The direction functor for Schreier extensions of monoids

THE DIRECTION FUNCTOR F OR SCHREIER EXTENSIONS OF MONOIDS STEF ANO AMBRA, ANDREA MONTOLI, AND DIANA RODELO Abstract. W e observe that the process of associating an action to any Schreier extension of monoids with commutativ e and cancellative k ernel is functorial. W e show that this functor is a generalisation of the direction functor, used to giv e a categorical description of non-abelian cohomology in terms of exten- sions. W e further pro v e that our functor is a conserv ative, pro duct preserving coﬁbration and from this we conclude that its ﬁbres are endow ed with a ca- nonical symmetric monoidal structure. The commutativ e monoids obtained as connected components of these symmetric monoidal categories are isomorphic to Patc hkoria’s second cohomology monoids of a monoid with co eﬃcients in semimodules. 1. Introduction The cohomology groups H n ( B , A ) of an ab elian group B with co eﬃcien ts in an ab elian group A hav e a classical description in terms of exact sequences. Indeed, the ﬁrst cohomology group H 1 ( B , A ) is isomorphic to the group of isomorphism classes of extensions of B b y A (namely short exact sequences starting with A and ending with B ) with the Baer sum. A similar description of higher cohomology groups b y means of n -extensions w as obtained b y Y oneda [28]. The same results hold in any abelian category . The situation for (non-ab elian) group cohomology is more diversiﬁed. In fact, the whole set Ext( G, A ) of isomorphism classes of extensions of a non-necessarily ab elian group G by an ab elian group A do es not admit a cohomological in ter- pretation. Ho wev er, every group extension of G b y an ab elian group A induces a G -mo dule structure on A ; the set Ext( G, A ) can then be partitioned in to subsets of the form Ext( G, A, φ ) , where φ : G → Aut( A ) is an action of G on A, and, as sho wn in [13], eac h set Ext( G, A, φ ) is isomorphic to the second Eilenberg-Mac Lane cohomology group H 2 ( G, A, φ ) of G with co eﬃcients in the G -mo dule ( A, φ ) . In terpretations of higher cohomology groups in terms of suitable exact sequences ha ve b een obtained in [16, 17]. Similar results hold for diﬀerent non-ab elian algeb- raic structures, lik e associative algebras, Lie algebras, Leibniz algebras and many others. The pro cess of asso ciating to a group extension with ab elian kernel its induced action is functorial. The prop erties of this functor ha v e been studied in detail in 2020 Mathematics Subje ct Classiﬁcation. 18G50, 20J06, 20M50, 18E13, 18C05. Key wor ds and phr ases. Schreier extension of monoids, direction functor, cohomology monoids, cancellative semimo dules. This w ork was supp orted by the Shota Rustav eli National Science F oundation of Georgia (SRNSFG), through grant FR-24-9660, “Categorical metho ds for the study of cohomology the- ory of monoid-like structures: an approach through Schreier extensions”. The ﬁrst and second authors are members of the Grupp o Nazionale p er le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA) dell’Istituto Nazionale di Alta Matematica “F rancesco Severi”. The third author acknowledges ﬁnancial supp ort by CIDMA (https://ror.org/05pm2mw36) un- der the Portuguese F oundation for Science and T echnology (F CT, https://ror.org/00snfqn58), Grants UID/04106/2025 (h ttps://doi.org/10.54499/UID/04106/2025) and UID/PRR/04106/2025 (https://doi.org/10.54499/UID/PRR/04106/2025). 1 2 S. AMBRA, A. MONTOLI, AND D. RODELO [5], where this functor is seen as a particular instance of a very general categorical construction, called the dir e ction functor . The direction functor has as domain the category of ob jects with global supp ort (meaning that the arro w to the terminal ob ject is a regular epimorphism) which are endow ed with an autonomous Mal’tsev op eration, in a Barr-exact category C ; the codomain is the category of internal ab elian groups in C . When applied to the slice category Gp /G, where Gp is the category of groups, the direction functor is precisely the functor sending ev ery extension of G with ab elian kernel to the corresp onding action. Indeed, the ob jects with global supp ort in Gp /G are the surjective group homomorphisms f : X → G, and suc h an f is endow ed with a, necessarily autonomous, Mal’tsev op eration if and only if its k ernel congruence E q ( f ) cen tralises itself in the sense of [25], whic h is equiv alent to sa ying that the kernel of f is ab elian. The double equiv alence relation determined b y the fact that E q ( f ) centralises itself, called the Chasles r elation in [5], plays a key role in the construction of the direction functor: in fact, the image of f under the direction functor is the split epimorphism with co domain G and domain the coequalizer of the Chasles relation. Suc h a split epimorphism, ha ving an abelian kernel, is an internal ab elian group in Gp /G. The action corresp onding to such split epimorphism, via the semidirect pro duct construction, is precisely the action, induced by f , of G on the kernel of f . It is shown in [5] that the direction functor has many go o d prop erties; in partic- ular, it is a coﬁbration, it is conserv ative and it preserves ﬁnite products. Thanks to these prop erties, the ab elian group structure on any ob ject in its co domain can b e lifted to a symmetric monoidal closed structure on its ﬁbre. On the sets of connected comp onen ts of such ﬁbres there result then canonical ab elian group structures. In the case C = Gp /G, these ab elian groups are precisely the cohomo- logy groups H 2 . In [11] a similar description of higher cohomology groups using higher dimensional direction functors has b een obtained for Barr-exact categories (actually Barr-exactness can b e weak ened, considering instead eﬃciently r e gular categories, including more examples, like the category of top ological groups; see [11]). There ha ve b een diﬀeren t attempts to give a cohomological description of ex- tensions of monoids, to o. A crucial notion in this resp ect is the one of Schreier extension of monoids, in tro duced in [26]. Sc hreier extensions of monoids retain sev eral go o d properties of group extensions. It was shown in [15, 20, 21] that the set of isomorphism classes of Schreier extensions, inducing the same action, of a monoid M by an abelian group A (called sp e cial Schr eier extensions in [8, 9]) has an ab elian group structure whic h extends the usual Baer sum of group extensions, and suc h ab elian groups are actually isomorphic to the cohomology groups H 2 ( M , A, φ ) of a monoid M with co eﬃcien ts in an M -module ( A, φ ) of a cohomology theory whic h generalises straigh tforwardly the Eilenberg-Mac Lane cohomology of groups. P atchk oria in tro duced in [23] a cohomology theory for monoids with co eﬃcien ts in semimodules (which is a v ariation on yet another cohomology theory he in tro- duced in [22]), in which one gets commutativ e monoids of cohomology , instead of ab elian groups. Moreo ver, he show ed in [24] that the second cohomology mo- noids H 2 ( M , K , φ ) of his theory , where M is a monoid and ( K, φ ) is a cancellativ e M -semimodule, describ e Schreier extensions of M b y K . In the presen t pap er w e sho w that, also in the case of monoids, the pro cess of asso ciating an action to a Sc hreier extension with comm utativ e k ernel is functorial and that this functor is, in a sense, a generalisation of the direction functor studied in [5]. The domain of our functor is the category cc - S E xt M whose ob jects are the Schreier extensions of a monoid M with comm utative and cancellative kernel. T o such an extension E : K / / k / / X f / / / / M we asso ciate a Schreier reﬂexive THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 3 relation (in the sense of [8, 9]) R E , whic h is automatically transitive (thanks to the results in [8, 9]) but not symmetric, in general: the congruence generated by R E is precisely the k ernel congruence E q ( f ) of f , and R E and E q ( f ) coincide if and only if the k ernel of f is a group. W e observ e that the Sc hreier relation R E cen tralises itself; the corresp onding double Schreier reﬂexive relation (whose existence and uniqueness are guaranteed by the results in [10]) is then a generalisation of the Chasles relation of [5], and our functor asso ciates with E a split epimorphism with co domain M and domain the co equalizer of the generalised Chasles relation. This image is an in ternal comm utativ e monoid in the category whose ob jects are Sc hreier extensions of a monoid M . The result is then a functor d : cc - S E xt M − → CMon ( cc - S E xt M ) whose co domain is the category of internal commutativ e monoids in cc - S E xt M . W e show that this functor still satisﬁes the key prop erties of the direction functor from [5]. This allows us to equip the ﬁbres of the functor with symmetric monoidal structures in a wa y that the sets of connected components of such ﬁbres turn out to b e endow ed with canonical commutativ e monoid structures. The so obtained monoids are precisely Patc hk oria’s second cohomology monoids built in [23]. The functor d can actually b e extended to a functor D : smod - S E xt M − → CMon ( smod - S E xt M ) , where smod - S E xt M is the category whose ob jects are the Sc hreier extensions of M inducing an M -semimo dule structure ( cc - S E xt M is a full sub category of smod - S E xt M ). This broader functor shares the same prop erties of d, hence the connected comp onents of its ﬁbres can also b e equipped with canonical comm utat- iv e monoid structures, that correspond to the second cohomology monoids of the cohomology theory considered by Patc hkoria in [22]. 2. Definition and first proper ties of Schreier extensions of monoids In this pap er we shall be in terested in monoid extensions, namely sequences of monoids and monoid homomorphisms (2.1) E : K / / k / / X f / / / / M in which ( K , k ) is a k ernel of f and f is a regular epimorphism (i.e. surjectiv e monoid homomorphism). F ollo wing the c hoice made in [18] for group extensions, w e shall adopt an additiv e notation on K and X and a multiplicativ e notation on M . It should be clear, nonetheless, that while w e will ev en tually require K to b e comm utative, no such assumption is ever made on X , in general. W e write K er ( f ) = k ( K ) = { x ∈ X : f ( x ) = 1 } . The category of monoids and monoid homomorphisms shall b e denoted b y Mon . Giv en a monoid extension (2.1), denote b y B m , for m ∈ M , the subset of f − 1 ( m ) deﬁned by B m = n u ∈ f − 1 ( m ) : ∀ x ∈ f − 1 ( m ) , ∃ ! a ∈ K such that x = k ( a ) + u o . Observ e that B m ma y very well b e empty for some m, but since k is a mono- morphism, one alwa ys has at least 0 ∈ B 1 . The elements of B m shall b e called the r epr esentatives of m, and we shall denote b y B ( E ) = S m ∈ M B m the set of all represen tatives. It is a subset of X, by construction. In the next lemma, we collect some useful prop erties which immediately stem out of these deﬁnitions. Lemma 2.1. In the ab ove notation: 4 S. AMBRA, A. MONTOLI, AND D. RODELO (1) F or every a, a ′ ∈ K, m ∈ M and u ∈ B m , the e quality k ( a ) + u = k ( a ′ ) + u holds if and only if a = a ′ . (2) If u, v ∈ B m , one has u = k ( a ) + v for a unique a ∈ U ( K ) (wher e U ( K ) ⊆ K denotes the sub gr oup of invertible elements). (3) Conversely, if u ∈ B m and a ∈ U ( K ) , then k ( a ) + u ∈ B m . Deﬁnition 2.2 ([26]) . W e shall sa y that (2.1) is a Schr eier extension if B m  = ∅ for every m ∈ M . If (2.1) is a Schreier extension, then for every ﬁxed m ∈ M and u ∈ B m the equation x = k  q m,u ( x )  + u deﬁnes a map of sets (denoted with a dashed arrow in this work to emphasise that it is not a morphism in Mon ) q m,u : f − 1 ( m ) / / K, x 7→ q m,u ( x ) . Corollary 2.3. If (2.1) is a monoid extension, for every m ∈ M and every u ∈ B m ther e is a bije ction ϑ = ϑ m,u : U ( K ) ∼ / / B m , a 7→ k ( a ) + u. Thus, for any u ∈ B m , ther e r esults a gr oup structur e on B m having u as neutr al element, which is ab elian if U ( K ) is ab elian. In p articular, if (2.1) is a Schr eier extension, for al l m, m ′ ∈ M ther e is a bije ction B m ∼ = B m ′ , which is a gr oup isomorphism with r esp e ct to the gr oup structur es induc e d by the ϑ ’s. Pr o of. By Lemma 2.1, ϑ m,u is injective (point (1)) and surjective (p oin t (2)). The desired group structure is then obtained b y transferring on B m the group structure of U ( K ) via ϑ m,u , i.e. by deﬁning the sum v + u w = ϑ m,u  ϑ − 1 m,u ( v ) + ϑ − 1 m,u ( w )  for all v , w ∈ B m . □ Some other immediate consequences of Deﬁnition 2.2 are collected in the follow- ing lemma. Lemma 2.4 (cf. [8, Prop osition 2.1.5]) . F or a Schr eier extension (2.1) : (1) q 1 , 0 k = id K ; (2) for every u ∈ B ( E ) , q f ( u ) ,u ( u ) = 0; (3) for every a ∈ K and u ∈ B ( E ) , u + k ( a ) = k  q f ( u ) ,u  u + k ( a )   + u ; (4) for every x, x ′ ∈ X, u ∈ B f ( x ) and u ′ ∈ B f ( x ′ ) , q f ( x ) ,u ( x ) + q f ( x ′ ) ,u ′ ( x ′ ) = q f ( x ′ ) ,u ′  k  q f ( x ) ,u ( x )  + x ′  . Pr o of. (1) Giv en a ∈ K, the element q 1 , 0  k ( a )  is uniquely determined b y k ( a ) = k  q 1 , 0  k ( a )   , and k is a monomorphism. (2) One has u ∈ B m for some m ∈ M , so that f ( u ) = m ; thus u is a represent- ativ e of f ( u ) . Also, u = k (0) + u and q f ( u ) ,u ( u ) is uniquely determined by u = k  q f ( u ) ,u ( u )  + u. The equality follows from the deﬁnition of a Sc hreier extension. (3) Immediate by observing that u is a representativ e of f  u + k ( a )  = f ( u ) . (4) The element q f ( x ′ ) ,u ′  k  q f ( x ) ,u ( x )  + x ′  ∈ K is uniquely determined by the equation k  q f ( x ) ,u ( x )  + x ′ = k  q f ( x ′ ) ,u ′  k  q f ( x ) ,u ( x )  + x ′  + u ′ . Then k  q f ( x ) ,u ( x ) + q f ( x ′ ) ,u ′ ( x ′ )  + u ′ = k  q f ( x ) ,u ( x )  + k  q f ( x ′ ) ,u ′ ( x ′ )  + u ′ = k  q f ( x ) ,u ( x )  + x ′ , and the desired equality holds from the deﬁnition of a Schreier extension. □ THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 5 F rom no w on, in order to simplify the notation, w e shall write u m to denote some representativ e of m ∈ M and q m,u = q for all m ∈ M and u ∈ B m . Thus, for example, giv en a Sc hreier extension (2.1) w e ha ve for every x ∈ X (and u f ( x ) ∈ B f ( x ) ) a unique q ( x ) ∈ K suc h that (2.2) x = k q ( x ) + u f ( x ) , and the equalit y in p oin t (4) of the previous lemma shall b e expressed simply as q ( x ) + q ( x ′ ) = q  k q ( x ) + x ′  . Since b y Lemma 2.1 any t wo represen tatives of the same m diﬀer b y an in vertible elemen t, we ha v e a c haracterisation of the Schreier extensions (2.1) in which the k ernel is a gr oup . Prop osition 2.5. Given a Schr eier extension (2.1) , the fol lowing ar e e quivalent: (1) B ( E ) = X , i.e. B m = f − 1 ( m ) for every m ∈ M ; (2) B 1 = K er ( f ); (3) K is a gr oup. Pr o of. It is clear that (1) implies (2) , and (3) follows from (2) by setting u = k ( a ) and v = 0 in Lemma 2.1(2) , for an y a ∈ K. Now, to prov e that (3) implies (1) , it is enough to show that if a ∈ K and u m is a represen tative, then k ( a ) + u m is also a representativ e: but if x ∈ f − 1 ( m ) , then x = k q ( x ) + u m b y (2.2), so that x = k ( a ′′ ) + ( k ( a ) + u m ) for a ′′ ∈ K uniquely determined as a ′′ = q ( x ) − a. □ Another relev an t consequence of Deﬁnition 2.2 is the follo wing one which prov es that Schreier extensions are, indeed, short exact sequences. Prop osition 2.6 (Cf. [24, Section 4]) . If (2.1) is a Schr eier extension, then f is a c okernel of k , so that (2.1) is a short exact se quenc e in Mon . Pr o of. Let g : X → Z b e a monoid homomorphism such that g k = 0 . Then g is constan t on ev ery ﬁbre f − 1 ( m ) of f . Indeed, since (2.1) is a Schreier extension w e hav e B m  = ∅ ; let u m ∈ B m . If x, x ′ ∈ f − 1 ( m ) , then x = k q ( x ) + u m and x ′ = k q ( x ′ ) + u m b y (2.2). It follows that g ( x ) = g ( u m ) = g ( x ′ ) . Note that, if u m , v m are tw o representativ es of m, then g ( u m ) = g ( v m ) , as u m , v m ∈ f − 1 ( m ) . Deﬁne φ : M → Z by setting φ ( m ) = g ( u m ) , where u m is any representativ e of m. W e prov ed abov e that φ is well deﬁned and, as a function, it clearly satisﬁes φf = g . T o see that it is also a monoid homomorphism, let m, m ′ ∈ M : then u m + u m ′ = k q ( u m + u m ′ ) + u m · m ′ , again by (2.2), and by using the fact that g k = 0 we ha v e φ ( m · m ′ ) = g ( u m · m ′ ) = g ( k q ( u m + u m ′ ) + u m · m ′ ) = g ( u m + u m ′ ) = g ( u m ) · g ( u m ′ ) = φ ( m ) · φ ( m ′ ) . The uniqueness of φ is immediate, since f is epimorphic. □ It will b e shown in Example 2.10.6 that the con v erse of this result do es not hold in general. The notion of Schreier extension comes with a coherent notion of morphism b et w een Schreier extensions: Deﬁnition 2.7. A morphism of Schr eier extensions E → E ′ is a morphism of monoid extensions (2.3) E : K (a) α 1   / / k / / X (b) α 2   f / / / / M α 3   E ′ : K ′ / / k ′ / / X ′ f ′ / / / / M ′ , 6 S. AMBRA, A. MONTOLI, AND D. RODELO meaning that α 1 , α 2 , α 3 are monoids homomorphisms suc h that the squares (a) and (b) commute, satisfying the additional condition α 2 ( B ( E )) ⊆ B ( E ′ ) . By the comm utativit y of (b) , the restriction of α 2 to B m giv es a map α 2 : B m → B ′ α 3 ( m ) (the latter denoting the set of representativ es of α 3 ( m ) for E ′ ), which is a group homomorphism with resp ect to the group structures of Corollary 2.3, because for every u ∈ B m the square U ( K ) α 1   ∼ ϑ m,u / / B m α 2   U ( K ′ ) ∼ ϑ α 3 ( m ) ,α 2 ( u ) / / B ′ α 3 ( m ) comm utes (using the commutativit y of (a) in (2.3)). It is a consequence of Lemma 2.1 that, in order to prov e that a morphism of monoid extensions (2.3) b etw een Schreier extensions is a morphism of Schreier extensions, it is enough to sho w that for all m ∈ M some representativ e u m of m in E is mapp ed b y α 2 to a representativ e in E ′ : Prop osition 2.8. Consider a morphism (2.3) of (not ne c essarily Sch r eier) monoid extensions and supp ose that α 2 ( u m ) ∈ B ′ α 3 ( m ) for some u m ∈ B m (wher e m ∈ M ). Then α 2 ( v m ) ∈ B ′ α 3 ( m ) for al l v m ∈ B m . Pr o of. If u m , v m ∈ B m , by Lemma 2.1(2) w e hav e v m = k ( a ) + u m for a unique a ∈ U ( K ) , and f ′  α 2 ( v m )  = α 3  f ( v m )  = α 3 ( m ) . Moreov er, α 1 ( a ) is inv ertible in K ′ b ecause a is in v ertible in K and α 1 is a monoid homomorphism. W e hav e α 2 ( v m ) = α 2  k ( a )  + α 2 ( u m ) = k ′  α 1 ( a )  + α 2 ( u m ) , whic h prov es that α 2 ( v m ) ∈ B ′ α 3 ( m ) b y Lemma 2.1(3) . □ One can prov e that the Short Five Lemma, whic h is not v alid in general for monoids (as the category Mon is not protomo dular [4]), do es hold for Schreier extensions. Prop osition 2.9 ([24, Prop osition 4.5]) . Consider a morphism (2.3) of Schr eier extensions. Then: (1) If α 1 and α 3 ar e monomorphisms (i.e. inje ctive monoid homomorphisms), α 2 is a monomorphism; (2) If α 1 and α 3 ar e r e gular epimorphisms (i.e. surje ctive monoid homomorph- isms), α 2 is a r e gular epimorphism. Thus, in p articular, if α 1 and α 3 ar e isomorphisms, so is α 2 . W e conclude this section with some concrete examples of Schreier and non- Sc hreier monoid extensions. Examples 2.10 . 1. Ev ery short exact sequence of groups K / / k / / X f / / / / G is a Sc hreier extension of monoids in whic h all x ∈ X are represen tativ es, b y Prop osition 2.5. F or any u ∈ X and any x ∈ f − 1 ( f ( u )) , one has indeed x = k ( a ) + u for a unique a ∈ K giv en by k ( a ) = x − u. 2. F or ev ery monoid M , the trivial extension M M / / / / 0 THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 7 is a Sc hreier extension whose representativ es are the inv ertible elements of M . Similarly , the extension 0 / / / / M M is a Schreier extension, with B m = { m } for all m ∈ M . 3. F or all monoids K and M , the monoid extension K / / ⟨ id K , 0 ⟩ / / K × M π M / / / / M (where π M ( a, m ) = m ) is a Schreier extension, having as representativ es the couples ( a, m ) with m ∈ M and a ∈ U ( K ) . 4. (Cf.[24, Example 4.2]) Let m > 1 b e a natural n um b er and consider the monoid extension ( N , + , 0) / / · m / / ( N , + , 0) f / / / / C m ( t ) , where C m ( t ) is the m ultiplicativ e cyclic group of order m with generator t, · m is the ordinary m ultiplication by m and f (1) = t. It is a Schreier extension, with representativ es B 1 = { 0 } , B t = { 1 } , . . . , B t m − 1 = { m − 1 } . 5. The monoid extension 0 / / / / N × N + / / / / N is not a Schreier extension by Prop osition 2.6, b ecause the regular epimor- phism + given b y the sum of natural num bers is not a normal epimorphism (otherwise it w ould be a cokernel of its kernel 0 , and thus an isomorphism, but + is not injective). 6. Let ( M 2 = { 0 , 1 } , · , 1) b e the comm utative monoid with unit 1 and 0 · 0 = 0 , and consider the monoid extension ( N \ { 0 } , · , 1) / / i / / ( N , · , 1) f / / / / ( M 2 , · , 1) where i is the inclusion morphism, f ( n ) = 1 if n  = 0 and f (0) = 0 . This extension is not a Schreier extension, b ecause there are inﬁnitely many a ∈ N \ { 0 } such that 0 = a · 0 . Even so, f is a normal epimorphism in Mon : indeed, if g : ( N , · , 1) → ( M , · , 1) is any monoid homomorphism such that g ( n ) = 1 for all n ∈ N \ { 0 } , g factors through f as g = g f (uniquely so, as f is epimorphic) with g : M 2 → M giv en by g (1) = 1 and g (0) = g (0) . Th us the conv erse of Prop osition 2.6 do es not hold. 3. A ction induced by a Schreier extension It is well known that if K / / k / / X f / / / / G is a short exact sequence of groups in which K is ab elian, then there results a (left) action of G on K given b y (3.1) G × K 99K K , ( g , a ) 7→ a ′ with k ( a ′ ) = s ( g ) + k ( a ) − s ( g ) , where s : G 99K X is an y set-theoretic section of f (i.e. s is any map of sets suc h that f s = id G ). W e wan t to sho w that something similar happ ens for Schreier extensions of monoids. Most facts in this section are already known, and the statements can b e found, for example, in [24], but since these prop erties will ha v e a central imp ortance in our main construction, we ﬁnd it appropriate to provide the reader with explicit pro ofs. Recall that a (left) action of a monoid ( M , · , 1) on a monoid ( A, + , 0) is a monoid homomorphism ( M , · , 1) → (End( A ) , ◦ , id A ) , where End( A ) denotes the set of mo- noid endomorphisms of A ; this is equiv alent to having a map of sets M × A 99K A, ( m, a ) 7→ m ∗ a, satisfying the four axioms: 8 S. AMBRA, A. MONTOLI, AND D. RODELO ( A 1 ) 1 ∗ a = a for all a ∈ A ; ( A 2 ) m ∗ 0 = 0 for all m ∈ M ; ( A 3 ) m ∗ ( a + a ′ ) = m ∗ a + m ∗ a ′ for all m ∈ M and a, a ′ ∈ A ; ( A 4 ) ( m ′ · m ) ∗ a = m ′ ∗ ( m ∗ a ) for all m, m ′ ∈ M and a ∈ A. If K / / k / / X f / / / / M is a Sc hreier extension of monoids with K commutativ e, w e claim that the map (3.2) M × K 99K K , ( m, a ) 7→ m ∗ a with u m + k ( a ) = k ( m ∗ a ) + u m is well deﬁned, meaning that it do es not dep end on the c hoice of the representativ e u m of m. Indeed, let u, v ∈ B m , a ∈ K , and let a u , a v ∈ K be suc h that u + k ( a ) = k ( a u ) + u and v + k ( a ) = k ( a v ) + v . As b oth u and v are representativ es of m, by Lemma 2.1(2) there exists a unique a ′ ∈ U ( K ) such that u = k ( a ′ ) + v . W e hav e u + k ( a ) = k ( a ′ ) + v + k ( a ) = k ( a ′ ) + k ( a v ) + v = k ( a ′ + a v ) + v , u + k ( a ) = k ( a u ) + u = k ( a u ) + k ( a ′ ) + v = k ( a u + a ′ ) + v , and by Lemma 2.1(1) , the c omm utativity of K and the inv ertibilit y of a ′ , it follo ws that a u = a v . Observ e that (3.2) can b e expressed by (3.3) m ∗ a = q  u m + k ( a )  , in the notation of the previous section. The key p oint here is that (3.2) almost deﬁnes an action of M on K , as shown next. Prop osition 3.1. If E : K / / k / / X f / / / / M is a Schr eier extension in which K is c ommutative, the e quation (3.2) always satisﬁes the axioms ( A 1 ) , ( A 2 ) and ( A 3 ) . If, mor e over, K is c anc el lative, then ( A 4 ) is satisﬁe d to o, and in this c ase (3.2) deﬁnes a monoid action of M on K . Pr o of. ( A 1 ) By deﬁnition, for a ∈ K, 1 ∗ a is the only element satisfying u 1 + k ( a ) = k (1 ∗ a ) + u 1 ; then by setting u 1 = 0 we get 1 ∗ a = a. ( A 2 ) F or m ∈ M , it follows from Lemma 2.1(1) that u m = u m + k (0) = k ( m ∗ 0) + u m en tails m ∗ 0 = 0 . ( A 3 ) Giv en m ∈ M and a, a ′ ∈ K , let m ∗ a = b and m ∗ a ′ = b ′ , so that u m + k ( a ) = k ( b ) + u m and u m + k ( a ′ ) = k ( b ′ ) + u m . Then u m + k ( a + a ′ ) = u m + k ( a ) + k ( a ′ ) = k ( b ) + u m + k ( a ′ ) = k ( b ) + k ( b ′ ) + u m = k ( b + b ′ ) + u m , whence m ∗ ( a + a ∗ ) = b + b ′ = m ∗ a + m ∗ a ′ . ( A 4 ) No w supp ose that K is also cancellative and deﬁne m ∗ a = b (so that u m + k ( a ) = k ( b ) + u m ), m ′ ∗ b = b ′ (so that u m ′ + k ( b ) = k ( b ′ ) + u m ′ ), and ( m ′ · m ) ∗ a = c (so that u m ′ · m + k ( a ) = k ( c ) + u m ′ · m ). W e wan t to pro ve THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 9 that c = b ′ . W rite u m ′ + u m = k ( l ) + u m ′ · m ; then k ( l + b ′ ) + u m ′ · m = k ( b ′ ) + k ( l ) + u m ′ · m = k ( b ′ ) + u m ′ + u m = u m ′ + k ( b ) + u m = u m ′ + u m + k ( a ) = k ( l ) + u m ′ · m + k ( a ) = k ( l ) + k ( c ) + u m ′ · m = k ( l + c ) + u m ′ · m , and the result follows from Lemma 2.1(1) and the cancellativity of K . □ W e stress the fact that the cancellativity of K is only a suﬃcient condition for (3.2) to b e a monoid action: indeed, observe that if k ( K ) is cen tral in X, then (3.2) is just the trivial assignment m ∗ a = a for all m ∈ M and a ∈ K , which is alwa ys an action (the trivial one) even if K is not cancellative. This is the case, for instance, of Example 2.10.2 (ﬁrst example) when M is commutativ e but not cancellativ e. Similarly , the Schreier extension of Example 2.10.3 alwa ys induces the trivial action when K is commutativ e (even if M is not commutativ e). Giv en a Sc hreier extension E as in Proposition 3.1 with comm utativ e and cancel- lativ e K , w e denote the monoid homomorphism corresp onding to the ab o ve action b y (3.4) η : M → End( K ) , m 7→ η ( m ) : K → K , where η ( m )( a ) = m ∗ a, as in (3.2). Observ e, moreo ver, that when K / / k / / X f / / / / G is a short exact sequence of groups with an ab elian kernel (which is alwa ys a Sc hreier extension of monoids, with K cancellativ e), the induced action (3.2) do es coincide with the group action (3.1). Since the request for K to b e commutativ e and cancellative will turn out re- p eatedly in the next sections, w e set for the sake of brevity the following deﬁnition. Deﬁnition 3.2. A c c-Schr eier extension is a Schreier extension K / / k / / X f / / / / M in which the kernel K is commutativ e and cancellativ e. W e conclude this section by p oin ting out a prop ert y of the map (3.2) which, despite its simplicity , will b e v ery useful in carrying out explicit computations. Lemma 3.3. If K / / k / / X f / / / / M is a Schr eier extension with K c ommutative, then (3.5) x + k ( a ) = k  f ( x ) ∗ a  + x for al l x ∈ X and a ∈ K. Pr o of. W e ha v e x = k q ( x ) + u f ( x ) , from (2.2). Then x + k ( a ) = k q ( x ) + u f ( x ) + k ( a ) (3.2) = k q ( x ) + k  f ( x ) ∗ a  + u f ( x ) = k  f ( x ) ∗ a  + k q ( x ) + u f ( x ) = k  f ( x ) ∗ a  + x, using the commutativit y of K . □ 4. Interlude on Schreier points and S -reflexive rela tions In this section we collect some of the main results concerning the so-called Sc hreier p oints , which shall b e needed in the course of the construction of our direction functor. The following notion was introduced in [19], Deﬁnition 2 . 6: 10 S. AMBRA, A. MONTOLI, AND D. RODELO Deﬁnition 4.1. A Schr eier split extension of monoids, or Schr eier p oint , is a split extension (4.1) K / / k / / B f / / / / M s o o in Mon (where the monoid homomorphism s is a ﬁxed section of f and ( K, k ) is a kernel of f ) for whic h there exists a unique map of sets q : B 99K K (called the Schr eier r etr action ) satisfying (4.2) b = k q ( b ) + sf ( b ) , for every b ∈ B . An extensive study of Schreier split extensions of monoids is carried out in [8]. R emark 4.2 . Observ e that any Schreier split extension of monoids (4.1) determines a Sc hreier extension K / / k / / B f / / / / M in the sense of Deﬁnition 2.2, with s ( m ) ∈ B m for all m ∈ M . Beware that B m need not b e reduced to the sole elemen t s ( m ) , as the case of split extensions of groups shows. Indeed, by Proposition 2.5 w e hav e B 1 = f − 1 (1) = k ( K )  = { 0 } when f is not monomorphic. Moreo ver, the unique element q ( b ) ∈ K satisfying (4.2) coincides with q f ( b ) ,sf ( b ) ( b ) using the notation of Section 2. Con versely , if (2.1) is a Schreier extension in the sense of Deﬁnition 2.2, whic h is split by a section s : M → X such that each s ( m ) ∈ B m , then K / / k / / X f / / / / M s o o is a Schreier split extension. The Schreier retraction is giv en by the map of sets q : X 99K K , where q ( x ) = q f ( x ) ,sf ( x ) ( x ) for every x ∈ X , as deﬁned after Deﬁnition 2.2. T o a v oid any confusion b etw een the tw o notions, we shall stick to the name Schr eier p oints , from no w on, to refer to (4.1). The relev ance of Schreier p oin ts is that they corresp ond to monoid actions in the same wa y as split extensions of groups are equiv alent to group actions, via the semidirect pro duct construction (see [8, Proposition 5.2.2], and also Section 5.2 of [3] for the classical case of groups). Prop osition 4.3 ([8, Theorem 5.1.2]) . Given a Schr eier p oint K / / k / / B q | | f / / / / M s o o with Schr eier r etr action q , the induc e d monoid action of M on K is given by (4.3) M × K 99K K , ( m, a ) 7→ m • a = q  s ( m ) + k ( a )  . Note that (4.3) is pr e cisely (3.3) , sinc e e ach s ( m ) is a r epr esentative of m. W e denote the monoid homomorphism corresp onding to this action by (4.4) σ : M → End( K ) , m 7→ σ ( m ) : K → K , where σ ( m )( a ) = m • a, as in (4.3). Consider a reﬂexive graph G : R ρ 1 / / ρ 2 / / X δ o o in Mon , so that ρ 1 δ = ρ 2 δ = id X . Deﬁnition 4.4 ([8, Deﬁnition 3.0.10]) . W e say that G is a Schr eier r eﬂexive gr aph , or an S -r eﬂexive gr aph , if K 1 / / k 1 / / R ρ 1 / / / / X δ o o is a Schreier point, where ( K 1 , k 1 ) is a kernel of ρ 1 . THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 11 Since an y t w o k ernels of ρ 1 are link ed b y a unique isomorphism of monoids, this do es not dep end on the c hoice of ( K 1 , k 1 ) . Of course, if G is a (reﬂexiv e) r elation on X (meaning that ( ρ 1 , ρ 2 ) are jointly monomorphic), we hav e a corresp onding notion of an S -reﬂexive relation. W e shall make use of the following result: Prop osition 4.5 ([8, Prop osition 3.1.5]) . Any S -r eﬂexive r elation G is tr ansitive, and it is symmetric if and only if K 1 is a gr oup. A relev ant asp ect here is that, in analogy with the case of Mal’tsev categories, there is a go od notion of c entr ality of ( S -)equiv alence relations. The study of cen trality has a long story that go es back to Smith [27] for Mal’tsev v arieties and to Carb oni, P edicchio, Pirov ano [12] and Pedicc hio [25] for the general context of Mal’tsev categories. Ev entually , the notion of c onne ctor was in tro duced in [6] and pro v en to b e equiv alent to the Smith-P edicchio deﬁnition of cen trality , given in terms of double centralizing relations (see also [7], Lemma 2 . 1). Connectors can b e deﬁned v ery generally in any ﬁnitely complete category C , but they b eha ve particularly w ell in the context of Mal’tsev categories, where the uniqueness of the (ev entual) connector betw een t wo equiv alence relations R and R ′ can b e established. Recall that if R : R ρ 1 / / ρ 2 / / X δ o o and R ′ : R ′ ρ ′ 1 / / ρ ′ 2 / / X δ ′ o o are equiv alence relations on an ob ject X in a ﬁnitely complete category C , a connector b et w een the tw o is a morphism p : R × X R ′ − → X in C satisfying certain equational axioms (see [6, Deﬁnition 2.2]), where R × X R ′ denotes a pullback R × X R ′   / / R ′ ρ ′ 1   R ρ 2 / / X. When C is a Mal’tsev category - meaning that any reﬂexive relation in C is an equiv alence relation - these axioms come down to the tw o set-theoretical equations p ( a, a, z ) = z and p ( b, y , y ) = b for all a, b, y , z ∈ X such that a R ′ z and b R y . Moreo ver, if a connector b et w een R and R ′ exists, it is necessarily unique ([6, Prop osition 4.1]). Observe that, in the Mal’tsev con text, a connector is simply a restricted Mal’tsev op eration, where the restriction is to the triples ( x, y , z ) suc h that x R y R ′ z . No w, the category of monoids is certainly not a Mal’tsev category , and ev en in the case of S -reﬂexiv e relations only transitivity , but not symmetry , is automatically guaran teed (Prop osition 4.5). Nev ertheless, it is a consequence of [8, Theorem 2.4.2] that when R and R ′ are reﬂexive relations in Mon and at le ast one of the two is a Schr eier r eﬂexive r elation , the same reduced deﬁnition of a connector can b e used to deﬁne centralit y: Deﬁnition 4.6 ([8, Deﬁnition 4.3.1]) . If R : R ρ 1 / / ρ 2 / / X δ o o and R ′ : R ′ ρ ′ 1 / / ρ ′ 2 / / X δ ′ o o are reﬂexiv e relations in Mon and R ′ is a Schreier reﬂexive relation, we sa y that R and R ′ c entr alise e ach other when there exists a monoid homomorphism p : R × X R ′ − → X which satisﬁes p ( a, a, z ) = z and p ( b, y , y ) = b for all a, b, y , z ∈ X such that a R ′ z and b R y . The morphism p is in this case necessarily unique, and it is called the c onne ctor of R and R ′ . The following characterisation will b e useful: 12 S. AMBRA, A. MONTOLI, AND D. RODELO Prop osition 4.7 ([8, Prop osition 4.3.3]) . Consider a r eﬂexive r elation R and an S -r eﬂexive r elation R ′ as ab ove, wher e K ′ / / k ′ / / R ′ q ′ z z ρ ′ 1 / / / / X δ ′ o o is a Schr eier p oint with Schr eier r etr action q ′ . Then, c onsidering K ′ ⊆ X and k ′ ( t ) = (0 , t ) for t ∈ K ′ , R and R ′ c entr alise e ach other if and only if for every t ∈ K ′ and every ( x, y ) ∈ R the e quality q ′ ( y , y + t ) + x = x + t holds in X ; in this c ase, the c onne ctor is given by p  x, y , z  = q ′ ( y , z ) + x (for al l ( x, y , z ) ∈ X × X × X such that x R y R ′ z ). These Mal’tsev asp ects of Sc hreier in ternal structures ha v e b een explored at a categorical level in [10]. 5. The Chasles rela tion of a cc-Schreier extension No w we resume the study of extensions and ﬁx a Sc hreier extension E : K / / k / / X f / / / / M , with K commu tative. Deﬁne a relation (5.1) R E : R E r 1 / / r 2 / / X s 0 o o on X in Mon b y ( x, z ) ∈ R E if and only if z = k ( a ) + x for some (not necessarily unique) a ∈ K, with r 1 ( x, z ) = x, r 2 ( x, z ) = z and s 0 ( x ) = ( x, x ) . Using (3.5), w e see that this is an in ternal relation in Mon , for w e can write  x, k ( a ) + x  +  y , k ( b ) + y  =  x + y , k ( a ) + x + k ( b ) + y  =  x + y , k ( a ) + k  f ( x ) ∗ b  + x + y  =  x + y , k  a + f ( x ) ∗ b  + x + y  . Moreo ver, R E is clearly reﬂexiv e and transitiv e. Since f  k ( a ) + x  = f ( x ) for all x ∈ X and all a ∈ K , there is an induced morphism j in the kernel pair of f R E r 2 ! ! r 1 $ $ j " " Eq( f ) f 1   f 2 / / X f   X f / / M ; j is a monomorphism by the equality ⟨ f 1 , f 2 ⟩ j = ⟨ r 1 , r 2 ⟩ and it can be realised as an inclusion map. Th us R E is a sub ob ject of Eq( f ) , and it follo ws from the next prop osition that the kernel pair relation of f , denoted by E q ( f ) , is the congruence generated by the relation R E (i.e., the smallest equiv alence relation on X in Mon con taining R E ): Prop osition 5.1. Given a Schr eier extension E with K c ommutative and the r e- lation R E (5.1) , the r e gular epimorphism f is a c o e qualiser of ( r 1 , r 2 ) . THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 13 Pr o of. W e ha ve already argued that f r 1 = f r 2 . Supp ose that h : X → Z is a monoid homomorphism satisfying hr 1 = hr 2 . Giv en ( x, y ) ∈ Eq( f ) , let f ( x ) = f ( y ) = m. Then, h ( x ) (2.2) = h ( k q ( x ) + u m ) = h ( u m ) , since hr 1 = hr 2 ; similarly , h ( y ) = h ( u m ) . W e get hf 1 = hf 2 , and h factors uniquely through f b ecause f is a co equaliser of its kernel pair. □ T o give a more precise account of the situation, denote by k 2 = ⟨ k , 0 ⟩ : K ↣ Eq( f ) the k ernel of the second pro jection f 2 of the kernel pair Eq( f ) . Then w e ha ve: Prop osition 5.2. Given a Schr eier extension E with K c ommutative and the r e- lation R E (5.1) , the morphisms j and k 2 ar e jointly extr emal epimorphic. Pr o of. F or all ( x, y ) ∈ Eq( f ) w e can write ( x, y ) (2.2) =  k q ( x ) + u m , k q ( y ) + u m  = ( k q ( x ) , 0) + ( u m , k q ( y ) + u m ) = k 2 ( q ( x )) + j ( u m , k q ( y ) + u m ) , where m = f ( x ) = f ( y ) . □ Hence the follo wing characterisation of the Sc hreier extensions E for which R E is a congruence: Corollary 5.3. Given a Schr eier extension E with K c ommutative and the r elation R E (5.1) , the fol lowing statements ar e e quivalent: (1) k 2 factors thr ough j ; (2) j is an isomorphism; (3) R E is symmetric; (4) K is a gr oup. Pr o of. (1) ⇒ (2) follows from the previous prop osition, considering the diagram R E   j   K / / k 2 / / 1 1 Eq( f ) R E . o o j o o (2) ⇒ (3) is clear. (3) ⇒ (4) and (4) ⇒ (1) come from the remark that an element a ∈ K is inv ertible if and only if  k ( a ) , 0  ∈ R E , whereas  0 , k ( a )  ∈ R E is alwa ys true. □ The fact that a ∈ K is in v ertible if and only if  k ( a ) , 0  ∈ R E means in particular that we hav e a pullback square U ( K ) / / i / /   k 2   K   k 2   R E / / j / / Eq( f ) , where i is the inclusion morphism and k 2 ( a ) =  k ( a ) , 0  . In the ab ov e notation, if