extra
coherence’’: one must not only know the relations in A, but also choose a compatible
retraction on the ambient group generated by A. In particular, even when a
splitting exists, there is typically no canonical choice; the set of
splittings (when nonempty) is a torsor for Hom(H, KA, G).
It is natural to regard these Ext- and
Hom-groups as secondary invariants of
Freiman data, computable from a presentation of Uk(A ⊆ G)
and H.
One recurring difficulty in applications is that a good ambient model
for A should be canonical
(hence functorial) rather than chosen ad hoc. The reflector U provides such a canonical model at
the level of universal ambient groups, but the passage from FRk∞ to Presk requires a section
s, and sections rarely exist
functorially. The previous paragraph explains why: functoriality would
amount to functorial choices of splittings of a family of extensions,
which is obstructed by nontrivial automorphisms and extension
classes.
Nonetheless, there are at least two meaningful weakenings. First, one can seek T-algebra structures after imposing additional structure (e.g. ordering on A, choice of basepoint, or a choice of decomposition of finitely generated abelian groups into invariant factors). Second, one can replace strict functorial splittings by data: the bar resolution of ηX is precisely the receptacle for such coherence, and suggests that the correct recipient of ``functorial splittings’’ is a higher-categorical enhancement of FRk∞ in which choices are tracked up to controlled homotopies.
Many questions in additive combinatorics are quantitative: one controls
doubling constants, additive energy, or sizes of sumsets, and wishes to
transport these bounds along Freiman maps. The present framework is
qualitative, but it is compatible with quantitative refinements. For
finite A, the group Uk(A ⊆ G)
is finitely generated (indeed presented by generators {ea}a ∈ A
and relations in Xk), so one can
attach computable numerical invariants such as rank Uk(A ⊆ G),
the torsion subgroup Tor(Uk(A ⊆ G)),
and the size or rank of KA, G.
These invariants measure, in different ways, the complexity of the k-term relation structure seen by
A. Because T is finitary, these computations
admit efficient reductions to finite relation extraction: one only needs
to enumerate those k-term
relations among elements of A
that actually hold in G.
A natural next step is to study how such invariants behave under passage to large structured subsets (e.g. modeling lemmas) and under operations such as products and iterated sumsets. Another direction is algorithmic: for finite A inside a finitely generated abelian group G, Uk(A ⊆ G) is accessible via integer linear algebra (Smith normal form applied to the matrix of relations), and the extension class in Ext1 becomes explicit. This suggests a computational toolkit for deciding whether a given Freiman configuration admits a T-algebra structure and for enumerating splittings when it does.
The reflector U isolates the
universal ambient group determined by k-term relations on A, while the Eilenberg–Moore
description identifies the additional data needed to rigidify Freiman
structure at the ambient-group level. The bar resolution and the
associated derived invariants provide a systematic way to measure, and
potentially quantify, the gap between an arbitrary Freiman object and
its universal approximation. We view these tools as a bridge between the
flexible combinatorial world of Freiman homomorphisms and the rigid
algebraic world of presentations, extensions, and homological
obstructions.