Categorical Aspects in Projective Geometry

Applied Categorical Structures 6: 87–103, 1998. c 1998 Kluwer Academic Publishers. Printed in the Netherlands. 87 Categorical Aspects in Projective Geometry CLAUDE-ALAIN FAURE and ALFRED FRÖLICHER Section de Mathématiques, Université de Genève, Case postale 240, CH-1211 Genève 24, Switzerland (Received: 8 December 1995; accepted: 5 June 1996) Abstract. After introducing morphisms between projective geometries, some categorical questions are examined. It is shown that there are three kinds of embeddings and two kinds of quotients. Furthermore the morphisms decompose in a canonical way into four factors. Mathematics Subject Classifications (1991). 18B99, 51A10. Key words: morphisms and homomorphisms of projective geometries, subgeometries and subspaces, quotient geometries, initial and final morphisms, sections and retractions, decomposition of a morphism. Introduction Categorical questions in projective geometry are new, since morphisms between projective geometries have been introduced only recently; cf. [3, 4] and [5]. Many questions about the category P roj of projective geometries ought to be studied; in this paper we examine three of these topics, namely embeddings, quotients and factorizations of morphisms. Since a morphism between projective geometries is a map between the underlying point sets which in general is not defined for all points of the source, the natural forgetful functor has its values in the so-called partial-map category P ar. The objects of P ar are the sets, and the morphisms from X to Y are the partial maps, i.e. the maps f : X \ N → Y where N ⊆ X . We call N the kernel of f and we write f : X− → Y and N = Ker f . P ar is known to be equivalent to the category Set⋆ of pointed sets; cf. [1]. By an embedding we understand a monomorphism which is initial with respect to the forgetful functor P roj → P ar. The embeddings turn out to be, up to isomorphy, the inclusions of the usual (projective) subgeometries. But subgeometries can have undesired features. For instance the dimension of a subgeometry can be bigger than the one of the ambiant geometry. It will be shown as an example that the usual real projective plane contains a subgeometry of dimension d for any d 6 ℵ0 . Several equivalent geometric conditions which prevent such a pathological behavior will be given. The respective embeddings will be called proper embeddings. Still smaller is the class of the embeddings which VTEX(ZJ) PIPS No.: 115592 MATHKAP APCS232.tex; 5/02/1998; 13:54; v.7; p.1 88 C.-A. FAURE AND A. FRÖLICHER are isomorphic to the inclusion of a subspace. These turn out to be exactly the sections of the category P roj . By a quotient we understand an epimorphism which is final with respect to the functor P roj → P ar. Among these one finds as special case the quotients by a given subspace which were considered in [4]; they are retractions of P roj . Every projective geometry is shown to be a quotient of the coproduct of its lines. An example will show that three points of a quotient geometry can be collinear even if no collinear preimages exist. In many categories the morphisms can be factorized in two special morphisms. For the category of projective geometries one even gets a factorization into four morphisms. 1. Morphisms of Projective Geometries In [4] we defined a projective geometry as a set G (called the point set) together with a ternary relation ℓ on G (called collinearity) satisfying the following three axioms: (L1 ) ℓ(a, b, a) ∀a, b ∈ G; (L2 ) ℓ(a, p, q), ℓ(b, p, q) and p 6= q ⇒ ℓ(a, b, p); (L3 ) ℓ(p, a, b) and ℓ(p, c, d) ⇒ ∃q with ℓ(q, a, c) and ℓ(q, b, d). A subspace of G is a subset E ⊆ G with the property a 6= b ∈ E and ℓ(a, b, c) ⇒ c ∈ E . A subgeometry of G is a subset G′ ⊆ G such that G′ together with the restriction of ℓ to G′ is itself a projective geometry. NOTATIONS 1.1. Let G be a projective geometry, A ⊆ G and a, b ∈ G; 1◦ C(A) will denote the smallest subspace of G containing A; 2◦ S(A) will denote the smallest subgeometry of G containing A; 3◦ a ⋆ b := C{a, b}. The subspaces of the form a ⋆ b where a 6= b are called lines; obviously a ⋆ a = {a}. There exist many equivalent axiomatic definitions of projective geometries, in particular by means of points, lines and an incidence relation (cf. e.g. [7]). We emphasize that we do not require that G is irreducible, i.e. that every line of G contains at least three points. G is called arguesian if it is irreducible, of dimension > 2 and has Desargues’ property. DEFINITION 1.2. A morphism from a projective geometry G1 to a projective geometry G2 is a map g: G1 \ N → G2 defined on the complement of a subset N ⊆ G1 and satisfying the following axioms: (M1 ) N is a subspace of G; APCS232.tex; 5/02/1998; 13:54; v.7; p.2 CATEGORICAL ASPECTS IN PROJECTIVE GEOMETRY 89 (M2 ) If a, b ∈ / N, c ∈ N and a ∈ b ⋆ c, then ga = gb; (M3 ) If a, b, c ∈ / N and a ∈ b ⋆ c, then ga ∈ gb ⋆ gc. The set N is called the kernel of g and will be denoted by Ker g; the set g(G1 \ N ) is called the image of g and will be denoted by Im g. The morphism