Theory and Applications of Categories, Vol. 11, No. 20, 2003, pp. 438–449.

THE STRONG AMALGAMATION PROPERTY AND (EFFECTIVE) CODESCENT MORPHISMS

DALI ZANGURASHVILI

ABSTRACT. Codescent morphisms are described in regular categories which satisfy the so-called strong amalgamation property. Among varieties of universal algebras possessing this property are, as is known, categories of groups, not necessarily associative rings, M-sets (for a monoid M), Lie algebras (over a field), quasi-groups, commutative quasi-groups, Steiner quasi-groups, medial quasigroups, semilattices, lattices, weakly associative lattices, Boolean algebras, Heyting algebras. It is shown that every codescent morphism of groups is effective.

1. Introduction

Throughout the paper, we use “(effective) codescent morphism” to mean “(effective) codescent morphism with respect to the basic fibration”. Namely, let C be a category with pushouts and let p : B ⟶ E be its morphism. There is an adjunction

\xymatrix{B/C \ar[r]_{p_{∗}} & E/C \ar[l]_{p^{!}}}

between coslice categories with the left adjoint p_{∗} pushing out along p and the right adjoint p^{!} composing with p from right. The induced comonad on E/C gives rise to the Eilenberg-Moore category of coalgebras (we denote it by Codes(p)) equipped with the comparision functor

Φ_{p} : B/C ⟶ Codes (p).

Recall that objects of Codes(p) (called codescent data with respect to p) are triples (C, γ, ξ), with C ∈ Ob C and γ, ξ being morphisms E ⟶ C and C ⟶ C ⊔_{B} E, respectively, such that the following equalities hold (see Figs.1 and 2):

ξ γ = π_{2},  (1.1)
(1_{C}, γ)ξ = 1_{C},  (1.2)
(π_{1} ⊔_{B} 1_{E})ξ = (ξ ⊔_{B} 1_{E})ξ,  (1.3)

The support rendered by INTAS Grant 97 31961 is gratefully acknowledged Received by the editors 2003-05-19 and, in revised form, 2003-12-05. Transmitted by Walter Tholen. Published on 2003-12-18. 2000 Mathematics Subject Classification: 18C20, 18A32, 20J15, 08B25. Key words and phrases: Strong amalgamation property, (effective) codescent morphism, group,

variety of universal algebras.

\begin{matrix}c \\ ◯ \end{matrix} Dali Zangurashvili, 2003. Permission to copy for private use granted.

while the functor Φ_{p} sends each f : B ⟶ D to

(D ⊔_{B} E, π′_{2}, π′_{1} ⊔_{B} 1_{E}),

where π′_{1} and π′_{2} are pushouts of p and f, respectively, along each other.

\xymatrix{B \ar[d]^{ph}_{p} \ar[r]^{p} & E \ar[dd]^{π_{2}} \ar[dddr]^{γ} &  & B \ar[d]_{p} \ar[r]^{p} & E \ar[dd]^{π_{2}} \ar[dddr] \\ E \ar[d]_{γ} &  &  & E \ar[d]_{γ} \\ C \ar[drr]_{1_{C}} \ar[r]^{π_{1}} \ar[r]_{ξ} & C ⊔_{B} E \ar[dr]_{(1_{C},γ)} &  & C \ar[d]_{π_{1}} \ar@{-->}[d]^{ξ} \ar[r]^{π_{1}} \ar[r]_{ξ} & C⊔_{B} E \ar[dr]_{π_{1}⊔_{B}1_{E}} \ar@{-->}[dr]^{ξ⊔_{B}1_{E}} \\  &  & C & C⊔_{B} E \ar[rr] &  & (C⊔_{B} E)⊔_{B} E}

Fig. 1  Fig. 2

p is called a codescent (effective codescent) morphism if Φ_{p} is full and faithful (an equivalence of categories), i.e. if p_{∗} is precomonadic (comonadic).