K is also cancellative, w e can prov e that R E is an S - reﬂexiv e relation (Deﬁnition 4.4) and that it can b e seen as a relation on the whole extension E . Lemma 5.4. If E is a c c-Schr eier extension as ab ove and ( x, y ) ∈ R E (se e (5.1) ), then y = k ( a ) + x for a unique a ∈ K. 14 S. AMBRA, A. MONTOLI, AND D. RODELO Pr o of. Indeed, supp ose that k ( a ) + x = k ( a ′ ) + x. Using (2.2) w e get k ( a ) + k q ( x ) + u f ( x ) = k ( a ′ ) + k q ( y ) + u f ( x ) . Applying Lemma 2.1(1) and the cancellativity of K, w e conclude that a = a ′ . □ Prop osition 5.5. If E is a c c-Schr eier extension as ab ove, then (5.2) K / / k 1 = ⟨ 0 ,k ⟩ / / R E r 1 / / / / X s 0 o o is a Schr eier p oint (se e (5.1) ); c onse quently, R E is an S -r eﬂexive r elation. Pr o of. It is easy to see that k 1 is the k ernel of r 1 . F or every  x, k ( a ) + x  ∈ R E w e can write  x, k ( a ) + x  =  0 , k ( a )  + ( x, x ) = k 1 ( a ) + s 0 r 1  x, k ( a ) + x  for an element a ∈ K which is unique b y Lemma 5.4, and w e conclude that (5.2) is a Sc hreier p oint. The corresponding Sc hreier retraction is the map of sets q 1 : R E 99K K, deﬁned by q 1 ( x, k ( a ) + x ) = a. □ W e denote by S E xt M the category whose ob jects are the Sc hreier extensions (2.1) with ﬁxed co domain M and whose morphisms are the morphisms (2.3) with M ′ = M and α 3 = id M . Since α 1 is uniquely determined by α 2 and the univ ersal prop- ert y of the kernel ( K ′ , k ′ ) of f ′ , we denote such morphisms as ( α 1 , α 2 ) : E → E ′ , or simply as α 2 : E → E ′ . Prop osition 5.6. If E is a c c-Schr eier extension as ab ove, then the r elation R E determines a r elation (5.3) K × K p 1   +   / / ˆ k / / R E r 1   r 2   f r 1 = f r 2 / / / / M K / / k / / X f / / / / M on E in S E xt M , wher e ˆ k ( a, b ) =  k ( a ) , k ( b ) + k ( a )  , p 1 ( a, b ) = a and + is the monoid op er ation on K. Pr o of. The fact that ˆ k is a kernel of f r 1 = f r 2 is immediate, as is the fact that + is a monoid homomorphism (b ecause K is comm utativ e). It is also easy to see that, since K is cancellative, the morphisms p 1 and + are join tly monomorphic in Mon . F or every  x, k ( b ) + x  ∈ R E w e hav e  x, k ( b ) + x  (2.2) =  k q ( x ) , k ( b ) + k q ( x )  +  u f ( x ) , u f ( x )  = ˆ k ( q ( x ) , b ) +  u f ( x ) , u f ( x )  for the unique q ( x ) ∈ K. The couple ( q ( x ) , b ) suc h that  x, k ( b ) + x  = ˆ k ( q ( x ) , b ) +  u f ( x ) , u f ( x )  is then unique b y the uniqueness of q ( x ) and the cancellativity of K (see Lemma 5.4). This prov es that the upp er row in (5.3) is a Schreier extension with represen tativ es ( u m , u m ) . It follo ws that ( p 1 , r 1 ) , (+ , r 2 ) are morphisms in S E xt M , and since b oth pairs ( p 1 , +) and ( r 1 , r 2 ) are jointly monomorphic, we conclude that (5.3) is a relation on E . □ The main p oin t, now, is the following: Prop osition 5.7. If E is a c c-Schr eier extension as ab ove, the S -r eﬂexive r elation R E (5.1) is self-c entr alizing in the sense of Deﬁnition 4.6. Pr o of. By Prop osition 4.7, it is enough to prov e that for every ( x, y ) ∈ R E and ev ery b ∈ K one has k q 1  y , y + k ( b )  + x = x + k ( b ) , where q 1 : R E 99K K is the THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 15 Sc hreier retraction of (5.2). Now, if ( x, y ) ∈ R E , w e hav e y = k ( a ) + x for a unique a ∈ K (see Lemma 5.4), and q 1  k ( a ) + x, k ( a ) + x + k ( b )  = q 1  k ( a ) + x, k ( a ) + k  f ( x ) ∗ b  + x  = q 1  k ( a ) + x, k  f ( x ) ∗ b  + k ( a ) + x  = f ( x ) ∗ b, using (3.5) and the commutativit y of K ; then indeed k q 1  k ( a ) + x, k ( a ) + x + k ( b )  + x = k  f ( x ) ∗ b  + x = x + k ( b ) , using again (3.5). □ The same Proposition 4.7 giv es us the explicit deﬁnition of the connector betw een R E and itself as p = p E : R E × X R E − → X, (5.4) p  x, k ( a ) + x, k ( b ) + k ( a ) + x  = k ( b ) + x, for any cc-Sc hreier extension E . W e stress the fact that the uniqueness of p entails that this construction dep ends only on E . R emark 5.8 . If X is a group, s o that E is a short exact sequence of groups with ab elian kernel K, by Corollary 5.3 the relation R E is simply E q ( f ) , the kernel pair relation of f , and by the ab ov e equation the connector p : Eq( f ) × X Eq( f ) ∼ = { ( x, y , z ) ∈ X × X × X : f ( x ) = f ( y ) = f ( z ) } − → X is given b y p ( x, y , z ) = p  x, k ( a ) + x, k ( b ) + k ( a ) + x  = k ( b ) + x = x − x − k ( a ) + k ( b ) + k ( a ) + x = x −  k ( a ) + x  +  k ( b ) + k ( a ) + x  = x − y + z (recalling that here w e can write y = k ( a ) + x and z = k ( b ) + y for unique elements a, b ∈ K ) . This means that in this case p is precisely the (unique, autonomous [5]) Mal’tsev op eration on the in ternal Mal’tsev algebra f in the slice category Gp / M (i.e. an ob ject endow ed with an internal Mal’tsev op eration in Gp / M ). W e shall denote for the sake of brevity P E = R E × X R E ∼ =  x, k ( a ) + x, k ( b ) + k ( a ) + x  : x ∈ X & a, b ∈ K  , so that we hav e a pullback P E p 2 / / p 1   R E r 1   R E r 2 / / X with pro jections p 1  x, k ( a ) + x, k ( b ) + k ( a ) + x  =  x, k ( a ) + x  and p 2  x, k ( a ) + x, k ( b ) + k ( a ) + x  =  k ( a ) + x, k ( b ) + k ( a ) + x  . No w we come to the main core of the construction. F or a cc-Schreier extension E , deﬁne a relation Ch E on R E b y: giv en ( x, k ( a ) + x ) , ( y , k ( b ) + y ) ∈ R E , let  x, k ( a ) + x  Ch E  y , k ( b ) + y  ⇐ ⇒  x, y , k ( b ) + y  ∈ P E & k ( a ) + x = p  x, y , k ( b ) + y  , whic h is equiv alent to  x, k ( a ) + x  Ch E  y , k ( b ) + y  ⇐ ⇒ ( x, y ) ∈ R E & k ( a ) + x = k ( b ) + x and to (5.5)  x, k ( a ) + x  Ch E  y , k ( b ) + y  ⇐ ⇒ ( x, y ) ∈ R E & a = b 16 S. AMBRA, A. MONTOLI, AND D. RODELO (using Lemma 5.4). It is an internal relation in Mon , b ecause so is R E and p is a monoid homomorphism, and it is clearly reﬂexive and transitive (using (5.5)). As a reﬂexive graph, it is represented by (5.6) Ch E : P E π 1 = ⟨ r 1 p 1 ,p ⟩ / / p 2 / / R E , σ 0 o o where π 1 = ⟨ r 1 p 1 , p ⟩ :  x, k ( a ) + x, k ( b ) + k ( a ) + x  7→  x, k ( b ) + x  and σ 0 :  x, k ( a ) + x  7→  x, x, k ( a ) + x  . W e shall call Ch E the Chasles r elation asso ciated with the cc-Schreier extension E , as it comes from Remark 5.8 that it is indeed a generalisation of the homonymous relation deﬁned in [5]. As for R E in the cancellative case, (5.6) turns out to b e an S -reﬂexiv e relation: Prop osition 5.9. If E is a c c-Schr eier extension as ab ove and R E is the r ela- tion (5.1) , then the split extension K / / κ 1 = ⟨ 0 ,k,k ⟩ / / P E π 1 / / / / R E σ 0 o o is a Schr eier p oint; c onse quently, Ch E is an S -r eﬂexive r elation. Pr o of. It is easy to see that κ 1 is the kernel of π 1 . F or ev ery  x, k ( a ) + x, k ( b ) + k ( a ) + x  ∈ P E w e can write  x, k ( a ) + x, k ( b ) + k ( a ) + x  =  x, k ( a ) + x, k ( a ) + k ( b ) + x  =  0 , k ( a ) , k ( a )  +  x, x, k ( b ) + x  = κ 1 ( a ) + σ 0  x, k ( b ) + x  = κ 1 ( a ) + σ 0 π 1  x, k ( a ) + x, k ( b ) + k ( a ) + x  , with a unique by Lemma 5.4. The corresp onding Schreier retraction is the map of sets q 1 : P E 99K K, deﬁned by q 1  x, k ( a ) + x, k ( b ) + k ( a ) + x  = a. □ By Prop osition 4.5 and Corollary 5.3, then, we hav e: Corollary 5.10. Given a c c-Schr eier extension E as ab ove, the Chasles r elation Ch E (5.6) is symmetric if and only if R E is symmetric (if and only if K is a gr oup). 6. Definition of th e direction functor for cc-Schreier extensions Fix a cc-Sc hreier extension of monoids E : K / / k / / X f / / / / M and consider the Chasles relation Ch E asso ciated with E as in (5.6), using the notation of the previous section. Let s 0 denote the unique morphism induced by the univ ersal prop ert y of the pullback ( P E , p 1 , p 2 ) as in the diagram R E s 0 r 2   s 0 ! ! P E p 2 / / p 1   R E r 1   R E r 2 / / X, s 0 O O s 0 ( x, k ( a ) + x ) = ( x, k ( a ) + x, k ( a ) + x ) . It is easily seen that P E p 1 / / π 2 = ⟨ p,r 2 p 2 ⟩ / / R E s 0 o o is also an S -reﬂexive relation on R E in Mon sharing all the prop erties of Ch E , and THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 17 that the diagram P E p 1   π 2   π 1 / / p 2 / / R E σ 0 o o r 1   r 2   R E s 0 O O r 1 / / r 2 / / X s 0 o o s 0 O O is commutativ e in the usual sense. W e know b y Prop osition 5.1 that f is a co equaliser of R E and, as Mon is co com- plete, we can also consider a co equaliser of the Chasles relation Ch E , which w e denote by γ = γ E = coeq ( π 1 , p 2 ) : R E / / / / d f ; th us (6.1) γ ( x, k ( b ) + x ) = γ ( k ( a ) + x, k ( a ) + k ( b ) + x ) , ∀ a, b ∈ K . W e get a commutativ e diagram (6.2) P E p 1   π 2   π 1 / / p 2 / / R E σ 0 o o r 1   r 2   γ / / / / d f f   R E s 0 O O r 1 / / r 2 / / X s 0 o o s 0 O O f / / / / M s O O in Mon , where f and s are uniquely induced by the universal prop ert y of the co equalisers γ and f , resp ectively , and are accordingly (w ell) deﬁned by (6.3) f  γ ( x, k ( a ) + x )  = f ( x ) and s ( m ) = γ ( x, x ) , where x ∈ X is an y elemen t satisfying f ( x ) = m. Clearly one has f s = id M , and it also follows from the commutativit y of the upw ard right-hand square in (6.2) that (6.4) γ ( x, x ) = γ ( y , y ) for all ( x, y ) ∈ Eq( f ) . Note that Ch E is not alwa ys a congruence (it is not symmetric if K is not a group), so that in general we only hav e an inclusion Ch E ⊆ E q ( γ ) . As a consequence, the implication γ  x, k ( a ) + x  = γ  y , k ( b ) + y  ⇒  x, k ( a ) + x  Ch E  y , k ( b ) + y  is, in general, false. Ho w ever, w e are able to describ e explicitly the congruence generated b y the Chasles relation, i.e. the k ernel pair relation E q ( γ ) of the co equaliser γ of Ch E : Lemma 6.1. Consider the diagr am (6.2) . The kernel p air r elation of γ c oincides with the r elation τ : T t 1 / / t 2 / / R E deﬁne d by   x, k ( a ) + x  ,  y , k ( b ) + y   ∈ T if and only if a = b and ( x, y ) ∈ Eq( f ) ; t 1 and t 2 ar e the ﬁrst and se c ond pr oje ctions, r esp e ctively. Pr o of. By Lemma 5.4, τ is w ell deﬁned. Supp ose no w that   x, k ( a ) + x  ,  y , k ( a ) + y   ∈ T and   x ′ , k ( a ′ ) + x ′  ,  y ′ , k ( a ′ ) + y ′   ∈ T . Using (3.5), we get  x, k ( a ) + x  +  x ′ , k ( a ′ ) + x ′  =  x + x ′ , k ( a ) + x + k ( a ′ ) + x ′  =  x + x ′ , k ( a ) + k ( f ( x ) ∗ a ′ ) + x + x ′  and similarly  y , k ( a ) + y  +  y ′ , k ( a ′ ) + y ′  =  y + y ′ , k ( a ) + k ( f ( y ) ∗ a ′ ) + y + y ′  , with k ( a ) + k ( f ( x ) ∗ a ′ ) = k ( a ) + k ( f ( y ) ∗ a ′ ); it is ob vious that ( x + x ′ , y + y ′ ) ∈ Eq( f ) . This 18 S. AMBRA, A. MONTOLI, AND D. RODELO pro ves that τ is an internal relation on R E in Mon . It is clear that τ is reﬂexiv e, symmetric, and transitive: thus τ is a congruence on R E . It is also easy to chec k that  x, k ( a ) + x  Ch E  k ( a ′ ) + x, k ( a ) + k ( a ′ ) + x  implies  x, k ( a ) + x  τ  k ( a ′ ) + x, k ( a ) + k ( a ′ ) + x  ; it follows that Ch E ⊆ τ , and th us E q ( γ ) ⊆ τ b y minimalit y of E q ( γ ) among all congruences on R E con taining Ch E . Con versely , consider  x, k ( a ) + x  τ  y , k ( a ) + y  . Supp ose that f ( x ) = f ( y ) = m and let u m ∈ B m . W e use (2.2), (6.4) and the commutativit y of K to get γ  x, k ( a ) + x  = γ  k q ( x ) + u m , k ( a ) + k q ( x ) + u m  = γ  k q ( y ) + k q ( x ) + u m , k q ( y ) + k ( a ) + k q ( x ) + u m  = γ  k q ( x ) + k q ( y ) + u m , k ( a ) + k q ( x ) + k q ( y ) + u m  = γ  k q ( x ) + y , k q ( x ) + k ( a ) + y  = γ  y , k ( a ) + y  . This gives τ ⊆ E q ( γ ) , so that τ = E q ( f ) . □ Corollary 6.2. Consider γ as in (6.2) . We have: (1) γ  x, k ( a ) + x  = γ  y , k ( b ) + y  if and only if a = b and ( x, y ) ∈ Eq( f ) ; (2) if ( x, y ) ∈ Eq( f ) , then γ  x, k ( a ) + x  = γ  y , k ( a ) + y  ; henc e in p articular (3) γ  x, k ( a ) + x  = γ  x, k ( b ) + x  if and only if a = b. Prop osition 6.3. Given a c c-Schr eier extension E as ab ove, the split extension (6.5) K / / κ / / d f q { { f / / / / M , s o o wher e κ ( a ) = γ  0 , k ( a )  , is a Schr eier p oint with Schr eier r etr action deﬁne d by q ( γ  x, k ( a ) + x  ) = a. Pr o of. It follows from Corollary 6.2(3) that κ in a monomorphism. Moreov er, it is easy to chec k that f κ = 0 , and if γ ( x, k ( a )+ x ) ∈ d f is such that f ( γ ( x, k ( a ) + x )) = 1 then f ( x ) = 1 = f (0) (see (6.3)): thus, by Corollary 6.2(2), w e ha v e γ ( x, k ( a ) + x ) = γ (0 , k ( a )) = κ ( a ) . The function q is well deﬁned b y Corollary 6.2(1). F or ev ery  x, k ( a ) + x  ∈ R E there is a unique decomp osition  x, k ( a ) + x  =  0 , k ( a )  +  