g is called constant if | Im g| 6 1 and empty if Im g = ∅ (i.e. if Ker g = G1 ). If g is a morphism from G1 to G2 we write g: G1 − → G2 , or also g: G1 → G2 if one knows that Ker g = ∅. A morphism g: G1 − → G2 is called a homomorphism if it satisfies also the axioms (M4 ) If a, b ∈ / N and ga 6= gb, then ga ⋆ gb ⊆ g(a ⋆ b); (M5 ) If a, b ∈ / N, a 6= b and ga = gb, then (a ⋆ b) ∩ N 6= ∅. One easily verifies that the composite of two morphisms in the sense of the above definition is a morphism. Hence one obtains a category; it will be noted P roj . Similarly, the projective geometries together with the homomorphisms form a subcategory of P roj . The axioms (M1 ), (M2 ), (M3 ) together are equivalent with each of the following axioms: (M) If F is a subspace of G2 , then g−1 F ∪ N is a subspace of G1 ; cf. 3.1.1 in [4]; f (M) g(C1 A \ N ) ⊆ C2 (g(A \ N )) for each A ⊆ G1 . Furthermore one has for any morphism g: if a, b ∈ / Ker g and ga 6= gb, then g(a ⋆ b) ⊆ ga ⋆ gb and hence g(a ⋆ b) = ga ⋆ gb if (M4 ) holds. PROPOSITION 1.3. A partial map g: G1 \ N → G2 between projective geometries Gi is a morphism if and only its restriction to every line δ of G1 is a morphism δ − → G2 . Proof. The restriction of g to δ is the composite g ◦ jδ , where jδ : δ → G1 is the inclusion. The stated condition is trivially necessary. Conversely, suppose that for every line δ ⊆ G1 the map g ◦ jδ : δ \ (N ∩ δ) → G2 is a morphism. Let F ⊆ G2 be a subspace. One has (g−1 F ∪ N ) ∩ δ = (g−1 F ∩ δ) ∪ (N ∩ δ) = jδ−1 (g−1 F ) ∪ (N ∩ δ) = (g ◦ jδ )−1 F ∪ (N ∩ δ). On the right-hand side one has by (M) a subspace of G1 . Hence the intersection of g−1 F ∪ N with every line δ is a subspace of G1 and this implies that g−1 F ∪ N is a subspace of G1 . So g 2 satisfies (M) and therefore is a morphism. PROPOSITION 1.4. The category P roj has arbitrary coproducts. Proof. Let Gi , i ∈ I be a family of projective geometries. We may assume that S the point sets Gi are pairwise disjoint and consider on G := i∈I Gi the relation ℓ defined as follows: ℓ(x, y, z) holds if there exists i ∈ I such that x, y, z ∈ Gi and ℓi (x, y, z) or if x, y, z are not all three different. One easily verifies that APCS232.tex; 5/02/1998; 13:54; v.7; p.3 90 C.-A. FAURE AND A. FRÖLICHER G together with ℓ is a projective geometry, that the inclusions ik : Gk → G are morphisms, and that the coproduct-property holds. Obviously G is not irreducible if at least two of the geometries are non-empty. 2 The two kinds of morphisms between projective geometries correspond to two kinds of morphisms between vector spaces. DEFINITION 1.5. Let V, W be vector spaces over (skew-) fields K, L. We call a map f : V → W semilinear (respectively quasilinear) if it is additive and if there exists a homomorphism (respectively isomorphism) of fields σ : K → L such that one has f (αx) = σ(α) · f (x) for all α ∈ K, x ∈ V . The category formed by the vector spaces as objects and the semilinear maps as morphisms will be denoted by V ec. To every vector space V one associates a projective geometry PV as follows: The points of PV are the 1-dimensional subspaces of V ; and three of them are collinear if they lie in a subspace of dimension 6 2. Every point of PV is determined (“represented”) by any of its non-zero vectors. Every semilinear map f : V →W induces a morphism Pf : PV − → PW whose kernel Ker(Pf ) is formed by the 1-dimensional subspaces of V which lie in ker f . If f is quasilinear, then Pf is a homomorphism. For the so obtained functor P: V ec → P roj the following result, called generalized fundamental theorem of projective geometry, holds. For a proof we refer to [5] and [6]. THEOREM 1.6. Let g: G1 − → G2 be a morphism of projective geometries. 1◦ If G1 , G2 are arguesian there exist isomorphisms ui : Gi → PVi . If in addition not all points of Im g are collinear, then there exists for given u1 , u2 a semilinear map f : V1 → V2 such that g = u−1 2 ◦ Pf ◦u1 . We will then shortly ∼ write g = Pf . 2◦ If g is non-constant, then the equation g = u−1 2 ◦ Pf ◦ u1 determines f up to a factor κ ∈ K2 , and f is quasilinear if and only if g is a homomorphism. PROPOSITION 1.7. For a morphism m: G1 − → G2 of projective geometries the following conditions are equivalent: 1◦ m is a monomorphism of P roj ; 2◦ Ker m = ∅ and the map m: G1 → G2 is injective. Proof. 1 ⇒ 2. Suppose that there would exist a point p ∈ Ker m. One chooses any non-empty projective geometry G and considers as g1 : G → G1 the morphism with constant value p (and empty kernel) and as g2 : G − → G1 the empty morphism. Then m ◦ g1 = m ◦ g2 but g1 6= g2 , in contradiction with the hypothesis. Hence Ker m = ∅. Suppose now that there would exist points a 6= b with ma = mb. One chooses any non-empty projective geometry G and considers as g1 : G → G1 APCS232.tex; 5/02/1998; 13:54; v.7; p.4 CATEGORICAL ASPECTS IN PROJECTIVE GEOMETRY 91 the morphism having the constant value a and as g2 : G → G1 the morphism with constant value b. One obtains the same contradiction as before and hence injectivity of m is proved. 