It is well-known that if C is finitely cocomplete, then codescent morphisms are precisely couniversal regular monomorphisms, i.e. morphisms whose pushouts along any morphisms are regular monomorphisms [JT].

One of the goals of descent theory is to characterize (effective) codescent morphisms in particular varieties of universal algebras. There are several results obtained along this line. For instance, this problem has been completely solved for commutative rings with units (Joyal-Tierney (unpublished), [M]).

In this paper we describe codescent morphisms in those varieties of algebras (more generally, regular categories) which satisfy the strong amalgamation property and investigate the problem of the effectiveness of codescent morphisms in the case of groups.

Recall that the strong amalgamation property means that for any pushout square

[figure: m; α; α′; m′]

with monomorphisms m, α, the morphisms m′, α′ are also monomorphisms and, moreover, we have

Im m′ ∩ Im α′ = Im m′α.

Among the varieties of universal algebras possessing this property are, as is known, categories of groups, not necessarily associative rings, M-sets (for a monoid M), Lie algebras (over a field), quasi-groups, commutative quasi-groups, Steiner quasi-groups, medial quasi-groups, semilattices, lattices, weakly associative lattices, Boolean algebras, Heyting algebras. We show that a monomorphism [figure: p : B; E] of a variety of this kind is codescent if and only if For any congruence R on B and its closure R′ in E one has

R′ ∩ (B × B) = R.

This in particular implies that every (regular) monomorphism of M-sets, semilattices, Boolean algebras and Heyting algebras is codescent. Note that the latter result related to the case of Boolean algebras was obtained earlier by Makkai (an unpublished work).

The main result of this paper asserts that

Every codescent morphism of groups is effective.

The author gratefully acknowledges useful discussions with George Janelidze on the subject of this paper.

2. The Strong Amalgamation Property and Codescent Morphisms

Before we begin our discussion, let us recall some definitions [KMPT].

Let C be a category equipped with some proper factorization system (𝔼, 𝕄), i.e., a factorization system which satisfies the conditions 𝔼 ⊂ Epi C and 𝕄 ⊂ Mon C.

C is said to satisfy the amalgamation (transferability; congruence extension) property if for each span

[figure: m; α]  (2.1)

with m, α ∈ 𝕄 (m ∈ 𝕄; m ∈ 𝕄 and α ∈ 𝔼) there exists a commutative square

[figure: m; α; α′; m′]  (2.2)

with m′, α′ ∈ 𝕄 (m′ ∈ 𝕄; m′ ∈ 𝕄). C is said to satisfy the strong amalgamation property (the intersection property of amalgamation) if any span (2.1) with m, α ∈ 𝕄 (any commutative diagram (2.2) with m, α, m′, α′ ∈ 𝕄) admits a pullback

[figure: m; α; α′′; m′′]  (2.3)

with m′′, α′′ ∈ 𝕄. In the case of a variety C of universal algebras and 𝕄 = Mon C, this definition is equivalent to that of the strong amalgamation property given in the Introduction, as follows from the arguments given below.

Clearly, C possesses the strong amalgamation property if and only if it satisfies both the amalgamation and the intersection property of amalgamation. If C admits finite products, then the transferability property is equivalent to the amalgamation and congruence extension properties ^{2}. The latter statement remains true if we replace “finite products” by “pushouts”. In that case all the definitions presented above can be reformulated. Namely, the amalgamation (transferability; congruence extension) property is equivalent to the requirement that the class 𝕄 be stable under pushouts along 𝕄-morphisms (any morphisms; 𝔼-morphisms), while the strong amalgamation property is satisfied if and only if the amalgamation property is fulfilled and pushout (2.2) is also a pullback square for any m, α ∈ 𝕄.

In what follows we make use of the following known result.

2.1. Theorem. Let C admit pushouts. Then the following conditions are equivalent:

(i) C satisfies the intersection property of amalgamation;

(ii) 𝕄 consists of all regular monomorphisms.