x, x  , so that γ  x, k ( a ) + x  = γ  0 , k ( a )  + γ  x, x  = κ ( a ) + sf  γ  x, k ( a ) + x  (using the fact that γ is a monoid homomorphism and (6.3)). The uniqueness of a in γ  x, k ( a ) + x  = κ ( a ) + sf  γ  x, k ( a ) + x  is the conten t of Corollary 6.2(3). □ Denote b y S P t M the category having as ob jects all Sc hreier points (4.1), with M ﬁxed (we shall say that (4.1) is a Schreier p oint on M ), and as morphisms the morphisms of split exact sequences S : K λ 1   / / k / / B q y y λ   f / / / / M s o o S ′ : K ′ / / k ′ / / B ′ q ′ x x f ′ / / / / M s ′ o o in Mon , i.e., λ 1 , λ are monoid homomorphisms such that k ′ λ 1 = λk , f ′ λ = f and λs = s ′ . As observed in [8, Prop osition 2.3.1], it follows that λ 1 q = q ′ λ, namely the THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 19 morphism ( λ 1 , λ ) preserv es the Schreier retraction. As in the case of the morphisms in S E xt M , we shall denote these as ( λ 1 , λ ) : S → S ′ , or simply as λ : S → S ′ . By the previous prop osition, ( f , s ) is an ob ject in S P t M and f has a comm utativ e and cancellativ e k ernel K ; the next result allo ws us to understand ho w these ob jects relate to the Schreier extensions (2.1). Denote by c - S E xt M (resp., cc - S E xt M ) the full sub category of S E xt M of all Sc hreier extensions with co domain M and commutativ e (resp., comm utativ e and cancellativ e) k ernel, and similarly b y c - S P t M (resp., cc - S P t M ) the full sub category of S P t M whose ob jects are the Schreier p oints on M having a commutativ e (resp., comm utative and cancellative) k ernel. No w, recall the following classical deﬁnition: Deﬁnition 6.4. Given a category C with ﬁnite products, we say that an ob ject X of C has an internal c ommutative monoid structur e in C when there exist morphisms ω : X × X → X (called the multiplic ation ) and ε : 1 → X (called the unit ) suc h that ω ( ω × id X ) = ω ( id X × ω ) associativity axiom , ω ⟨ p 2 , p 1 ⟩ = ω comm utativity axiom , ω ⟨ ε ! X , id X ⟩ = id X unit axiom . (Here ! X stands for the terminal morphism X → 1 . ) A morphism of internal comm utative monoids betw een tw o in ternal commutativ e monoids ( X, ω , ε ) and ( X ′ , ω ′ , ε ′ ) is a morphism f : X → X ′ in C suc h that f ε = ε ′ ( f preserves the unit) and f ω = ω ′ ( f × f ) ( f preserves the multiplication). W e denote by CMon ( C ) the category of internal comm utative monoids in C . Then we hav e: Theorem 6.5. Ther e ar e e quivalenc es of c ate gories c - S P t M ∼ = CMon ( S E xt M ) ∼ = CMon ( c - S E xt M ) . When the c anc el lativity of the kernel is c onsider e d, this e quival- enc e r estricts to an e quivalenc e cc - S P t M ∼ = CMon ( cc - S E xt M ) . In order to pro ve the theorem, w e shall need the following result, whic h app ears as Prop osition 2 . 1 . 5 in [8] and which is the version for Schreier p oints of our Lemma 2.4: Lemma 6.6. Given a Schr eier p oint K / / k / / B q | | f / / / / M s o o with Schr eier r etr action q , the fol lowing pr op erties hold true: (1) q k = id K ; (2) q s = 0; (3) k q  s ( m ) + k ( a )  + s ( m ) = s ( m ) + k ( a ) for al l m ∈ M and a ∈ K ; (4) q ( b + b ′ ) = q ( b ) + q  sf ( b ) + k q ( b ′ )  for al l b, b ′ ∈ B . Pr o of of The or em 6.5. Let K / / k / / B q | | f / / / / M s o o b e a Sc hreier p oin t on M with K commutativ e. W e wan t to prov e that there is a natural commutativ e monoid structure on the extension E : K / / k / / B f / / / / M in ( c -) S E xt M . The pro duct E × E in ( c -) S E xt M can b e realised by E × E : K × K p 1   p 2   / / k × M k / / Eq( f ) f 1   f 2   f f 1 = f f 2 / / / / M E : K / / k / / B f / / / / M , 20 S. AMBRA, A. MONTOLI, AND D. RODELO where  Eq( f ) , f 1 , f 2  is a kernel pair of f and p 1 , p 2 are the obvious pro jections. It is easy to c hec k that E × E is indeed a Sc hreier extension, with represen tatives of the type ( u m , v m ) ∈ B m × B m , for every m ∈ M (here B m denotes the set of represen tatives of m for E , as usual). W e deﬁne a map µ : Eq( f ) − → B by µ ( x, y ) = k q ( x ) + y . It is a monoid ho- momorphism, b ecause µ (0 , 0) = 0 , µ ( x, y ) + µ ( x ′ , y ′ ) = k q ( x ) + y + kq ( x ′ ) + y ′ , and µ  ( x, y ) + ( x ′ , y ′ )  = µ  x + x ′ , y + y ′  = k q ( x + x ′ ) + y + y ′ = k q ( x ) + k q  sf ( x ) + k q ( x ′ )  + k q ( y ) + sf ( y ) + y ′ = k q ( x ) + k q ( y ) + k q  sf ( y ) + k q ( x ′ )  + sf ( y ) + y ′ = k q ( x ) + k q ( y ) + sf ( y ) + k q ( x ′ ) + y ′ = k q ( x ) + y + k q ( x ′ ) + y ′ , using Lemma 6.6(4), Lemma 6.6(3), (4.2) and the commutativit y of K. W e claim that (6.6) E × E : K × K (a) +   / / k × M k / / Eq( f ) (b) µ   f f 1 = f f 2 / / / / M E : K / / k / / B f / / / / M , where + denotes the monoid op eration on K (whic h is a monoid homomorphism b e- cause K is comm utativ e), is a morphism of Schreier extensions. The comm utativit y of (b) is immediate b y the deﬁnition of µ, and (a) comm utes b y Lemma 6.6(1). T o pro ve that µ preserv es the represen tatives, let ( u m , v m ) ∈ B m × B m b e a represen t- ativ e of m ∈ M for E × E . W e w ant to show that µ ( u m , v m ) = k q ( u m ) + v m ∈ B m , i.e. that for an y b ∈ f − 1 ( m ) one can write b = k ( a ) + kq ( u m ) + v m for a unique a ∈ K. As u m and v m are represen tatives of m, we hav e b = k ( a ′ ) + u m and s ( m ) = k ( a ′′ ) + v m for unique elements a ′ , a ′′ ∈ K, so that b = k ( a ′ ) + u m (4.2) = k ( a ′ ) + k q ( u m ) + s ( m ) = k ( a ′ ) + k q ( u m ) + k ( a ′′ ) + v m = k ( a ′ + a ′′ ) + k q ( u m ) + v m ( K is commutativ e) = k ( a ) + k q ( u m ) + v m with a = a ′ + a ′′ . Even tually , if k ( a ) + k q ( u m ) + v m = k ( a ) + k q ( u m ) + v m for some other a ∈ K, then a + q ( u m ) = a + q ( u m ) by Lemma 2.1(1), and hence a = a b ecause q ( u m ) is in vertible in K , b y Lemma 2.1(2) (as u m = k q ( u m ) + s ( m ) and b oth u m and s ( m ) are representativ es of m ). W e now prov e that the morphism µ : E × E → E gives a m ultiplication on E in ( c -) S E xt M . First, µ is associative and commutativ e. Indeed, denote µ ( x, y ) = x · y for all ( x, y ) ∈ Eq( f ); then we hav e: • Asso ciativity : x · ( y · z ) = k q ( x ) +  k q ( y ) + z  and ( x · y ) · z = kq  k q ( x ) + y  + z =  k q ( x ) + k q ( y )  + z (b y Lemma 6.6); • Commutativity : x · y = k q ( x ) + y (4.2) = k q ( x ) + k q ( y ) + sf ( y ) = k q ( y ) + k q ( x ) + sf ( x ) (4.2) = k q ( y ) + x = y · x (b y the commutativit y of K and the fact that f ( x ) = f ( y )). THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 21 The unit on the extension E is given b y s : 1 → E (6.7) 1 : 0 0 K   / / / / M s   M E : K / / k / / B f / / / / M (observ e that 0 / / / / M M is a terminal ob ject in ( c -) S E xt M and that s preserv es the representativ es { m } of m for 1 ; cf. Example 2.10.2). Indeed, the section s of f acts as a neutral element for µ : for any b ∈ B , µ ⟨ sf , id B ⟩ ( b ) = µ ( sf ( b ) , b ) = k q sf ( b ) + b = 0 + b = b, by applying Lemma 6.6 (see the unit axiom in Deﬁnition 6.4). W e conclude that E is an internal commutativ e monoid in ( c -) S E xt M with mul- tiplication (6.6) and unit (6.7). The functoriality of this construction is readily established by observing that if K λ 1   / / k / / B q { { λ   f / / / / M s o o K ′ / / k ′ / / B ′ q ′ z z f ′ / / / / M s ′ o o is a morphism of Schreier p oin ts, then K λ 1   / / k / / B λ   f / / / / M K ′ / / k ′ / / B ′ f ′ / / / / M is a morphism of Schreier extensions (by Prop osition 2.8, since λs = s ′ ), which is a morphism in CMon (( c -) S E xt M ) with respect to the monoid structures that w e ha ve deﬁned ab o ve. Indeed, λs = s ′ , so that the unit is preserv ed, and for all ( x, y ) ∈ Eq( f ) λ  µ ( x, y )  = λ  k q ( x ) + y  = k ′ λ 1 q ( x ) + λ ( y ) = k ′ q ′ λ ( x ) + λ ( y ) = µ ′  λ ( x ) , λ ( y )  = µ ′ ( λ × M λ )( x, y ) , so that the multiplication is preserved; cf. Deﬁnition 6.4. Con versely , supp ose that E : K / / k / / B f / / M is an internal commutativ e monoid in S E xt M with multiplication ω : E × E → E E × E : K × K ω 1   / / k × M k / / Eq( f ) ω   f f 1 = f f 2 / / / / M E : K / / k / / B f / / / / M 22 S. AMBRA, A. MONTOLI, AND D. RODELO and unit s : 1 → E 1 : 0   / / / / M s   M E : K / / k / / B f / / / / M . Then, as f s = id M and s preserves the representativ es { m } of m for 1 , the split extension K / / k / / B f / / / / M s o o is a Schreier p oin t on M with Schreier re- traction q : B 99K K deﬁned by q ( b ) = q f ( b ) ,sf ( b ) ( b ) , for all b ∈ B (see Remark 4.2). Moreo ver, for all a ∈ K w e hav e ω  0 , k ( a )  = ω  sf k ( a ) , k ( a )  = k ( a ) by the unit ax- iom of Deﬁnition 6.4. The comm utativit y of ω gives ω  k ( a ) , 0  = ω  0 , k ( a )  = k ( a ) . W e deduce that ω 1 (0 , a ) = ω 1 ( a, 0) = a. Then, as ω 1 is a monoid homomorphism, b y the Ec kmann-Hilton argument w e conclude that ω 1 = + , the monoid op eration on K , which is thus commutativ e; consequen tly , E is an ob ject of c - S E xt M . If ( x, y ) ∈ Eq( f ) , f ( x ) = f ( y ) = m, we ha v e ω ( x, y ) = ω  k q ( x ) + s ( m ) , y  = ω   k q ( x ) , 0  +  s ( m ) , y   = ω  k q ( x ) , 0  + ω  s ( m ) , y  = k q ( x ) + ω  sf ( y ) , y  = k q ( x ) + y = µ ( x, y ); here we used (4.2) and the unit axiom of Deﬁnition 6.4. This prov es that the ab ov e given m ultiplication ω : E × E → E coincides with the deduced one in (6.6). Ev entually , it is clear that when the cancellativity of the kernels is also con- sidered, this equiv alence restricts to an equiv alence cc - S P t M ∼ = CMon ( cc - S E xt M ) . □ R emark 6.7 . Observe that if K / / k / / B q | | f / / / / M s o o is a Schreier p oin t on M with K commutativ e and cancellative, the internal multiplication µ ( x, y ) = k q ( x ) + y deﬁned on the cc-Sc hreier extension E : K / / k / / B f / / / / M giv en in Theorem 6.5 can b e expressed in terms of the connector p : P E − → X as µ ( x, y ) = p  y , k q ( y ) + y , k q ( x ) + k q ( y ) + y  (see (5.4)). Corollary 6.8. Given the Schr eier p oint (6.5) , ther e r esults a natur al internal c ommutative monoid structur e on the Schr eier extension E : K / / κ / / d f f / / / / M as in (6.5) , whose unit is s : 1 → E and whose multiplic ation is the morphism µ : E × E → E , wher e µ : Eq( f ) − → d f is given by (6.8) µ  γ ( x, k ( a ) + x ) , γ ( y, k ( b ) + y )  = γ  x, k ( a )+ k ( b ) + x  (= γ  y , k ( a )+ k ( b ) + y  ) . Pr o of. The result follows immediately from Prop osition 6.3, Theorem 6.5 and Co- rollary 6.2(2). □ When no am biguity o ccurs, w e will denote this monoid op eration on d f simply b y µ  γ ( x, y ) , γ ( z , w )  = γ ( x, y ) · γ ( z , w ) . THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 23 R emark 6.9 . It is easy to prov e that with respect to this monoid operation µ on d f , γ satisﬁes the so-called Chasles identities γ ( x, x ) = 1 and γ ( x, y ) · γ ( y, z ) = γ ( x, z ) (cf. [5]). Indeed, the former is immediate by the deﬁnition of the unit s : 1 → E (since s ( m ) = γ ( x, x ) , for any m = f ( x ) ∈ M ), and for the latter we ha ve γ  x, k ( a ) + x  · γ  k ( a ) + x, k ( b ) + k ( a ) + x  = γ  x, k ( a ) + x  · γ  k ( a ) + x, k ( a ) + k ( b ) + x  = γ  x, k ( a ) + x  · γ  x, k ( b ) + x  = γ  x, k ( a ) + k ( b ) + x  , using the commutativit y of K , and the equalities (6.1) and (6.8). The construction (6.2) results in an asso ciation of ob jects (6.9) d : cc - S E xt M − → CMon ( cc - S E xt M ) , E 7− → d ( E ) = ( E , µ : E × E → E , s : 1 → E ) whic h we claim to b e functorial. Observe that for every morphism α : E → E ′ (6.10) E : K α 1   / / k / / X α   f / / / / M E ′ : K ′ / / k ′ / / X ′ f ′ / / / / M , w e deduce tw o monoid homomorphisms R ( α ) : R E − → R E ′ ,  x, k ( a ) + x  7− →  α ( x ) , αk ( a ) + α ( x )  =  α ( x ) , k ′ α 1 ( a ) + α ( x )  and P ( α ) : P E − → P E ′ ,  x, k ( a ) + x, k ( b ) + k ( a ) + x  7− →  α ( x ) , αk ( a ) + α ( x ) , αk ( b ) + αk ( a ) + α ( x )  = =  α ( x ) , k ′ α 1 ( a ) + α ( x ) , k ′ α 1 ( b ) + k ′ α 1 ( a ) + α ( x )  making the left-hand side of the following diagram commute P E P ( α )   π 1 / / p 2 / / R E R ( α )   γ / / / / d f d ( α )   P E ′ π ′ 1 / / p ′ 2 / / R E ′ γ ′ / / / / d f ′ . By the universal prop ert y of the co equaliser γ , this induces a unique morphism d ( α ) : d f → d f ′ in Mon such that d ( α ) γ = γ ′ R ( α ) . In terms of elemen ts, w e hav e d ( α )  γ ( x, k ( a ) + x )  = γ ′  α ( x ) , k ′ α 1 ( a ) + α ( x )  . Observ e now that with this deﬁnition of d ( α ) the diagram K α 1   / / κ / / d f d ( α )   f / / / / M s o o K ′ / / κ ′ / / d