2 ⇒ 1. Let m ◦ g1 = m ◦ g2 . Then, using the hypothesis Ker m = ∅, one gets Ker g1 = Ker(m ◦ g1 ) = Ker(m ◦ g2 ) = Ker g2 =: N . Furthermore, for x ∈ G \ N one obtains from mg1 x = mg2 x by injectivity of m the equality 2 g1 x = g2 x. Hence g1 = g2 . PROPOSITION 1.8. For a morphism e: G1 − → G2 of projective geometries the following conditions are equivalent: 1◦ e is an epimorphism of P roj ; 2◦ e is surjective, i.e. Im e = G2 . Proof. 1 ⇒ 2. Suppose that a point p ∈ G2 \ Im e would exist. One chooses a projective geometry G containing exactly one point q and considers the map g1 : G2 → G (with Ker g1 = ∅) and the map g2 : G2 \ {p} → G. They both are (constant) morphisms. In order to show that g1 ◦ e = g2 ◦ e it is enough to verify that the kernels are the same: Ker(gi ◦ e) = Ker e ∪ e−1 (Ker gi ) = Ker e for i = 1, 2. Since g1 6= g2 one has a contradiction with the hypothesis and hence e is surjective. 2 ⇒ 1. Let g1 , g2 be morphisms with g1 ◦ e = g2 ◦ e. Suppose x ∈ Ker g1 . Choose y ∈ G1 \ Ker e with x = ey . Then y ∈ Ker(g1 ◦e) = Ker(g2 ◦e) = Ker e∪ e−1 (Ker g2 ). Therefore y ∈ e−1 (Ker g2 ) and hence x ∈ Ker g2 . By symmetry one obtains Ker g1 = Ker g2 . Finally, if x ∈ / Ker g1 = Ker g2 one chooses z such that x = ez . Then g1 x = (g1 ◦ e)z = (g2 ◦ e)z = g2 x and g1 = g2 follows. 2 2. Embeddings PROPOSITION 2.1. For a morphism g: G1 − → G2 of projective geometries the following conditions are equivalent: 1◦ g is an embedding, i.e. an initial monomorphism of the category P roj ; 2◦ g is a monomorphism and ℓ2 (gx, gy, gz) ⇔ ℓ1 (x, y, z); 3◦ Ker g = ∅ and if a1 , a2 resp. a1 , a2 , a3 are independent points of G1 , then ga1 , ga2 resp. ga1 , ga2 , ga3 are independent points of G2 ; 4◦ g is an isomorphism of G1 onto a subgeometry G′ of G2 , i.e. one has g = j ◦u where j : G′ → G2 is the inclusion of a subgeometry and u: G1 → G′ an isomorphism; 5◦ g is an initial morphism of the category P roj . Proof. 1 ⇒ 2. Suppose that one has points x, y, z ∈ G1 with ℓ2 (gx, gy, gz). One chooses as G a projective geometry consisting of three different collinear points a, b, c and considers the map h: G → G1 sending them respectively APCS232.tex; 5/02/1998; 13:54; v.7; p.5 92 C.-A. FAURE AND A. FRÖLICHER to x, y, z . Then g ◦ h: G → G2 is a morphism since the images of a, b, c are collinear. This implies by hypothesis that h: G → G1 is a morphism and therefore ℓ(a, b, c) ⇒ ℓ1 (x, y, z). The converse is trivial. 2 ⇒ 3. This follows from the fact that in a projective geometry two points are independent iff they are different and three points are independent iff they are non-collinear. 3 ⇒ 4. The map g: G1 → G2 can be decomposed as g = j ◦ u where u: G1 → g(G1 ) is the map ux := gx and j : g(G1 ) → G2 the inclusion map. Since g is injective, u is bijective. And since one gets ℓ1 (x, y, z) ⇔ ℓ2 (ux, uy, uz) it follows that G′ := g(G1 ) is a subgeometry of G2 and u an isomorphism. 4 ⇒ 5. Any inclusion j of a subgeometry is trivially an initial morphism. Therefore also j ◦ u since u is an isomorphism. 5 ⇒ 1. (a) Suppose that there would exist a point p ∈ Ker g. One chooses a projective geometry G with a subset A which is not a subspace and considers the map h: G \ A → G1 having the constant value p. Then g ◦ h is the empty morphism, but h is not a morphism since its kernel is not a subspace. This contradicts the hypothesis and hence Ker g = ∅. (b) Suppose now that there would exist points a 6= b with ga = gb. One chooses G, A as before and considers the map h: G → G1 defined by hx := a if x ∈ A and hx := b if x ∈ / A. Then the map g ◦ h: G → G2 is constant and thus a morphism, but h is not a morphism since h−1 a = A is not a subspace of G. This contradicts the hypothesis and hence g is injective. By 1.7 one concludes that g is a monomorphism. 2 PROPOSITION 2.2. For a monomorphism g: G1 → G2 of P roj the following conditions are equivalent: 1◦ 2◦ 3◦ 4◦ 6◦ There exists a basis B of G1 such that gB ⊆ G2 is independent; A ⊆ G1 independent ⇒ gA ⊆ G2 independent; For the rank r(E) of any subspace E ⊆ G1 one has: r(E) = r(C2 (gE)); For every subspace E ⊆ G1 one has: E = g−1 C2 (gE); For every subspace E ⊆ G1 there exists a subspace E2 ⊆ G2 with E = g−1 E2 ; 5◦ For every subset A ⊆ G1 one has: C1 (A) = g−1 C2 (gA). Proof. 1 ⇒ 2. We may assume that A is finite. Then there exists a finite subset B ′ of B with A ⊆ C1 B ′ . By the Steinitz exchange theorem there exists a basis of C1 B ′ of the form A ∪˙ B ′′ with B ′′ ⊆ B ′ . Then gB ′ ⊆ g(C1 (A ∪˙ B ′′ )) ⊆ C2 (gA ∪˙ gB ′′ ). Since the set gB ′ is independent and since |gA ∪˙ gB ′′ | = |A ∪˙ B ′′ | = |B ′ | = |gB ′ | < ∞, the preceding inclusion shows that gA ∪˙ gB ′′ and hence also gA is independent. e ) one gets C2 (gA) ⊆ C2 (gC1 A) ⊆ 2 ⇒ 3. Let A be a basis of E . By (M C2 C2 (gA) and hence C2 (gE) = C2 (gA). Being independent, gA is therefore a basis of C2 (gE) and so one gets r(C2 (gE)) = |gA| = |A| = r(E). APCS232.tex; 5/02/1998; 13:54; v.7; p.6 CATEGORICAL ASPECTS IN PROJECTIVE GEOMETRY 93 3 ⇒ 4. Let E be a subspace of G1 . Then F := g−1 C2 (gE) is also a subspace of G1 . Since E ⊆ g−1 gE ⊆ g−1 C2 (gE) = F and gF ⊆ C2 (gE) one has C2 (gE) = C2 (gF ) and hence r(C2 (gE)) = r(C2 (gF )). By hypothesis 3 this is equivalent with r(E) = r(F ). From this and from E ⊆ F the assertion E = F follows immediately in the case where r(E) < ∞. The case r(E) = ∞ then follows also, since the operator C2 is finitary. 