If, in addition, C satisfies the amalgamation property, then each of (i), (ii) is equivalent to

(iii) every epimorphism lies in 𝔼, and hence 𝔼 = Epi C.

The implication (i)⇒(ii) is obvious. (ii)⇒(i) is proved in [K], [Ri], [T], while the equivalence (i)⇔(iii) is proved in [Ri].

Most of investigations related to the question whether concrete categories satisfy the above-discussed properties deal with the case, where

𝕄 = all monomorphisms.  (2.4)

From now on we shall follow assumption (2.4).

2.2. Proposition. Let C be a regular category with pushouts and let C satisfy the strong amalgamation property. Then the following conditions are equivalent for a monomorphism [figure: f : B; E] :

(i) f is a codescent morphism;

(ii) f is a couniversal regular monomorphism;

(iii) f is a {Regular epis}-couniversal regular monomorphism^{3};

(iv) f is an {Epis}-couniversal monomorphism; (v) for any kernel pair \xymatrix{R \ar[r]^{α_{1}} \ar[r]_{α_{2}} & B} on B and the kernel pair \xymatrix{R′ \ar[r]^{α′_{1}} \ar[r]_{α′_{2}} & E} of the coequalizer

2We have to apply the transferability property to both “sides” and after that to take the product.

3Similarly to the notion of a couniversal regular monomorphism, for arbitrary morphism classes E and M one can define a E-couniversal M-morphism as a morphism whose pushout along any E-morphism lies in M.

of (fα_{1}, fα_{2}), the square

[figure: R; (α_{1},α_{2}); R′; (α′_{1},α′_{2}); B × B; f×f; E × E]  (2.5)

is a pullback.

Each (regular) monomorphism of C is a codescent morphism if and only if C satisfies additionally the congruence extension (or, equivalently, the transferability property).

If C is (Barr-)exact, then one can replace the first “kernel pair” in (v) by “equivalence relation”.

Proof. Let us first indicate some properties of the considered categories C. The pair (Regular epis, Monos) is a factorization system on C. Moreover, by Theorem 2.1 each monomorphism as well as each epimorphism is regular and for any diagram

[figure: e; β; α]  (2.6)

with e being an epimorphism there exists a diagram

[figure: \overline{β}; \overline{α}; \overline{e}; e; β; α]  (2.7)

such that \overline{e} is also an epimorphism and e\overline{α} = α\overline{e}, e\overline{β} = β\overline{e} ^{4}.

The equivalences (ii)⇔(iii)⇔(iv) obviously follow from the presented properties. (i)⇔(ii) is the well-known fact proved, for instance, in [JT].

(iv)⇒(v): Let e be a coequalizer of (α_{1}, α_{2}). Then e′, the pushout of e along f, is a coequalizer of (fα_{1}, fα_{2}). Consider the diagram

4We first take a pullback e_{1} of e along α and then the pullback e_{2} of e along βe_{1}. e′ is given by the composition e_{2}e_{1}.

[figure: A; γ; β_{1}; β_{2}; R; f′; α_{1}; α_{2}; R′; α′_{1}; α′_{2}; B; f; e; E; e′; C; g; C′]  (2.8)

with any β_{1}, β_{2}, γ such that fβ_{1} = α′_{1}γ and fβ_{2} = α′_{2}γ. Then geβ_{1} = e′fβ_{1} = e′fβ_{2} = geβ_{2}, whence eβ_{1} = eβ_{2}, since g being the pushout of f along a (regular) epimorphism is a monomorphism. Since (α_{1}, α_{2}) is a kernel pair of e, we have a unique morphism δ : A ⟶ R such that β_{1} = α_{1}δ and β_{2} = α_{2}δ. Moreover, α′_{1}f′δ = fα_{1}δ = fβ_{1} = α′_{1}γ and, similarly, α′_{2}f′δ = α′_{2}γ, so that f′δ = γ.