f ′ f ′ / / / / M s ′ o o is commutativ e, so that d ( α ) is a morphism in cc - S P t M , and consequently , by The- orem 6.5, it determines a morphism d ( α ) : d ( E ) → d ( E ′ ) in CMon ( cc - S E xt M ) . It is clear that this deﬁnition of d on the morphisms α : E → E ′ preserv es comp ositions and iden tities, allowing us to conclude that (6.9) is indeed a functor, which w e call 24 S. AMBRA, A. MONTOLI, AND D. RODELO the dir e ction functor for cc-Schreier extensions. W e are en titled to call it so, b e- cause we kno w by Remark 5.8 that when E is an extension of groups, the connector p = p E is the unique autonomous Mal’tsev operation associated with f ∈ Gp / M , and in this case our functor d coincides, by construction, with the classical direction functor [5] applied to the slice category Gp / M . As it happ ens for the aforementioned case of groups, it follows b y Theorem 6.5 that the functor (6.9) admits an explicit description in terms of monoid actions . Recall by Prop osition 4.3 that any Sc hreier p oint K / / k / / B q | | f / / / / M s o o determines an action σ : M → End( K ) of M on K deﬁned b y σ ( m )( a ) = m • a = q  s ( m ) + k ( a )  (see (4.4)), which allows to describ e B as the semidirect pro duct B ∼ = K ⋊ σ M thanks to the Split Short Five Lemma (v alid for Sc hreier points, see [8, Corollary 2.3.8]). More precisely , we ha ve an isomorphism in S P t M K / / ⟨ id K , 0 ⟩ / / K ⋊ σ M π K y y φ   π M / / / / M ⟨ 0 ,id M ⟩ o o K / / k / / B q y y f / / / / M , s o o where K ⋊ σ M = ( K × M , +) (the monoid op eration is giv en by ( a, m ) + ( a ′ , m ′ ) = ( a + m • a ′ , m · m ′ ) , for a, a ′ ∈ K and m, m ′ ∈ M ), and the morphism φ : K ⋊ σ M → B , φ ( a, m ) = k ( a ) + s ( m ) , is an isomorphism in Mon . When K is commutativ e, one says that ( K , σ ) is an M -semimo dule . W e denote b y S mod M the category of M -semimodules a nd action preserving homomorphisms: h : ( K, σ ) → ( K ′ , σ ′ ) is a morphism in S mod M if h : K → K ′ is a monoid homo- morphism such that the following diagram commutes M × K / / id M × h   K h   M × K ′ / / K ′ , i.e., h ( σ ( m )( a )) = σ ′ ( m )( h ( a )) for all a ∈ K and m ∈ M . It is easy to chec k that c - S P t M ∼ = S mod M . The equiv alence is obtained b y asso ciating with every Schreier p oin t K / / k / / B q | | f / / / / M s o o with comm utativ e K the M -semimodule ( K , σ ) , where σ is deﬁned in (4.4), and, conv ersely , with every M -semimodule ( K , σ ) the Schreier point K / / ⟨ id K , 0 ⟩ / / K ⋊ σ M π K y y π M / / / / M ⟨ 0 ,id M ⟩ o o (with π K and π M pro jections). In our case, we hav e: Prop osition 6.10. Given a c c-Schr eier extension E as ab ove, the monoid action of M on K induc e d by the Schr eier p oint (6.5) c oincides with the monoid action (3.2) induc e d by E . Pr o of. By Prop osition 6.3, the Schreier retraction of (6.5) is given by q ( γ ( x, k ( a ) + x )) = a. Hence, for all m = f ( x ) ∈ M and a ∈ K, the asso ciated M -action on K is m • a = q  s ( m ) + κ ( a )  = q γ  x, x + k ( a )  = q γ  x, k ( m ∗ a ) + x  = m ∗ a (using (3.5)), the same as in (3.2). □ THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 25 If we denote by η : M → End( K ) the monoid homomorphism which giv es the action (3.2) induced by E (see (3.4)), it follows that d f ∼ = K ⋊ η M via φ : K ⋊ η M − → d f , ( a, m ) 7→ κ ( a ) + s ( m ) = γ  x, k ( a ) + x  , where x ∈ X is any elemen t suc h that f ( x ) = m. W e conclude that the direction functor deﬁned in (6.9) can b e in terpreted as the functor d : cc - S E xt M − → S mod M asso ciating with eac h cc- Sc hreier extension E the M -semimo dule ( K, η ) , and with each morphism (6.10) of cc-Sc hreier extensions, the morphism of M -semimo dules α 1 : ( K, η ) → ( K ′ , η ′ ) . Indeed, this morphism is action preserving, meaning that α 1 ( m ∗ a ) = m ∗ ′ α 1 ( a ) , for all a ∈ K and m ∈ M , thanks to the fact that in (6.10) α preserves the represen tatives. 7. Proper ties of the direction functor Here we collect the main prop erties of the direction functor d introduced in the previous section. W e shall freely use b oth descriptions of d, either as the functor d : cc - S E xt M − → CMon ( cc - S E xt M ) ∼ = cc - S P t M E : K / / k / / X f / / / / M 7− → K / / κ / / d f q { { f / / / / M , s o o attac hing the Sc hreier point (6.5) to any cc-Schreier extension E or as the action functor d : cc - S E xt M − → S mod M E : K / / k / / X f / / / / M 7− → ( K, η : M → End( K )) asso ciating with E the M -semimo dule structure on K given by the monoid ac- tion (3.2). Observ e that the functor d : cc - S E xt M − → S mod M can b e deﬁned, more gen- erally , on the full sub category smod - S E xt M ⊆ c - S E xt M of all Schreier extensions inducing an M -semimo dule structure, namely Schreier extensions (2.1) with com- m utative K such that the map (3.2) is an action (indeed, w e kno w from Section 3 that the cancellativity of the kernel K is only a suﬃcient condition for this to hap- p en). In the remainder of the section, we shall p oin t out the go od prop erties of b oth d and this broadened functor D : smod - S E xt M − → S mod M (suc h that d = D on cc - S E xt M ). Prop osition 7.1. The functors d and D ar e c onservative. Pr o of. Consider a morphism α : E → E ′ as in (6.10) in cc - S E xt M suc h that K α 1   / / κ / / d f d ( α )   f / / / / M s o o K ′ / / κ ′ / / d f ′ f ′ / / / / M s ′ o o is an isomorphism in cc - S P t M . Then, in particular, α 1 is an isomorphism of mo- noids, and it follo ws from the Short Fiv e Lemma (Prop osition 2.9) applied to (6.10) that α is also an isomorphism. Observ e that if u ′ m = α ( u ) ∈ X ′ is a representativ e, then u ∈ X is also a representativ e, so that α − 1 preserv es the representativ es. This sho ws that d is conserv ative; the same pro of works for D , as well. □ In fact, the conserv ativit y of d and D can b e seen as a consequence of the follo wing result: 26 S. AMBRA, A. MONTOLI, AND D. RODELO Prop osition 7.2. The functors d and D pr eserve and r eﬂe ct monomorphisms and r e gular epimorphisms. The pro of is just an application of the Short Five Lemma in c - S E xt M and the Split Short Fiv e Lemma in c - S P t M , once that the nature of monomorphisms and regular epimorphisms in b oth categories is established. The full detailed pro of is giv en in App endix A. Prop osition 7.3. The dir e ction functors d and D pr eserve ﬁnite pr o ducts. Pr o of. Recall that the trivial extension 1 : 0 / / / / M M is a terminal ob ject in cc - S E xt M , and observ e that giv en t wo cc-Schreier extensions E : K / / k / / X f / / / / M , E ′ : K ′ / / k ′ / / X ′ f ′ / / / / M , their pro duct in cc - S E xt M is given by (7.1) E × E ′ : K × K ′ / / k × M k ′ / / X × M X ′ f ′ p 2 = f p 1 / / / / M , with the obvious pro jections on E and E ′ , where X × M X ′ p 2 / / p 1   X ′ f ′   X f / / M is a pullbac k. (By realising X × M X ′ ∼ =  ( x, x ′ ) ∈ X × X ′ : f ( x ) = f ′ ( x ′ )  , (7.1) is a Sc hreier extension with representativ es ( u m , u ′ m ) , where u m is a represen tative of m for E and u ′ m is a representativ e of m for E ′ . ) No w, consider the direction functor in its action guise d : cc - S E xt M − → S mod M , so that d ( E ) = ( K , η ) , where η is the induced monoid homomorphism action (3.4); let d ( E ′ ) = ( K ′ , η ′ ) . Then d ( 1 ) is the trivial M -semimo dule (0 , ! M ) , whic h is clearly a terminal ob ject in S mod M . Supp ose next that d ( E × E ′ ) = ( K × K ′ , ν ) , where ν : M → End( K × K ′ ) is the monoid homomorphism induced by the pro duct Sc hreier extension (7.1): thus, for any m ∈ M , ν ( m ) : K × K ′ − → K × K ′ , ( a, a ′ ) 7→ ( b, b ′ ) , with ( u m , u ′ m ) + k × M k ′ ( a, a ′ ) = k × M k ′ ( b, b ′ ) + ( u m , u ′ m ) ⇔ ( u m , u ′ m ) +  k ( a ) , k ′ ( a ′ )  =  k ( b ) , k ′ ( b ′ )  + ( u m , u ′ m ) . It follo ws that u m + k ( a ) = k ( b ) + u m and u ′ m + k ′ ( a ′ ) = k ′ ( b ′ ) + u ′ m , i.e. b = η ( m )( a ) and b ′ = η ′ ( m )( a ′ ) . W e conclude that ν ( m )( a, a ′ ) = ( η ( m )( a ) , η ′ ( m )( a ′ )) , which means that ν is the pro duct action η × η ′ giv en, more precisely , by the monoid homomorphism M ⟨ id M ,id M ⟩ / / M × M η × η ′ / / End( K ) × End( K ′ ) / / / / End( K × K ′ ) . This prov es that d ( E × E ′ ) ∼ = d ( E ) × d ( E ′ ) . Observe that cancellativit y is used no where in the pro of, so the same holds for D . □ W e come to the main property of d and D , namely the fact that they are coﬁbra- tions. Recall the general deﬁnition of the latter: THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 27 Deﬁnition 7.4. Let F : U → V b e a functor. (1) W e say that a morphism f : Y → X in U is c o c artesian over a morphism α : J → I in V if F f = α and, for every morphism g : Y → Z in U such that F g = β α in V for some β : I → F Z Y f / / g   X Z J F g   α = F f / / I ∃ β ~ ~ F Z , one has g = hf for a unique h : X → Z in U such that F h = β Y f / / g   X ∃ ! h ~ ~ Z J F g   α = F f / / I β = F h } } F Z . A morphism f in U is c o c artesian (for F ) if it is co cartesian ov er α = F f . (2) F : U → V is a c oﬁbr ation if for every morphism α : J → I in V and for ev ery ob ject Y in the ﬁbre F − 1 ( J ) there exists a co cartesian morphism in U ov er α whose domain is Y . (This notion, together with its dual notion of ﬁbr ation , go es back to A. Grothen- diec k [14]. See also [3], App endix A. 7 . ) Fix a Schreier extension E : K / / k / / X f / / / / M in smod - S E xt M , inducing the action η as in (3.4), and consider a morphism of M -semimo dules α 1 : ( K , η ) = D ( E ) − → ( K ′ , η ′ ): th us, α 1 is a monoid homomorphism satisfying α 1  η ( m )( a )  = η ′ ( m )  α 1 ( a )  for every m ∈ M and a ∈ K. W e shall pro ceed by constructing the so-called pushforwar d of E along α 1 , whic h will giv e a Schreier extension E ′ and a co cartesian morphism α : E → E ′ in smod - S E xt M ab o v e α 1 : D ( E ) → ( K ′ , η ′ ) (see (7.3)). First, observe that b y composing the action η ′ : M − → End( K ′ ) with f we get an action ψ = η ′ f : X − → End( K ′ ) of X on K ′ suc h that (7.2) ψ ( x )( a ′ ) = η ′ ( f ( x ))( a ′ ) , for all x ∈ X and a ′ ∈ K ′ . W e can then form the semidirect pro duct K ′ ⋊ ψ X and consider the square K k / / α 1   X ⟨ 0 ,id X ⟩   K ′ ⟨ id K ′ , 0 ⟩ / / K ′ ⋊ ψ X, whic h is not commutativ e, in general. Deﬁne  X ′ , r : K ′ ⋊ ψ X / / / / X ′ ) to b e a co equaliser of ⟨ id K ′ , 0 ⟩ α 1 and ⟨ 0 , id X ⟩ k in Mon , so that X ′ ∼ = ( K ′ ⋊ ψ X ) / ∼ where ∼ is the in ternal equiv alence relation on K ′ ⋊ ψ X generated by  α 1 ( a ) , 0  ∼  0 , k ( a )  for all a ∈ K. W e ha ve: Lemma 7.5 (Cf. [24]) . The ab ove r elation ∼ c oincides with the r elation ρ on K ′ ⋊ ψ X deﬁne d by  a ′ , x  ρ  b ′ , y  if and only if ( x, y ) ∈ Eq( f ) and a ′ + α 1 ( q ( x )) = b ′ + α 1 ( q ( y )) , wher e x = k q ( x ) + u m and y = k q ( y ) + u m with r esp e ct to some r epr esentative u m of m = f ( x ) = f ( y ) (using (2.2) ). 28 S. AMBRA, A. MONTOLI, AND D. RODELO Pr o of. First observ e that ρ is well deﬁned, meaning that it do es not dep end up on the choice of u m , b ecause an y t w o representativ es diﬀer by an in vertible element of K (Lemma 2.1) and the image of an inv ertible element under the monoid homo- morphism α 1 is still inv ertible in K ′ . Moreo ver, ρ is clearly an equiv alence relation on K ′ ⋊ ψ X, and it is sho wn in [24, Prop osition 4.7] that ρ is an internal relation in Mon (using (3.5)). Observe also that for every a ∈ K one has  α 1 ( a ) , 0  ρ  0 , k ( a )  , b y choosing 0 ∈ X as a representativ e of 1 ∈ M , so that ∼ is contained in ρ. Con versely , supp ose that  a ′ , x  ρ  b ′ , y  and write x = k q ( x ) + u m , y = k q ( y ) + u m , where m = f ( x ) = f ( y ) . Then  a ′ + α 1 ( q ( x )) , u m  ∼  b ′ + α 1 ( q ( y )) , u m  b ecause ∼ is reﬂexive, and  a ′ + α 1 ( q ( x )) , u m  =  α 1 ( q ( x )) , 0  +  a ′ , u m  ∼  0 , k q ( x )  +  a ′ , u m  =  a ′ , x  (using the commutativit y of K ′ , the generators of ∼ and the deﬁnition of the monoid op eration on K ′ ⋊ ψ X ). Similarly ,  b ′ + α 1 ( q ( y )) , u m  ∼  b ′ , y  , which allows us to conclude that  a ′ , x  ∼  b ′ , y  . □ Then, we can compute the co equaliser ( X ′ , r ) as K ′ ⋊ ψ X r − − → X ′ = ( K ′ ⋊ ψ X ) /ρ, ( a ′ , x ) 7→ [( a ′ , x )] ρ , and we ha v e a commutativ e square K / / k / / α 1   X α = r ⟨ 0 ,id X ⟩   K ′ k ′ = r ⟨ id K ′ , 0 ⟩ / / X ′ . Observ e that the morphism k ′ = r ⟨ id K ′ , 0 ⟩ : K ′ → X ′ is a monomorphism, b e- cause it follo ws from the deﬁnition of ρ that ( a ′ , 0) ρ ( b ′ , 0) if and only if a ′ = b ′ . Moreo ver, if π X denotes the pro jection morphism K ′ ⋊ ψ X → X , ( a ′ , x ) 7→ x, the equalit y f π X ⟨ 0 , id X ⟩ k = f k = 0 = f π X ⟨ id K ′ , 0 ⟩ α 1 and the universal property of the co equaliser guarantee that a unique f ′ exists as in the diagram K ⟨ 0 ,id X ⟩ k / / ⟨ id K ′ , 0 ⟩ α 1 / / K ′ ⋊ ψ X f π X     r / / / / X ′ f ′ w w M satisfying f π X = f ′ r . Explicitly , f ′ is deﬁned by f ′  [( a ′ , x )] ρ  = f ( x ) , and observe that as f and π X are regular epimorphisms in Mon , so is f ′ . Consequently , w e ha v e K er ( f ′ ) = { [( a ′ , x )] ρ ∈ X ′ : x ∈ K er ( f ) } = k ′ ( K ′ ) , where k ′ = r ⟨ id K ′ , 0 ⟩ , b ecause  a ′ , k ( a )  ρ =  a ′ + α 1 ( a ) , 0  ρ = k ′  a ′ + α 1 ( a )  for every a ∈ K and a ′ ∈ K ′ , and as we already know that k ′ is a monomorphism we conclude that ( K ′ , k ′ ) is a k ernel of f ′ . W e obtain a monoid extension E ′ : K ′ / / k ′ / / X ′ f ′ / / / / M (whic h is the push- forw ard of E along α 1 ) and we see that the diagram (7.3) E : K α 1   / / k / / X α   f / / / / M E ′ : K ′ / / k ′ / / X ′ f ′ / / / / M comm utes (b ecause f ′ α = f ′ r ⟨ 0 , id X ⟩ = f π X ⟨ 0 , id X ⟩ = f ). Lemma 7.6. The se quenc e E ′ and t he morphism (7.3) b elong to smod - S E xt M and, mor e over, D ( E ′ ) = ( K ′ , η ′ ) . THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 29 Pr o of. Every [( a ′ , x )] ρ ∈ X ′ , where x = k q ( x ) + u f ( x ) b y (2.2), can be written as [( a ′ , x )] ρ = [( a ′ + α 1 ( q ( x )) , u f ( x ) )] ρ = [( a ′ + α 1 ( q ( x )) , 0)] ρ + [(0 , u f ( x ) )] ρ = k ′ ( a ′ + α 1 ( q ( x ))) + [(0 , u f ( x ) )] ρ , and it follows from the deﬁnition of ρ that if [( a ′ , x )] ρ = k ′ ( b ′ ) + [(0 , u f ( x ) )] ρ = [( b ′ , u f ( x ) )] ρ for some other b ′ ∈ K ′ , then a ′ + α 1 ( q ( x )) = b ′ . Th us E ′ is a Schreier extension with representativ es [(0 , u m )] ρ = α ( u m ); con- sequen tly α : E → E ′ is a morphism of Sc hreier extensions (see Deﬁnition 2.7). The only thing that w e are left to pro v e is that the map M × K ′ 99K K ′ asso ciated with E ′ as in (3.2) corresp onds to the one induced from η ′ . But this is true, b ecause M × K ′ 99K K ′ is deﬁned b y ( m, a ′ ) 7→ a ′′ , where a ′′ is the unique element of K ′ suc h that [(0 , u m )] ρ + k ′ ( a ′ ) = k ′ ( a ′′ ) + [(0 , u m )] ρ . This gives [(0 , u m )] ρ + [( a ′ , 0)] ρ = [( a ′′ , 0)] ρ + [(0 , u m )] ρ ⇔ [( ψ ( u m )( a ′ ) , u m )] ρ = [( a ′′ , u m )] ρ ⇔ ψ ( u m )( a ′ ) = a ′′ ⇔ η ′ ( m )( a ′ ) = a ′′ , using the deﬁnition of ρ in Lemma 7.5 and (7.2). □ W e claim that (7.3) is a co cartesian morphism ov er α 1 , in the sense of Deﬁn- ition 7.4, for the functor D : smod - S E xt M − → S mod M . Indeed, supp ose that a morphism E : K λ 1   / / k / / X λ   f / / / / M F : L / / l / / Y g / / / / M is given in smod - S E xt M , such that λ 1 = β 1 α 1 for some β 1 : ( K ′ , η ′ ) → ( L, ν ) in S mod M ( K, η ) α 1 / / λ 1   ( K ′ , η ′ ) β 1 z z ( L, ν ) , where ( L, ν ) = D ( F ) . W e deﬁne h : K ′ ⋊ ψ X − → Y , ( a ′ , x ) 7→ lβ 1 ( a ′ ) + λ ( x ) , which is a monoid homomorphism b ecause h  ( a ′ , x ) + ( b ′ , y )  = h  a ′ + ψ ( x )( b ′ ) , x + y  = h  a ′ + η ′ ( f ( x ))( b ′ ) , x + y  = lβ 1  a ′ + η ′ ( f ( x ))( b ′ )  + λ ( x + y ) = lβ 1 ( a ′ ) + l β 1  η ′ ( f ( x ))( b ′ )  + λ ( x ) + λ ( y ) = lβ 1 ( a ′ ) + l  ν ( f ( x ))( β 1 ( b ′ ))  + λ ( x ) + λ ( y ) = lβ 1 ( a ′ ) + l  ν ( g λ ( x ))( β 1 ( b ′ ))  + λ ( x ) + λ ( y ) = lβ 1 ( a ′ ) + λ ( x ) + l β 1 ( b ′ ) + λ ( y ) = h ( a ′ , x ) + h ( b ′ , y ) (using (7.2), the fact that β 1 is a morphism in S mod M and (3.5)). W e ha ve then h ⟨ 0 , id X ⟩ k = h ⟨ id K ′ , 0 ⟩ α 1 , b ecause λk = l λ 1 = lβ 1 α 1 . By the universal prop ert y of 30 S. AMBRA, A. MONTOLI, AND D. RODELO the co equaliser r, a unique β : X ′ → Y is induced (7.4) K ⟨ 0 ,id X ⟩ k / / ⟨ id K ′ , 0 ⟩ α 1 / / K ′ ⋊ ψ X h   r / / / / X ′ β w w Y satisfying h = β r, explicitly (well) deﬁned by β  [( a ′ , x )] ρ  = h ( a ′ , x ) = lβ 1 ( a ′ ) + λ ( x ) . The diagram (7.5) E ′ : K ′ β 1   / / k ′ / / X ′ β   f ′ / / / / M F : L / / l / / Y g / / / / M is also commutativ e, b ecause β k ′ ( a ′ ) = β  [( a ′ , 0)] ρ  = lβ 1 ( a ′ ) and g β  [( a ′ , x )] ρ  = g  lβ 1 ( a ′ ) + λ ( x )  = 0 + g λ ( x ) = f ( x ) = f ′  [( a ′ , x )] ρ  for every a ′ ∈ K ′ and x ∈ X . Moreo ver, β preserv es the representativ es because, by assumption, λ does and β  [(0 , u m )] ρ  = λ ( u m ): thus (7.5) is a morphism in smod - S E xt M , and as for ev ery x ∈ X the equality β α ( x ) = β  [(0 , x )] ρ  = λ ( x ) holds (so that β α = λ ), the triangle (7.6) E ( α 1 ,α ) / / ( λ 1 ,λ )   E ′ ( β 1 ,β ) } } F comm utes in smod - S E xt M . T o pro ve the uniqueness of β in (7.6), supp ose that for some other morphism E ′ : K ′ β 1   / / k ′ / / X ′ β   f ′ / / / / M F : L / / l / / Y g / / / / M in smod - S E xt M w e hav e β α = λ. Then for all ( a ′ , x ) ∈ K ′ ⋊ ψ X β r ( a ′ , x ) = β  [( a ′ , x )] ρ  = β  [( a ′ , 0)] ρ + [(0 , x )] ρ  = β  [( a ′ , 0)] ρ  + β  [(0 , x )] ρ  = β k ′ ( a ′ ) + β α ( x ) = lβ 1 ( a ′ ) + λ ( x ) = h ( a ′ , x ) , so that β = β by the uniqueness of β in (7.4). Th us we hav e prov ed: Theorem 7.7. The functors D : smod - S E xt M − → S mod M and d : cc - S E xt M − → c - S mod M , wher e c - S mod M ⊆ S mod M is the ful l sub c ate gory of al l M -semimo dules ( K, η ) with K c anc el lative, ar e c oﬁbr ations. THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 31 (It is immediate that the same pro of given for D : smod - S E xt M − → S mod M also works for d : cc - S E xt M − → c - S mod M . ) The imp ortance of this fact comes from the following classical result: Theorem 7.8. L et U b e a c ate gory with ﬁnite pr o ducts and F : U − → V a pr o duct pr eserving c oﬁbr ation. Supp ose also that the c o c artesian morphisms in U ar e stable under ﬁnite pr o ducts. If ( M , ω : M × M − → M , ε : 1 − → M ) is an internal monoid in V , ther e r esults a monoidal structur e on the ﬁbr e F − 1 ( M ) , which is symmetric as so on as the internal monoid M is c ommutative. (See for example [5, Theorem 9] for a pro of.) Observ e that both functors D : smod - S E xt M − → S mod M and d : cc - S E xt M − → c - S mod M fall under the scope of Theorem 7.8, because, for a coﬁbration, b eing con- serv ative is equiv alent to the fact that every morphism in the domain category is co cartesian. Moreov er, observe that S mod M ∼ = CMon ( S mod M ) , b ecause every M -semimodule ( K , η ) is an internal monoid in S mod M whose in ternal monoid op- eration in S mod M is the monoid op eration + : K × K → K of the comm utativ e monoid K itself (the commutativit y of K guarantees that + is a monoid homomor- phism, and it is action-preserving by the action axioms on η ). Similarly , w e ha v e c - S mod M ∼ = CMon ( c - S mod M ) . By Theorem 7.8, we hav e: Corollary 7.9. F or every obje ct ( K, η ) in S mod M (r esp., in c - S mod M ) ther e r esults a symmetric monoidal structur e on the ﬁbr e D − 1 ( K, η ) of the dir e ction functor D : smod - S E xt M − → S mod M (r esp., of the functor d : cc - S E xt M − → c - S mod M ). Since it is kno wn from [5, Section VI] that, in the case of group extensions of a group G by a G -mo dule ( A, η ) , the resulting tensor pro duct on the ﬁbres of the corresponding direction functor for groups coincides with the usual Baer sum of extensions, we are entitled to call the tensor pro duct E ⊗ E ′ resulting on the ﬁbres d − 1 ( K, η ) of the functor d : cc - S E xt M − → c - S mod M the Baer sum of the cc-Sc hreier extensions E , E ′ (and similarly for D : smod - S E xt M − → S mod M ). Denoting by SExt( M , K, η ) the set of connected comp onents of D − 1 ( K, η ) , which is nothing but the set of isomorphism classes of Sc hreier extensions of M by K inducing the action η, w e conclude that the tensor product on D − 1 ( K, η ) endo ws the set SExt( M , K , η ) with the structure of a comm utative monoid. Thanks to [24, Theorem 4.19], this comm utative monoid SExt( M , K, η ) is isomorphic to the second cohomology monoid H 2 ( M , K , η ) of the cohomology theory introduced by P atchk oria in [22]. Moreo v er, when K is cancellative, the ﬁbres D − 1 ( K, η ) and d − 1 ( K, η ) coincide, and it follo ws from [24, Theorem 4.29] that the corresp onding comm utative monoid SExt( M , K, η ) is also isomorphic to the second cohomology monoid H 2 ( M , K , η ) of the cohomology theory introduced by P atchk oria in [23]. The in terpretation of the higher cohomology monoids of suc h cohomology the- ories is material for future w ork. Appendix A. Monomorphisms and regular epimorphisms in cc - S E xt M W e aim to prov e that the monomorphisms (resp., regular epimorphisms) in the category cc - S E xt M of cc-Sc hreier extensions on a monoid M are precisely the morphisms (A.1) E : K ( ∗ ) α 1   / / k / / X α   f / / / / M E ′ : K ′ / / k ′ / / X ′ f ′ / / / / M 32 S. AMBRA, A. MONTOLI, AND D. RODELO in cc - S E xt M suc h that α 1 and α are monomorphisms (resp., regular epimorphisms) in Mon . This characterisation allows us to give the full details of the pro of of Prop osition 7.2. Existence of k ernel pairs in cc - S E xt M . Recall the following v ery general fact: Lemma A.1. Consider a c ommutative diagr am A α   l / / B β   g / / C A ′ / / k / / B ′ f / / C in a p ointe d c ate gory C , wher e ( A ′ , k ) is a kernel of f . Then ( A, l ) is a kernel of g if and only if the left squar e is a pul lb ack. Consider then the k ernel pairs of α 1 and α in Mon , together with the induced morphism k × X ′ k : Eq( α 1 ) → Eq( α ) , to obtain the commutativ e diagram (A.2) E : E : E ′ : Eq( α 1 ) ( † ) π 1   π 2   k × X ′ k / / Eq( α ) ρ 1   ρ 2   f ρ 1 = f ρ 2 / / M K ( ∗ ) α 1   / / k / / X α   f / / / / M K ′ / / k ′ / / X ′ f ′ / / / / M . It follows that the tw o commutativ e squares ( † ) are pullbacks, since ( ∗ ) is a pullback. By Lemma A.1, w e conclude that E is an extension of monoids in the sense of (2.1). (Observ e that f ρ 1 (= f ρ 2 ) is a regular epimorphism, b ecause so are f and ρ 1 and Mon is a regular category .) As K is comm utative and cancellative and Eq( α 1 ) is (isomorphic to) a submonoid of K × K, it is clear that Eq( α 1 ) is also comm utative and cancellative, and w e hav e: Prop osition A.2. The se quenc e E is a c c-Schr eier extension and (A.2) is a kernel p air of ( α 1 , α ) : E → E ′ in cc - S E xt M . Pr o of. If u m is a represen tativ e of m ∈ M for E then ( u m , u m ) is a representa- tiv e of m for E . Indeed, observ e that if ( x, y ) ∈ Eq( α ) , then f ( x ) = f ′ α ( x ) = f ′ α ( y ) = f ( y ) , so that x = k q ( x ) + u m and y = k q ( y ) + u m using (2.2) (where m = f ( x ) = f ( y )) . Since α ( x ) = α ( y ) , it follows that α ( k q ( x ))+ α ( u m ) = α ( k q ( y ))+ α ( u m ); consequently , α ( k q ( x )) = α ( k q ( y )) b ecause of Lemma 2.1(1) and the fact that α preserves representativ es. The couple ( q ( x ) , q ( y )) is in Eq( α 1 ) , b ecause k ′ α 1 ( q ( x )) = α ( k q ( x )) = α ( k q ( y )) = k ′ α 1 ( q ( y )) and k ′ is monomorphic. Then ( x, y ) = k × X ′ k ( q ( x ) , q ( y )) + ( u m , u m ) , for the unique element ( q ( x ) , q ( y )) ∈ Eq( α 1 ) . Moreo ver, b y Prop osition 2.8 the morphisms ρ 1 and ρ 2 preserv e all represen tativ es. Next, we need to prov e that the commutativ e square E ( π 1 ,ρ 1 )   ( π 2 ,ρ 2 ) / / E ( α 1 ,α )   E ( α 1 ,α ) / / E ′ THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 33 is a pullbac k in cc - S E xt M . Supp ose that t w o morphisms ( t 1 , t ) , ( h 1 , h ) of cc-Sc hreier extensions F : E : L t 1   h 1   / / l / / Y t   h   g / / / / M K / / k / / X f / / / / M are such that α 1 t 1 = α 1 h 1 and αt = αh. Then, b y the univ ersal prop erty of the k ernel pairs, we get tw o unique monoid homomorphisms (A.3) L h 1 ! ! t 1 $ $ φ 1 = ⟨ t 1 ,h 1 ⟩ " " Eq( α 1 ) π 1   π 2 / / K α 1   K α 1 / / K ′ , Y h ! ! t $ $ φ = ⟨ t,h ⟩ " " Eq( α ) ρ 1   ρ 2 / / X α   X α / / X ′ . Since ρ 1 and ρ 2 are jointly monomorphic and ( ρ 1 k × X ′ k φ 1 = k π 1 φ 1 = k t 1 = tl = ρ 1 φl ρ 2 k × X ′ k φ 1 = k π 2 φ 1 = k h 1 = hl = ρ 2 φl, it follows that k × X ′ k φ 1 = φl. Moreov er, f ρ 1 φ = f t = g , so that F : E : L φ 1   / / l / / Y φ   g / / / / M Eq( α 1 ) / / k × X ′ k / / Eq( α ) f ρ 1 / / / / M is a morphism of monoid extensions. It is a morphism of Schreier extensions, b ecause φ = ⟨ t, h ⟩ and, by assumption, t and h preserve the represen tatives. If ( ψ 1 , ψ ) : F − → E is another morphism of Schreier extensions suc h that π 1 ψ 1 = t 1 , ρ 1 ψ = t and π 2 ψ 1 = h 1 , ρ 2 ψ = h, it immediately follo ws that ψ 1 = φ 1 and