4 ⇒ 5. E2 := C2 (gE) is a subspace and by the hypothesis one gets E = −1 g E2 . 5 ⇒ 6. From A ⊆ g−1 gA ⊆ g−1 C2 (gA) one deduces C1 A ⊆ g−1 C2 (gA). Furthermore one has by hypothesis C1 A = g−1 E2 for some subspace E2 of G2 . From A ⊆ g−1 E2 follows gA ⊆ E2 and hence C2 (gA) ⊆ E2 and finally g−1 C2 (gA) ⊆ g−1 E2 = C1 A. 6 ⇒ 1. Let B be a basis of G1 . Proceeding indirectly we suppose that gB is dependent, i.e. that there exists b ∈ B such that gb ∈ C2 (gB \ gb). Then g−1 C2 g(B \ b) = C1 (B \ b), and this contradicts the independence of B . 2 DEFINITION 2.3. 1◦ A proper embedding is a monomorphism which satisfies the conditions of the preceding proposition (remark that every proper embedding is an embedding; and that an embedding is proper iff it is an embedding for the category of closure spaces). 2◦ A subspace-embedding is a proper embedding for which the image is a subspace. PROPOSITION 2.4. For a morphism g: G1 − → G2 of P roj the following conditions are equivalent: 1◦ g is a homomorphism with empty kernel; 2◦ g is a subspace-embedding; 3◦ g is a section of the category P roj . Proof. 1 ⇒ 2. From Ker g = ∅ and (M5 ) one deduces that g is injective and hence a monomorphism. Hence one gets E = g−1 gE for any subspace E of G1 . Since (M4 ) implies that gE is a subspace of G2 , condition 5 of 2.2 holds and hence g is a subspace-embedding. 2 ⇒ 3. By 4 of 2.1 one has g = j ◦ u where j : g(G1 ) → G2 is the inclusion map and u is an isomorphism. Moreover, g(G1 ) is a subspace of G2 and it is known, cf. 6.1.3 in [4], that inclusions of subspaces are sections of P roj . With j also j ◦ u is a section. 3 ⇒ 1. The equation r ◦ g = IdG1 implies Ker g = ∅ and hence also (M5 ) holds. In order to verify (M4 ), let ga 6= gb and z ∈ ga⋆gb. Then rz ∈ r(ga⋆gb) ⊆ rga ⋆ rgb = a ⋆ b and hence z ′ := grz ∈ g(a ⋆ b). From ga 6= gb and rz = rz ′ one deduces by (M3 ) that z = z ′ and hence z ∈ g(a ⋆ b). 2 APCS232.tex; 5/02/1998; 13:54; v.7; p.7 94 C.-A. FAURE AND A. FRÖLICHER COROLLARY 2.5. Let f : V → W be a semilinear map and g := Pf the associated morphism of projective geometries. 1◦ g is an embedding of P roj if and only if for 1 6 n 6 3 one has x1 , . . . , xn linearly independent in V ⇒ f x1 , . . . , f xn linearly independent in W; 2◦ g is a proper embedding of P roj if and only if for all n ∈ N one has x1 , . . . , xn linearly independent in V ⇒ f x1 , . . . , f xn linearly independent in W . Proof. One uses that points of PV are independent if and only if representing vectors are linearly independent. 2 If f is injective and quasilinear the above conditions hold. We give a more precise result: PROPOSITION 2.6. Let, as before, f : V → W be a semilinear map and g := Pf . If dim(V ) > 2, then the following conditions are equivalent: 1◦ g is a subspace-embedding (i.e. a section of P roj , cf. 2.4); 2◦ f is injective and quasilinear; 3◦ f is section of V ec. Proof. 1 ⇒ 2. Since PV contains at least two points, the morphism g is non-constant. So one can use 2 of 1.6 in order to obtain the quasilinearity of f . 2 ⇒ 3. One verifies that f (V ) is a vector-subspace of W and that f : V → f (V ) is an isomorphism of V ec. The claim follows since the inclusion of a vector-subspace obviously is a section of V ec. 3 ⇒ 1. This follows trivially from 2.4 2 EXAMPLE 2.7. An embedding of P(Q(N) ) into P(R3 ). One chooses α, β, γ ∈ R such that for every N ∈ N one has aijk ∈ Q and N X aijk αi β j γ k = 0 ⇒ aijk = 0 for all i, j, k i,j,k=1 and one considers the map f : Q(N) → R3 defined as follows: f (x1 , x2 , x3 , . . .) := X i i xi α , X j j xj β , X xk γ k k  . This map f is obviously semilinear. We show that it satisfies condition 1◦ of 2.5. For n = 1 the condition means injectivity of f which certainly holds. Suppose now that x, y are vectors of Q(N) such that f x, f y are linearly dependent. Then P P X xi xj xi αi j xj β j i P αi β j = 0. j = i P i yi α j yj β i,j yi yj APCS232.tex; 5/02/1998; 13:54; v.7; p.8 CATEGORICAL ASPECTS IN PROJECTIVE GEOMETRY 95 This implies that xi xj = 0 for all yi yj i, j and this is exactly the condition for x, y to be linearly dependent in Q(N) . Let finally x, y, z be three vectors in Q(N) such that f x, f y, f z are linearly dependent. Then one has P P P x γk x βj x αi X xi xj xk Pi i i Pj j j Pk k k yi yj yk αi β j γ k = 0. y γ y β y α = Pi i i Pj j j Pk k k i zi α j zj β k zk γ i,j,k zi zj zk Since this implies that all the determinants in the sum are zero, the vectors x, y, z ∈ Q(N) are linearly dependent. By 1 of 2.5 one now deduces that Pf : P(Q(N) ) → P(R3 ) is an embedding of P roj . The image is a subgeometry of P(R3 ) which is isomorphic to P(Q(N) ). Obviously the embedding cannot be proper. As an immediate consequence one gets: For every n ∈ N there exists a subgeometry of P(R3 ) which is isomorphic to P(Qn ). For n > 3 such an embedding