(v)⇒(iv): Let gα = gβ and consider \overline{e}, \overline{α}, \overline{β} from diagram (2.7). Clearly, one has e′f\overline{α} = e′f\overline{β}, which gives a morphism θ with \overline{α} = α_{1}θ and \overline{β} = α_{2}θ. Therefore e\overline{α} = e\overline{β}, from which it follows that α = β.

For varieties of universal algebras the condition (v) of Proposition 2.2 clearly takes the form:

For any congruence R on B and its closure R′ in E one has

R′ ∩ (B × B) = R.

2.3. Example. As is known, the varieties of M-sets (for a given monoid M) [KMPT], groups [S], not necessarily associative rings [Di], Lie algebras (over a given field) [R], quasi-groups [YK1], commutative quasi-groups [YK], Steiner quasi-groups [KMPT], medial quasi-groups [JK], semilattices [HK], lattices [Y], [G], weakly associative lattices [FG], Boolean algebras [DY] and Heyting algebras [D] satisfy the strong amalgamation property. Moreover, the categories of M-sets [KMPT], semillatices [KMPT], [BL], Boolean algebras [Si] and Heyting algebras [D] possess the congruence extension property as well. In each case we give the reference (according to [KMPT]) where the corresponding property is determined for the first time.

3. Effective Codescent Morphisms of Groups

Proposition 2.2 gives rise to 3.1. Theorem. Let [figure: p : B; E] be a monomorphism of groups. p is a codescent morphism if and only if for any normal subgroup N of B and its normal closure N′ in E one has

N′ ∩ B = N.

When B is a normal subgroup of E, this is equivalent to the requirement that any normal subgroup of B be normal in E as well.

Before giving our main result, let us recall some well-known facts related to free products of groups with an amalgamated subgroup [Ku]. Let H be a group, and ( [figure: H; φ_{i}; G_{i}] )_{i∈I} be a family of groups equipped with monomorphisms φ_{i}. Further, let G be a group containing (isomorphic copies of) all G_{i}, and let

G_{i} ∩ G_{j} = H  (3.1)

for i ≠ j (here we identify every h ∈ H with all φ_{i}(h)). For any i ∈ I and any right coset of G_{i} by H, except for H itself, we choose a representative. We denote the set of all chosen representatives by A. G is a free product of (G_{i})_{i∈I} with the amalgamated subgroup H if and only if every element of G can be uniquely written as a product

h a_{1} a_{2} ⋯ a_{n},  (3.2)

where n ≥ 0, h ∈ H, all a_{j} lie in A and no two a_{j}, a_{j+1} belong to the same G_{i} ^{5}. Form (3.2) is called canonical. It is easy to see how an element

a′_{1} a′_{2} ⋯ a′_{m}  (3.3)

of G (taken in the uncancellable form) can be reduced to the canonical form. Indeed, if m = 1 and a′_{m} ∈ H, then (3.3) is already the desired one. If again m = 1, but a′_{m} ∈ G_{i_{m}}\H, then we consider the representation

a′_{m} = h \overline{a′_{m}}  (3.4)

with h ∈ H and \overline{a′_{m}} ∈ G_{i_{m}} ∩ A; (3.4) is the canonical form for (3.3). Let m > 1 and a′_{2} a′_{3} ⋯ a′_{m} be already reduced to the required form

a′_{2} a′_{3} ⋯ a′_{m} = h′ a_{1} a_{2} ⋯ a_{n}

with a_{j} ∈ G_{i′_{j}} ∩ A. If a′_{1} ∈ H, then

(a′_{1} h′)a_{1} a_{2} ⋯ a_{n}

5In fact, this criterion remains true even without requirement (3.1)—we only have to replace “set of representatives” by “system of representatives” and then to specify what kind of products (3.2) and the uniqueness is meant here.