ψ = φ b y the uniqueness of φ 1 and φ in (A.3). □ Characterisation of monomorphisms and regular epimorphisms. Now w e can prov e that (A.1) is a monomorphism in cc - S E xt M if and only if α 1 and α are monomorphisms in Mon , and similarly for regular epimorphisms. W e shall proceed b y steps. Prop osition A.3. If α 1 and α ar e r e gular epimorphisms in Mon , then (A.1) is a r e gular epimorphism in cc - S E xt M . Pr o of. W e show that under our assumptions ( α 1 , α ) is a co equaliser of its kernel pair (A.2). Supp ose that ( β 1 , β ) : E − → F E : E : F : Eq( α 1 ) π 1   π 2   k × X ′ k / / Eq( α ) ρ 1   ρ 2   f ρ 1 = f ρ 2 / / M K β 1   / / k / / X β   f / / / / M L / / l / / Y g / / / / M 34 S. AMBRA, A. MONTOLI, AND D. RODELO is a morphism of cc-Sc hreier extensions such that ( β 1 , β )( π 1 , ρ 1 ) = ( β 1 , β )( π 2 , ρ 2 ) . Then, as α 1 is a co equaliser of ( π 1 , π 2 ) and α is a co equaliser of ( ρ 1 , ρ 2 ) , there are unique factorisations Eq( α 1 ) π 1 / / π 2 / / K α 1 / / / / β 1   K ′ γ 1 ~ ~ L, Eq( α ) ρ 1 / / ρ 2 / / X α / / / / β   X ′ γ ~ ~ Y . It is easy to chec k that ( γ 1 , γ ) : E ′ → F is a morphism of monoid extensions such that ( β 1 , β ) = ( γ 1 , γ )( α 1 , α ) . The uniqueness of such a morphism ( γ 1 , γ ) follows from the fact that α 1 and α are (regular) epimorphisms. W e are left to prov e that ( γ 1 , γ ) : E ′ − → F is a morphism of Sc hreier extensions, i.e. that γ preserves the represen tatives. By Prop osition 2.8, it is enough to sho w that for every m ∈ M some representativ e of m for E ′ is preserved by γ . Since γ  α ( u m )  = β ( u m ) for ev ery represen tativ e u m of m for E and, by assumption, b oth α and β preserv e the represen tatives, the result follows. □ Concerning monomorphisms, it is clear that if α 1 and α are monomorphisms in Mon then (A.1) is a monomorphism in cc - S E xt M . The conv erse is also true (as w e sho w next), and we can complete our characterisation of monomorphisms and regular epimorphisms in cc - S E xt M as follows: Prop osition A.4. Consider a morphism (A.1) of c c-Schr eier extensions. Then: (1) ( α 1 , α ) is a monomorphism in cc - S E xt M if and only if α 1 and α ar e mo- nomorphisms in Mon ; (2) ( α 1 , α ) is a r e gular epirmorphism in cc - S E xt M if and only if α 1 and α ar e r e gular epimorphisms in Mon . Pr o of. Consider a (regular epimorphism, monomorphism) factorisation α = ne X α / / e X ′ I α > > n > > of α in Mon , and let ( P , n 1 , λ ) b e a pullback of n along k ′ . Then in the diagram (A.4) E : P : E ′ : K ( a ) α 1   / / k / / e 1   X e     α   f / / / / M P / / λ / /   n 1   I α   n   f ′ n / / M K ′ / / k ′ / / X ′ f ′ / / / / M , where e 1 is the unique morphism induced by α 1 and ek through the pullbac k P , the square ( a ) is also a pullbac k (using the fact that the square ( ∗ ) in (A.1) is a pullbac k, as we already argued). Thus e 1 is a regular epimorphism in Mon , and α 1 = n 1 e 1 is a (regular epimorphism, monomorphism) factorisation of α 1 . Moreov er, observ e that f ′ n is a regular epimorphism, b ecause so is ( f ′ n ) e = f , and by Lemma A.1 ( P , λ ) is a kernel of f ′ n. THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 35 W e pro ve that the extension of monoids P is a cc-Sc hreier extension and that ( e 1 , e ) , ( n 1 , n ) are morphisms of Schreier extensions. First observe that P is com- m utative and cancellativ e (as, b y the monomorphism n 1 , it can be realised as a sub- monoid of K ′ ). Next, we claim that, for every representativ e u m of m for E , e ( u m ) w orks as a represen tativ e of m for P . Indeed, given y ∈ I α w e ha ve y = e ( x ) for some x ∈ X (b ecause e is a regular epimorphism in Mon , i.e. a surjective monoid homo- morphism). Using (2.2), we get y = e ( x ) = e  k q ( x ) + u m  = e ( k q ( x )) + e ( u m ) = λ ( e 1 q ( x )) + e ( u m ) . Supp ose that y = λ ( z ) + e ( u m ) for some other z ∈ P. W e ha v e z = e 1 ( b ) for some b ∈ K (because e 1 is a regular epimorphism), thus e  k q ( x ) + u m  = y = λe 1 ( b ) + e ( u m ) = e  k ( b ) + u m  ⇒ αk q ( x ) + α ( u m ) = ne  k q ( x ) + u m  = ne  k ( b ) + u m  = αk ( b ) + α ( u m ) ⇒ αk q ( x ) = αk ( b ) ⇒ k ′ α 1 ( q ( x )) = k ′ α 1 ( b ) ⇒ α 1 ( q ( x )) = α 1 ( b ) ⇒ e 1 ( q ( x )) = e 1 ( b ) = z , using the fact that α ( u m ) is a representativ e, Lemma 2.1(1) and that k ′ and n 1 are monomorphic. Then P is a Sc hreier extension and e preserv es the representativ es b y construction. Since α = ne and α preserves the represen tatives, we conclude that n preserves the representativ es as well (using Prop osition 2.8). By Prop osition A.3, ( e 1 , e ) : E − → P is a regular epimorphism in cc - S E xt M , and as n 1 and n are monomorphisms of monoids, ( n 1 , n ) is monomorphic in cc - S E xt M . F rom ( α 1 , α ) = ( n 1 , n )( e 1 , e ) , it follows that if ( α 1 , α ) is a monomorphism in cc - S E xt M , so is ( e 1 , e ) , which implies that ( e 1 , e ) is an isomorphism in cc - S E xt M . This, in turn, implies that e 1 and e are isomorphisms in Mon , and we conclude that α 1 = n 1 e 1 and α = ne are monomorphic in Mon . Similarly , if ( α 1 , α ) is a regular epimorphism in cc - S E xt M , then n 1 and n are isomorphisms of monoids in Mon , and we conclude that α 1 = n 1 e 1 and α = ne are regular epimorphisms in Mon . □ R emark A.5 . By the previous prop osition, every morphism (A.1) in cc - S E xt M admits a factorisation (A.4) as a regular epimorphism follo wed by a monomorphism, and we kno w from Prop osition A.2 that kernel pairs exist. Moreov er, it is not diﬃcult to pro ve that regular epimorphisms are preserved by pullbacks in cc - S E xt M whenev er these exis t, so that cc - S E xt M is a regular category in the sense of [1, I.1.3] and [2, Deﬁnition 2.1.1]. Beware, though, that cc - S E xt M is not ﬁnitely complete, in general: indeed, the prop erty of being closed under pullbacks fails ev en in the case of group extensions. The problem here is that the pullback of tw o morphisms α : f → g , β : h → g X f α / / Y g   Z h   β o o G in Gp /G is constructed by means of a pullback X × Y Z π 2 / / π 1   Z β   X α / / Y 36 S. AMBRA, A. MONTOLI, AND D. RODELO of β along α in Gp , but even if f , g and h are surjective, the comp osite morphism f π 1 = hπ 2 need not b e surjective, in general. The same situation happ ens in cc - S E xt M . The direction functor preserv es and reﬂects monomorphisms and regular epimorphisms. W e can now pro v e that the functor d : cc - S E xt M − → cc - S P t M preserv es and reﬂects monomorphisms and regular epimorphisms. By the ab ov e results, a morphism (A.1) in cc - S E xt M is a monomorphism if and only if b oth α 1 and α are monomorphisms in Mon , and similarly for regular epimorhisms. The same characterisation holds for monomorphisms and regular epimorphisms in cc - S P t M . Supp ose that by applying d to a morphism (A.1) in cc - S E xt M w e obtain a monomorphism (A.5) K α 1   / / κ / / d f d ( α )   f / / M s o o K ′ / / κ ′ / / d f ′ f ′ / / M s ′ o o of cc-Schreier p oints. Then, in particular, α 1 is monomorphic in Mon , and the Short Fiv e Lemma (Prop osition 2.9) for cc-Sc hreier extensions guaran tees that α is also a monomorphism of monoids; this en tails that (A.1) is a monomorphism in cc - S E xt M , so that d reﬂects monomorphisms. Conv ersely , if (A.1) is a monomor- phism in cc - S E xt M , by Proposition A.4 α 1 is a monomorphism in Mon . By the Split Short Five Lemma for Sc hreier p oints (see [8], Prop osition 2 . 3 . 10), d ( α ) is also a monomorphism of monoids, and we conclude that (A.5) is a monomorphism in cc - S P t M , i.e. that d preserv es monomorphisms. A similar argument shows that d reﬂects and preserv es regular epimorphisms, which completes the pro of of Prop osition 7.2. W e conclude by observing that the characterisation of monomorphisms and reg- ular epimorphisms we got for cc - S E xt M do es not mak e use of cancellativit y , so the same pro of works for the category smod - S E xt M . References [1] M. Barr, Exact c ate gories , in L ecture Notes in Mathematics 236, Springer-V erlag, 1971, 1-120. [2] F. Borceux, Handb o ok of Cate goric al Algebr a , V ol. 2, Cambridge Univ. Press, 1994. [3] F. Borceux, D. Bourn, Mal’c ev, Pr otomo dular, Homolo gic al and Semi-A b elian Cate gories , Mathematics and its Applications 566, Kluw er, 2004. [4] D. Bourn, Normalization e quivalenc e, kernel equivalenc e and aﬃne c ategories , in Lecture Notes in Mathematics 1488, Springer-V erlag, 1991, 43-62. [5] D. Bourn, Baer sums and ﬁber e d asp e cts of Mal’cev op er ations , Cah. T opol. G´ eom. Diﬀ´ er. Cat´ eg. 40 n.4 (1999), 297-316. [6] D. Bourn, M. Gran, Centrality and normality in pr otomodular c ate gories , Theory and Ap- plications of Categories, 9 (2002), 151–165. [7] D. Bourn, M. Gran, Centr ality and c onne ctors in Maltsev c ate gories , Algebra Universalis 48 (2002), 309-331. [8] D. Bourn, N. Martins-F erreira, A. Montoli, M. Sobral, Schr eier split epimorphisms in monoids and in semirings , T extos de Matem´ atica (S´ erie B), vol. 45, Departamento de Matem´ atica da Univ ersidade de Coimbra, 2013. [9] D. Bourn, N. Martins-F erreira, A. Montoli, M. Sobral, Schr eier split epimorphisms b etween monoids , Semigroup F orum 88 (2014), 739–752. [10] D. Bourn, N. Martins-F erreira, A. Montoli, M. Sobral, Monoids and p ointe d S -pr otomodular c ategories , Homology , Homotopy and Applications 18(1) (2016), 151–172. [11] D. Bourn, D. Ro delo, Cohomolo gy without proje ctives , Cah. T op ol. G ´ eom. Diﬀ ´ er. Cat´ eg. 48 (2007), 104–153. THE DIRECTION FUNCTOR FOR SCHREIER EXTENSIONS 37 [12] A. Carb oni, M. C. Pedicchio, N. Pirov ano, Internal gr aphs and internal gr oup oids in Mal’c ev c ategories , CMS Conf. Pro c. 13 (1992), 97–109. [13] S. Eilenberg, S. Mac Lane, Cohomolo gy Theory in Abstr act Gr oups, I , Annals of Math. 48 (1947), 51-78. [14] A. Grothendieck, T e chnique de desc ente et th ´ eor` emes d’existence en g ´ eom´ etrie alg´ ebrique. I. G´ en ´ er alit´ es. Desc ente p ar morphismes ﬁd` element plats , S´ eminaire Nicolas Bourbaki exp. 190 (1960), 299-327. [15] G. Hoﬀ, On the cohomolo gy of c ate gories , Rend. Mate. 7 (1974), 169–192. [16] D. Holt, An interpr etation of the c ohomology groups H n ( G, M ), J. Algebra 60 (1979), 307- 318. [17] J. Huebsc hmann, Crosse d n -fold extensions of groups and c ohomolo gy , Comment. Math. Helv. 55 (1980), 302-314. [18] S. Mac Lane, Homolo gy , Springer-V erlag, 1963. [19] N. Martins-F erreira, A. Montoli, M. Sobral, Semidir e ct pr o ducts and cr osse d mo dules in monoids with op er ations , J. Pure Appl. Algebra 217 (2013), 334-347. [20] N. Martins-F erreira, A. Montoli, M. Sobral, Baer sums of sp eci al Schr eier extensions of monoids , Semigroup F orum 93 (2016), 403–415. [21] N. Martins-F erreira, A. Montoli, M. Sobral, The nine lemma and the push forwar d c onstruc- tion for sp e cial Schr eier extensions of monoids with op er ations , Semigroup F orum 97 (2018), 325-352. [22] A. Patc hkoria, Cohomology of monoids with c o eﬃcients in semimo dules , Bull. Georgian Acad. Sci. 86 (1977), 545–548 (in Russian). [23] A. Patc hkoria, Cohomolo gy monoids of monoids with c o eﬃcients in semimo dules I , J. Ho- motopy Relat. Struct. 9 (2014), 239–255. [24] A. Patc hkoria, Cohomolo gy monoids of monoids with co eﬃcients in semimo dules II , Semig- roup F orum 97 (2018), 131–153. [25] M. C. Pedicc hio, A c ate goric al appr o ach to c ommutator the ory , J. Algebra 177 (1995), 647- 657. [26] L. R´ edei, Die V er al lgemeinerung der Schr eierischen Erweiterungsthe orie , Acta Sci. Math. Szeged 14 (1952), 252-273. [27] J. D. H. Smith, Mal’c ev V arieties , Lecture Notes in Mathematics 554, Springer-V erlag, 1976. [28] N. Y oneda, On Ext and exact se quenc es , J. F ac. Sci. T okyo, Sec. I 8 (1960), 507-526. (Stefano Ambra) Dip ar timento di Ma tema tica “Federigo Enriques”, Universit ` a degli Studi di Milano, Via Saldini 50, 20133 Milano, It al y Email address : stefano.ambra@unimi.it (Andrea Montoli) Dip ar timento di Ma tema tica “Federigo Enriques”, Universit ` a degli Studi di Milano, Via Saldini 50, 20133 Milano, It al y Email address : andrea.montoli@unimi.it (Diana Rodel o) Dep ar tment of Ma thema tics, University of the Algar ve, 8005-139 F aro, Por tugal and Center for Research and Development in Ma thema tics and Applic- a tions (CIDMA), Dep ar tment of Ma thema tics, University of A veiro, 3810-193 A veiro, Por tugal Email address : drodelo@ualg.pt

The direction functor for Schreier extensions of monoids

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