cannot be proper. EXAMPLE 2.8. P(Qn ) is isomorphic to a proper subgeometry of P(Rn ). One chooses as f : Qn → Rn the natural inclusion and uses 2 of 2.5. Obviously the embedding is not a subspace-embedding. More general examples of proper embeddings are obtained by choosing any extension of fields K ⊆ L and as f the natural inclusion K (N) → L(N) or K N → LN . PROPOSITION 2.9. Let W be a vector space over L and G ⊆ P(W ) an irreducible subgeometry of dimension > 2. Then there exists a subfield K of L and a vector subspace V of WK , where WK is W considered as vector space over K , such that G ∼ = P(V ). Proof. G is arguesian since it is irreducible and inherits from P(W ) the property of Desargues. Let j : G → P(W ) be the inclusion. Since Im j contains non-collinear points, there exist by the Fundamental Theorem 1.6 a vector space V ′ over some field K ′ and a semilinear map f ′ : V ′ → W such that G ∼ = P(V ′ ) ′ ′ ∼ and j = Pf . Using that f is injective one easily shows that V := f ′ (V ′ ) is a vector subspace of WK for K := σ(K ′ ) and that f ′ can be factorized as f ′ = i ◦ f where i: V → W is the inclusion and f : V ′ → V an isomorphism of 2 V ec. So one has G ∼ = P(V ′ ) ∼ = P(V ). Let us remark that 2.5 implies for n 6 3 the following property: if x1 , . . . , xn ∈ V are linearly independent over K , then also over L. Moreover, this property holds for every n ∈ N if and only if G is a proper subgeometry of P(W ). APCS232.tex; 5/02/1998; 13:54; v.7; p.9 96 C.-A. FAURE AND A. FRÖLICHER 3. Quotients We saw in 2.1 that every initial morphism is an embedding, i.e. an initial monomorphism. However, not every final morphism is a quotient, i.e. a final epimorphism. EXAMPLE 3.1. Let G1 be any and G2 a non-empty discrete projective geometry ` (i.e. G2 is a coproduct of singletons). Then the canonical map j1 : G1 → G1 G2 is a final morphism of P roj but not an epimorphism. ` Proof. Let g: G1 G2 − → G be a partial map into a projective geometry G for which g◦j1 : G1 − → G is a morphism. Since G2 is discrete, g◦j2 : G2 − → G ` is also a morphism. Hence g: G1 G2 − → G is a morphism. Since j1 is not surjective, it is not an epimorphism by 1.8. 2 More general examples are obtained as j1 ◦ f¯ where j1 is as before and f¯ is a final morphism. PROPOSITION 3.2. For a morphism f : G1 \ N → G2 of projective geometries the following conditions are equivalent: 1◦ f is a final morphism of the category P roj ; 2◦ If E ⊆ G2 and f −1 E ∪ N is a subspace of G1 , then E is a subspace of G2 . Proof. 1 ⇒ 2. Let f −1E ∪ N be a subspace of G1 . One chooses a projective geometry G with a point p ∈ G and considers the map h: G2 \ E → G having constant value p. Then the map h ◦ f : G1 \ (f −1 E ∪ N ) → G is constant and its kernel f −1 E ∪ N is by hypothesis a subspace of G1 . This implies that h ◦ f and hence also h is a morphism, and therefore E = Ker h is a subspace. 2 ⇒ 1. Let G be a projective geometry and h: G2 \ M → G be a map such that h ◦ f : G1 − → G is a morphism. Then for every subspace F of G one knows that (h ◦ f )−1 F ∪ Ker(h ◦ f ) = f −1 (h−1 F ∪ M ) ∪ N is a subspace of G1 and by hypothesis this implies that h−1 F ∪ M is a subspace of G2 . This shows that h is a morphism. 2 COROLLARY 3.3. Let f : G′ − → G be a final morphism of`P roj . Then G1 := Im g and G2 := G \ Im g are subspaces of G with G = G1 G2 . Furthermore, G2 is discrete and f factors as f = j1 ◦ f¯ where f¯ is a final epimorphism and j1 : G1 → G is the inclusion which is a final morphism by 3.1. COROLLARY 3.4. If G is irreducible, then every non-empty final morphism f : G′ − → G is an epimorphism. PROPOSITION 3.5. Every projective geometry G which contains at least two points is a quotient of the coproduct of its lines. APCS232.tex; 5/02/1998; 13:54; v.7; p.10 97 CATEGORICAL ASPECTS IN PROJECTIVE GEOMETRY ` set of ∆ := Proof. Let ∆ be the set of all lines of G. The underlying ` δ is the disjoint union of the lines δ . By i : δ → ∆ and jδ : δ → G δ δ∈∆ we denote the respective embeddings. By the coproduct-property there exists a ` unique morphism k: ∆ → G such that k ◦ iδ = jδ for all δ ∈ ∆. Since every point of G lies on some line, k is an epimorphism by 1.8. Furthermore, Proposition 1.3 implies that k is a final morphism. 2 ` DEFINITION 3.6. A morphism g: G − → G′ of projective geometries is called reflective if for any collinear points a′ , b′ , c′ in the image Im g there exist preimages a, b, c which are collinear. The quotients encountered in 3.5 are reflective. The same holds for the quotients considered in 3.7 and for many other quotients. In 3.8 we give an example of a quotient which is not reflective. We found no categorical characterization of the reflective quotients. PROPOSITION 3.7. Let g: G − → G′ be a morphism of projective geometries and let us consider the following conditions: 1◦ g is a 2◦ g is a 3◦ There that g 4◦ g is a 5◦ g is a reflective epimorphism and satisfies (M5 ); final epimorphism and satisfies (M5 ); exists a subspace N ⊆ G and an isomorphism u: G/N → G′ such = u ◦ π where π : G − → G/N is the canonical projection; surjective homomorphism; retraction of the category P roj . One has 1 ⇔ 2 ⇔ 3 ⇔ 4 ⇒ 5; and if G is irreducible and G′ contains more that one point, then also 4 ⇐ 5 holds. Proof. 