is clearly the canonical form for (3.3). If a′_{1} ∈ G_{i_{1}}\H, then we consider the representation

a′_{1} h′ = h′′\overline{a′_{1}},

where h′′ ∈ H and \overline{a′_{1}} ∈ G_{i_{1}} ∩ A. If i_{1} ≠ i′_{1}, then

h′′ \overline{a′_{1}}a_{1} a_{2} ⋯ a_{n}

is the canonical form for (3.3). If i_{1} = i′_{1} and \overline{a′_{1}} a_{1} ∈ H, then

(h′′ \overline{a′_{1}} a_{1}) a_{2} ⋯ a_{n}

is the required representation for (3.3). If i_{1} = i′_{1} and \overline{a′_{1}} a_{1} ∉ H, then we have

\overline{a′_{1}} a_{1} = h′′′ \overline{a′_{1}a_{1}}

for some h′′′ ∈ H and \overline{a′_{1}a_{1}} ∈ G_{i_{1}} ∩ A; the representation

h′′′ \overline{a′_{1}a_{1}} a_{2} ⋯ a_{n}

is the desired form for (3.3).

Now it is clear that if a′_{j_{1}}, a′_{j_{2}}, …, a′_{j_{k}} are precisely those elements in (3.3) which do not lie in H and if i_{j_{r}} ≠ i_{j_{r+1}} (!) for any 1 ≤ r ≤ k − 1, then in the canonical form of (3.3) we have exactly k factors (except, perhaps, for a coefficient from H) and these factors belong to G_{i_{j_{1}}}, G_{i_{j_{2}}}, …, G_{i_{j_{k}}}, respectively. Indeed, in each step of the reduction we either pick out an H-coefficient from some a′_{j} or perform the relevant multiplication. Clearly, each a′_{j_{r}} is changed again by an element from G_{i_{jr}} \H, while no two a_{j_{r}}, a_{j_{r+1}} can “cancel” each other.

3.2. Proposition. For any monomorphism [figure: p : B; E] the action of Φ_{p} on objects is surjective up to an isomorphism.

Proof. Let (C, γ, ξ) be codescent data with respect to p. The pushout in Fig. 1 is the concatenation of the following two pushouts:

[figure: B; p; E; γ(B); p′; i; E/(Ker γ ∩ B)′; i′; C; π_{1}; C ⊔_{γ(B)} E/(Ker γ ∩ B)′]  (3.5)

Here (Ker γ ∩ B)′ is the normal closure of (Ker γ ∩ B) in E. From (1.2) we conclude that π_{1} and thus both p′ and i′ are monomorphisms.

From (1.1) we have

ξ(γ(E)) = π_{2}(E).  (3.6)

On the other hand, by (1.2) ξ is a split monomorphism. Therefore the homomorphism γ(E) ⟶ π_{2}(E) ≈ E/(Ker γ ∩ B)′ is an isomorphism and its inverse is the morphism induced by γ. This in particular implies that (Ker γ ∩ B)′ = Ker γ and thus diagram (3.5) takes the form

[figure: B; p; E; γ(B); p′; i; γ(E); i′; C; π_{1}; C ⊔_{γ(B)} γ(E)]

If C = γ(E), then (C, γ, ξ) ≈ Φ^{p}(B ⟶ γ(B)).

Suppose now that C ≠ γ(E). Without loss of generality it can be assumed that p is not an isomorphism. Since we intend to work with the free product of the groups G_{1} = C and G_{2} = γ(E) (as well as with the free product of the groups G_{1} = C, G_{2} = γ(E) and G_{3} being another copy of γ(E)) with the amalgamated subgroup γ(B), we choose a representative from each right coset (different from γ(B)) of G_{1} and G_{2} (in G_{3} we choose the same representatives as in G_{2}). Let c ∈ C\γ(E) and let the canonical form of ξ(c) be written as

ξ(c) = γ(b) a_{1} a_{2} ⋯ a_{n}.  (3.7)

We show that if a_{i} ∈ G_{1} for some i, then

ξ(a_{i}) ∈ G_{1},

whence, by (1.2)

ξ(a_{i}) = a_{i}.