1 ⇒ 2. It is enough to show that every reflective epimorphism is final, i.e. satisfies condition 2 of 3.2. So we suppose that E ⊆ G′ is such that g−1 E ∪ Ker g is a subspace of G. Let x′ 6= y ′ ∈ E and z ′ ∈ x′ ⋆ y ′ . By 1.8 g is surjective and so there exist x, y, z ∈ G \ Ker g with x′ = gx, y ′ = gy, z ′ = gz . Since g is reflective we can choose x, y, z such that ℓ(x, y, z). Then one has x 6= y ∈ g−1 E ⊆ g−1 E ∪ Ker g and hence z ∈ g−1 E ∪ Ker g. Since z ∈ / Ker g one has z ∈ g−1 E and hence z ′ = gz ∈ E . So we have proved that E is a subspace of G′ . 2 ⇒ 3. We consider the canonical projection π : G \ N → G/N , where N := Ker g. For x, y ∈ G \ N one has gx = gy ⇒ πx = πy by (M5 ), and πx = πy ⇒ gx = gy by (M2 ). Therefore there exists a (unique) injective map u: G/N → G′ such that g = u ◦ π . Since g is surjective, u is also surjective, i.e. u is a bijection. One has u−1 ◦ g = π and hence the finality of g implies that u−1 is a morphism. Since π is a reflective epimorphism satisfying (M5 ) it is final by 1 ⇒ 2. Hence g = u ◦ π implies that u is a morphism. 3 ⇒ 4. We first show that the canonical projection π : G \ N → G/N is a homomorphism. The verification of (M5 ) is trivial. We verify (M4 ). So let APCS232.tex; 5/02/1998; 13:54; v.7; p.11 98 C.-A. FAURE AND A. FRÖLICHER πa 6= πb and C ∈ πa ⋆ πb. Since π is surjective, C = πc for some c ∈ G \ N . By 5.1.4 in [4] one has c ∈ (a ⋆ b) ∨ N . Hence c ∈ c′ ∨ N for some c′ ∈ a ⋆ b and so one has C = πc = πc′ ∈ π(a ⋆ b). Knowing now that u and π are surjective homomorphisms, the same holds for g = u ◦ π . 4 ⇒ 1. The only property of g which needs verification is the reflectivity. So suppose that one has ℓ′ (gx, gy, gz). We may assume that gx 6= gy . Then one gets gz ∈ gx ⋆ gy ⊆ g(x ⋆ y) and hence gz = gze for some ze ∈ x ⋆ y . 3 ⇒ 5. Since π is known to be a retraction, cf. 6.1.4 in [4], also g = u ◦ π is a retraction. 5 ⇒ 4 is known to hold under the assumptions that G is irreducible and that G′ contains more than one point, cf. 6.2.3 in [4]. 2 EXAMPLE 3.8. A final epimorphism with empty kernel, which however is not reflective. We consider g := Pf : P(F52 ) − → P(F34 ) for the semilinear map f defined by f (α, β, γ, δ, ε) := (α + βi, γ + δi, ε), where {1, i} is a basis of F4 over F2 . We remark that i2 = 1 + i. The point of P(K n ) represented by (α1 , α2 , . . . , αn ) ∈ K n \0 will be denoted by (α1 : α2 : . . . : αn ). The verification that g is surjective and hence by 1.8 an epimorphism is straightforward. Since g is not injective (e.g. (1 : 0 : 0 : 0 : 0) and (0 : 1 : 0 : 0 : 0) have the same image) and since Ker g = ∅ one concludes that (M5 ) does not hold. Moreover, the three points (1 : 0 : 1) = g(1 : 0 : 0 : 0 : 1) (0 : i : 1) = g(0 : 0 : 0 : 1 : 1) (i : 1 : 1) = g(0 : 1 : 1 : 0 : 1) are collinear, but their unique preimages are not collinear. Therefore g is not reflective. In order to show that g is final, let H1 := P(F42 × {0}) and H2 := P(F24 × {0}). For x ∈ P(F34 ) the inverse image g−1 x is a line if x ∈ H2 and a singleton otherwise. Moreover one has g(H1 ) = H2 . Let now E ⊆ P(F34 ) be a subset such that g−1 E is a subspace of P(F52 ). By 3.2 it is enough to show that then E is a subspace of P(F34 ). Case 1. card(E ∩ H2) = 0. Since H1 is a hyperplane, g−1 E contains at most one point and hence card(E) 6 1. Case 2. E ∩ H2 = {x} is a singleton. Then either g−1 E = g−1 x or g−1 E = / H1 . In the second case, using that card(C(g−1 x ∪ a)) = C(g−1 x ∪ a) where a ∈ 7, one gets card(E) = 1 + 4 = 5, and E ⊆ C(x, g(a)). Therefore E is equal to the subspace x ⋆ g(a). APCS232.tex; 5/02/1998; 13:54; v.7; p.12 CATEGORICAL ASPECTS IN PROJECTIVE GEOMETRY 99 Case 3. card(E ∩ H2 ) > 2. Then g−1 E contains two disjoint lines of H1 . Hence either g−1 E = H1 or g−1 E = P(F52 ) and therefore either E = H2 or E = P(F34 ). PROPOSITION 3.9. Let f : V → W be a semilinear map. Then the morphism Pf : PV − → PW is a quotient if and only if for the subsets A of W one has: (λ · A ⊆ A for all λ ∈ L and f −1A + f −1A ⊆ f −1A) ⇒ A + A ⊆ A. Proof. Suppose first that g := Pf is a quotient, and let A ⊆ W satisfy λ·A ⊆ A for all λ ∈ L and f −1 A + f −1 A ⊆ f −1A. Let E := {[a] ∈ P(W )/a ∈ A \ 0} where [a] denotes the point of P(W ) represented by a. One shows that then g−1 E ∪ N , where N := P(ker f ), is a subspace of G1 . This implies by 3.2 that E is subspace of P(V ), and from this A + A ⊆ A follows easily. For the converse one proceeds analogously. 2 COROLLARY 3.10. Let f : V → W be a surjective semilinear map. Then P(f ): P(V ) − → P(W ) is a final morphism of the category P roj . Proof. Let A ⊆ W be such that f −1 A + f −1 A ⊆ f −1 A. Then one gets A + A = f f −1A + f f −1A = f (f −1A + f −1 A) ⊆ f (f −1A) = A. From this the assertion follows by the preceding proposition. 2 PROPOSITION 3.11. Let f : V → W be a semilinear map and dim(W ) > 2. Then the following conditions are equivalent: 1◦ P(f ) is a retraction of P roj ; 2◦ f is surjective and quasilinear; 3◦ f is a retraction of V ec. Proof. 1 ⇒ 2. Since P(W ) contains at least two points, the morphism g := P(f ) is non-constant. So one can use 2 of 1.6 in order to obtain the quasilinearity of f (g is a homomorphism by 3.7). 2 ⇒ 3. One verifies that the induced map f : V / ker f → W is an isomorphism of V ec and that any projection V → V /V ′ is a retraction of V ec. 3 ⇒ 1 is trivial. 4. Factorization of Morphisms LEMMA 4.1. Let h: G1 \ N → G2 be a morphism satisfying (M5 ) and E ⊆ G1 any subspace containing N . Then one has h−1 (h(E \ N )) = E \ N . Proof. Let a ∈ h−1 (h(E \ N )) be given. Then ha = hb for some point b ∈ E \ N , and one may assume a 6= b. By hypothesis there exists a point c ∈ (a ⋆ b) ∩ N . So one obtains a ∈ b ⋆ c ⊆ b ∨ N ⊆ E , and the equality follows. 2 APCS232.tex; 5/02/1998; 13:54; v.7; p.13 100 C.-A. FAURE AND A. FRÖLICHER PROPOSITION 4.2. Let h: G0 − → G1 be a surjective homomorphism and g: G0 − → G any morphism. Then there exists a morphism f : G1 − → G such that g = f ◦ h if and only if Ker h ⊆ Ker g. Moreover, the morphism f is unique and its kernel is given by Ker f = h(Ker g \ Ker h). Proof. We use that h is a final epimorphism satisfying (M5 ); cf. 3.7. If g = f ◦ h one gets Ker g = (Ker h) ∪ h−1 (Ker f ) ⊇ Ker h. Conversely, suppose that Ker h ⊆ Ker g. By the lemma one knows that Ker g = (Ker h) ∪ h−1 (N1 ), where N1 := h(Ker g \ Ker h). Now let a, b ∈ G1 \ Ker g with a 6= b and ha = hb. By (M5 ) there exists a point c ∈ (a ⋆ b) ∩ Ker h. So one obtains ga = gb by (M2 ), and this shows that there exists a partial map f : G1 \ N1 → G such that g = f ◦ h. This partial map is a morphism because h is final, and it is unique because h is an epimorphism. 2 COROLLARY 4.3. Every morphism g: G0 − → G can be decomposed as g = f ◦ π where π is the canonical projection G0 − → G0 / Ker g. Moreover, the induced morphism f : G0 / Ker g → G has empty kernel. LEMMA 4.4. Let h1 : G0 − → G1 and h2 : G0 − → G2 be two surjective homomorphisms. If Ker h1 = Ker h2 , then there exists an isomorphism ϕ: G1 → G2 such that h2 = ϕ ◦ h1 . Proof. By 4.2 there exist a morphism ϕ: G1 − → G2 such that h2 = ϕ ◦ h1 and a morphism ψ: G2 − → G1 such that h1 = ψ ◦ h2 . From h2 = ϕ ◦ ψ ◦ h2 one obtains ϕ ◦ ψ = id, because h2 is an epimorphism. Similarly, one has ψ ◦ ϕ = id, and this shows that ϕ is an isomorphism. 2 DEFINITION 4.5. Let C1 , . . . , Cn be n classes of morphisms of a category X . We say that X has (C1 , . . . , Cn )-factorization if every morphism g: G0 → Gn can be written as g = cn ◦· · ·◦c1 with morphisms ci : Gi−1 → Gi belonging to Ci , and if this decomposition is essentially unique: that is, if g = c′n ◦· · ·◦c′1 is another such decomposition, then there exist isomorphisms ϕi : Gi − → G′i (i = 1, . . . , n − 1) such that the following diagram commutes: G0 c1 G1 / c2 ϕ1 id G2 / ··· Gn−1 cn Gn / ϕn−1 ϕ2 id     G0  / c′1 G′1 / c′2 G′2 ··· G′n−1 / c′n Gn PROPOSITION 4.6. The category P roj has (H, K)-factorization, where H is the class of all surjective homomorphisms and K is the class of all morphisms having empty kernel. Proof. The existence of a decomposition g = k ◦ h follows directly from 4.3. Now let g = k1 ◦ h1 and g = k2 ◦ h2 be two such decompositions. We have to show that there exists an isomorphism ϕ such that h2 = ϕ ◦ h1 and k2 ◦ ϕ = k1 . APCS232.tex; 5/02/1998; 13:54; v.7; p.14 CATEGORICAL ASPECTS IN PROJECTIVE GEOMETRY 101 Clearly, one has Ker h1 = Ker h2 = Ker g. Therefore by 4.4 there exists an isomorphism ϕ such that h2 = ϕ ◦ h1 . Finally, one gets k2 ◦ ϕ = k1 because k2 ◦ ϕ ◦ h1 = k1 ◦ h1 and h1 is an epimorphism. 2 PROPOSITION 4.7. Let i: G1 − → G0 be an initial morphism and g: G − → G0 any morphism. Then there exists a morphism f : G − → G1 such that g = i◦f if and only if Im g ⊆ Im i. Moreover, such a morphism f is unique; its image is given by Im f = i−1 (Im g). Proof. We put N := Ker g. If g = i ◦ f , then Im g = i(f (G \ N )) is contained in Im i. Conversely, suppose that Im g ⊆ Im i. Then for any point a ∈ G \ N there exists a unique point b ∈ G1 such that ga = ib. Defining f a := b one gets a partial map f : G \ N − → G1 with i ◦ f = g. This partial map is a morphism because i is initial, and it is unique because i is a monomorphism, cf. 2.1. 2 COROLLARY 4.8. Every morphism g: G − → G0 can be decomposed as g = i ◦ f where i is the inclusion of the subgeometry S(Im g) generated by Im g, and also as g = s ◦ f where s is the inclusion of the subspace C(Im g). DEFINITION 4.9. A morphism g: G1 − → G2 of projective geometries is called S -dense if G2 = S(Im g). It is called C -dense if G2 = C(Im g). The morphism f of the preceding corollary is S -dense in the first case, and C -dense in the second one. REMARK 4.10. It is easy to verify that the class of C -dense morphisms is closed under composition. But we do not know whether the same holds for the class of S -dense morphisms. This is the reason why we use a definition of factorizations which is slightly different from the classical one; cf. Definition 15.1 and Proposition 15.5 in [2]. LEMMA 4.11. Let i1 : G1 → G0 and i2 : G2 → G0 be two initial morphisms. If Im i1 = Im i2 , then there exists an isomorphism ϕ: G1 → G2 such that i2 ◦ϕ = i1 . Proof. By 4.7 there exist two morphisms ϕ: G1 − → G2 and ψ: G2 − → G1 such that i2 ◦ ϕ = i1 and i1 ◦ ψ = i2 . From i2 = i2 ◦ ϕ ◦ ψ one obtains ϕ ◦ ψ = id, because i2 is a monomorphism. Similarly one has ψ ◦ ϕ = id, and hence ϕ is an isomorphism. 2 The proposition and its corollary give rise to two different decompositions of a morphism g: G1 − → G2 . We first need some elementary results. LEMMA 4.12. Let i: G1 → G2 be an initial morphism. Then i(S1 ) is a subgeometry of G2 for every subgeometry S1 ⊆ G1 , and i−1 (S2 ) is a subgeometry of G1 for every subgeometry S2 ⊆ G2 . Proof. By 2.1 we may assume that i is the inclusion of a subgeometry. Then i(S1 ) = S1 and i−1 (S2 ) = G1 ∩ S2 are subgeometries. 2 APCS232.tex; 5/02/1998; 13:54; v.7; p.15 102 C.-A. FAURE AND A. FRÖLICHER COROLLARY 4.13. Let i: G1 → G2 be an initial morphism. Then for any subset A ⊆ G1 one has i(S(A)) = S(i(A)). Proof. Clearly, one has S(i(A)) ⊆ i(S(A)) because i(S(A)) is subgeometry containing i(A). Similarly, one has S(A) ⊆ i−1 (S(i(A))) because i−1 (S(i(A))) is a subgeometry containing A. So the second inclusion follows. 2 PROPOSITION 4.14. The category P roj has (DS , I )-factorization, where DS is the class of all S -dense morphisms and I is the class of all initial morphisms. Proof. The existence of a decomposition g = i ◦ d follows from 4.8. Now let g = i1 ◦ d1 and g = i2 ◦ d2 be two such decompositions. By 4.13 one has i1 (S(Im d1 )) = S(i1 (Im d1 )) = S(Im g) = S(i2 (Im d2 )) = i2 (S(Im d2 )), and this implies that Im i1 = Im i2 . By 4.11 there exists an isomorphism ϕ such that i2 ◦ ϕ = i1 . Finally, one has ϕ ◦ d1 = d2 because i2 ◦ ϕ ◦ d1 = i2 ◦ d2 and i2 2 is a monomorphism. So the decomposition is essentially unique. LEMMA 4.15. Let s: G1 → G2 be a section. Then for any subset A ⊆ G1 one has s(C(A)) = C(s(A)). Proof. The image s(E1 ) of a subspace E1 ⊆ G1 is a subspace of G2 because s is a homomorphism. The inverse image s−1 (E2 ) of a subspace E2 ⊆ G2 is a subspace of G1 because s is a morphism. So one concludes as in 4.13. 2 PROPOSITION 4.16. The category P roj has (DC , S )-factorization, where DC is the class of all C -dense morphisms and S is the class of all sections. Proof. One proceeds as in Proposition 4.14. 2 By combining the three Propositions 4.6, 4.14 and 4.16, we conclude that any morphism g: G1 − → G2 can be decomposed in an essentially unique way into four factors: a surjective homomorphism h, an S -dense morphism k having empty kernel, a C -dense initial morphism i and a section s. DECOMPOSITION THEOREM 4.17. The category P roj of projective geometries has (H, K ∩ DS , I ∩ DC , S )-factorization, where • • • • H is the class of all surjective homomorphisms, K ∩ DS is the class of all S -dense morphisms having empty kernel, I ∩ DC is the class of all C -dense initial morphisms, S is the class of all sections. Moreover, one has for any morphism g: G1 − → G2 a canonical decomposition as follows: h k i s G1 − → G1 / Ker g −→ S(Im g) −→ C(Im g) −→ G2 APCS232.tex; 5/02/1998; 13:54; v.7; p.16 CATEGORICAL ASPECTS IN PROJECTIVE GEOMETRY 103 where h is the canonical projection and i, s are the respective inclusion morphisms. Proof. The preceding results imply that the factorization g = sikh exists and is of the claimed type. We now verify the uniqueness. Let g = s1 i1 k1 h1 and g = s2 i2 k2 h2 be two such decompositions. Clearly, s ◦ i ◦ k has empty kernel. Therefore there exists by 4.6 an isomorphism ϕ1 such that ϕ1 h1 = h2 . Now one remarks that s ◦ i is an initial morphism and that k ◦ h is S -dense (because Im(k ◦ h) = Im k). By 4.14 there exists an isomorphism ϕ2 such that ϕ2 k1 h1 = k2 h2 and s2 i2 ϕ2 = s1 i1 . One has ϕ2 k1 = k2 ϕ1 because ϕ2 k1 h1 = k2 ϕ1 h1 (and h1 is an epimorphism). Finally, one remarks that i ◦ k ◦ h is C -dense, which follows from the equality C(Im(i ◦ k ◦ h)) = C(Im(i ◦ k)) = CS(i(Im k)) = C(i(S(Im k))) = C(Im i). By 4.16 there exists an isomorphism ϕ3 such that s2 ϕ3 = s1 . Using the equality s2 ϕ3 i1 = s2i2 ϕ2 one deduces that ϕ3 i1 = i2 ϕ2 . Therefore the decomposition of a morphism g: G1 − → G2 is essentially unique. 2 References 1. Adámek, J.: Theory of Mathematical Structures, D. Reidel Publishing Company, Dordrecht, 1983. 2. Adámek, J., Herrlich, H. and Strecker, G. E.: Abstract and Concrete Categories, Wiley, New York, 1990. 3. Crapo, H. H. and Rota, G.-C.: On the Foundations of Combinatorial Theory: Combinatorial Geometries, MIT Press, 1970. 4. Faure, C.-A. and Frölicher, A.: Morphisms of projective geometries and of corresponding lattices, Geom. Dedicata 47 (1993), 25–40. 5. Faure, C.-A. and Frölicher, A.: Morphisms of projective geometries and semilinear maps, Geom. Dedicata 53 (1994), 237–262. 6. Faure, C.-A. and Frölicher, A.: Dualities for infinite-dimensional projective geometries, Geom. Dedicata 56 (1995), 225–236. 7. Hermes, H.: Einführung in die Verbandstheorie, Grundlehren Band 73, Springer-Verlag, 1967. APCS232.tex; 5/02/1998; 13:54; v.7; p.17