Let the canonical form of ξ(a_{i}) have the form

ξ(a_{i}) = γ(b′)a′_{1} a′_{2} ⋯ a′_{m}.

Clearly, by (1.2) m > 0. Suppose that at least one a′_{j} lies in G_{2}. From (1.3) we obtain

γ(b) ⋯ ◯ a_{i−1} a_{i} ◯ a_{i+1} ⋯
= γ(b) ⋯ ◯ a_{i−1} γ(b′) a′_{1} a′_{2} ⋯ a′_{m} ◯ a_{i+1} ⋯,  (3.8)

where the elements in circles are considered as representatives of G_{3}. The left and right neighbors of any γ(b′′) (if they exist) in the right part of (3.8) lie in different G_{i} (1 ≤ i ≤ 3). Therefore, according to the observation preceding this proposition, the canonical form of the right part of (3.8) necessarily contains a representative from G_{2}. On the other hand, the left part of (3.8) is already canonical and does not contain any representative from G_{2}, which is a contradiction.

Let C′ be the smallest subgroup of C containing γ(B) and all elements of G_{1} which take part in representation (3.7) for some c ∈ C\γ(E). From (1.2) we obtain

c = γ(b) a′_{1}a′_{2} ⋯ a′_{n},

where each a′_{i} coincides either with a_{i} or with γ(a_{i}). Thus C is generated by C′ and γ(E). Moreover, as we have just proved

ξ|_{C′} = π_{1}|_{C′}.

Therefore, after choosing a set of representatives from the right cosets of C′ and γ(E) by γ(B), we conclude that each element of C has precisely one canonical form. Thus C = C′ ⊔_{γ(B)} γ(E) and (C, γ, ξ) ≈ Φ^{p}(γ′), where γ′ : B ⟶ C′ is the homomorphism induced by γ.

Proposition 3.2 immediately gives rise to

3.3. Theorem. In the category of groups every codescent morphism is effective.

3.4. Remark. It seems natural to ask what classes of effective codescent morphisms various comonadicity criteria determine in this case. In this connection, we only observe that for a monomorphism [figure: p : B; E] , the change-of-cobase functor p_{∗} preserves (all) equalizers if and only if the intersection of any right and left cosets of E by B which are different from B, contains at most one element. The “if” part is easy to verify. For the “only if” part we consider the equalities tb = b′s and tb_{1} = b′_{1}s with t, s ∈ E\B, b, b′, b_{1}, b′_{1} ∈ B and choose a diagram

[figure: B; i_{1}; i_{2}; i_{3}; C; γ; D; α; β; K]

with monomorphisms i_{1}, i_{2}, i_{3} such that the bottom part is an equalizer diagram and, moreover, there exist d, d′ ∈ D\B with b^{−1}α(d) = b_{1}^{−1}β(d) and α(d′)b′ = β(d′)b′_{1}. It is easy to see that for the element x = d′td of D ⊔_{B} E we have (α ⊔_{B} 1_{E})(x) = (β ⊔_{B} 1_{E})(x). Hence x = (γ ⊔_{B} 1_{E})(y) for some y ∈ C ⊔_{B} E. If we choose representatives of the right cosets of C by B, then their images under γ can be taken as representatives of (some) right cosets of D. The canonical form of y necessarily ends in an element c from C. We have γ(c) = b′′′d for some b′′′ ∈ B and therefore α(d) = β(d) and b = b_{1}, b′ = b′_{1}.

In particular, when B is normal in E, p_{∗} preserves equalizers if and only if B is trivial.

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Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org, Associate Managing Editor Lawrence Breen, Université Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of Wales Bangor: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Palo Alto Research Center: paiva@parc.xerox.com Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, University of Western Sydney: s.lack@uws.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca, Managing Editor Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca