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Electronic Notes in Theoretical Computer Science 301 (2014) 61-77

www.elsevier.com/locate/entcs

Characterising E-projectives via Co-monads

Weng Kin Ho 1,2

Mathematics and Mathematics Education, National Institute of Education Nanyang Technological University Singapore

Abstract

This paper demonstrates the usefulness of a comonadic approach to give previously unknown characterisation of projective objects in certain categories over particular subclasses of epimorphisms. This approach is a simple adaptation of a powerful technique due to M. Escardo which has been used extensively to characterise injective spaces and locales over various kinds of embeddings, but never previously for projective structures. Using some examples, we advertise the versatility of this approach - in particular, highlighting its advantage over existing methods on characterisation of projectives, which is that the comonadic machinery forces upon us the structural properties of projectives without relying on extraneous characterisations of the underlying object of the co-algebra arising from the comonad.

Keywords: E-projectives, KZ-comonads, Right U-quotients, Ordered monoids, Normal semi-rings, Semilattices, Z-frames

1 Introduction

The problem of characterizing projectives and their duals, injectives, in various categories has a long history in mathematics with its origin tracing back to module theory, e.g., characterizing projective modules in module theory. An object P of a category C is projective if for every epimorphism e : A —> B and every morphism f : P —> B, there is a C-morphism (not necessarily unique) f' : P —> A such that f = f' o e. In the category of sets, every object is projective, while the only projectives in the category of groups are the free ones. In algebra, projective objects are viewed as a generalization of free objects.

Much later on, attention with regards to the study of projectives and injectives was shifted from algebraic structures to ordered structures and topological spaces. R. Sikorski showed that the injectives in the category of boolean algebras are precisely the complete boolean algebras ([19]). Later, R. Balbes extended this result to

1 Supported by NTU AcRF Project No. RP/10HWK

2 Email: wengkin.ho@nie.edu.sg

http://dx.doi.Org/10.1016/j.entcs.2014.01.006 1571-0661/© 2014 Elsevier B.V. All rights reserved.

show that the injectives in the category of distributive lattices are also the complete boolean algebras ([2]). G. D. Crown in [5] characterised the projectives (and injectives) in the category of sup complete lattices as the completely distributive lattices. Another important example is D.S. Scott's discovery that the injective T0-spaces are exactly the continuous lattices with their Scott-topology [18]. Notably, in their effort of generalizing Scott's result to frames, B. Banaschewski and S. B. Niefeld in [4] reported that the only projective frame is 2 the two-element chain. Indeed, a similar result has been already been reported in [3] for the category of distributive lattices. For situations like these where projectives are scarce, it is natural to consider more general types of projective objects, whence the notion of E-projectives. To achieve this, one takes not all the epimorphisms but only a certain subclass E of epimorphisms. More precisely, an object P of a category C is E-projective or projective over the E-morphisms if for every C-morphism f : P —> A and every E-morphism e : A —> B, there is a C-morphism f' : P —> A such that f = f 'e. Dually, there is the notion of E-injectives.

In [4] B. Banaschewski and S. B. Niefeld showed that the projectives over E the collection of regular epimorphisms (i.e., morphisms which are co-equalisers of some parallel pair of morphisms) in the category of frames are exactly the stably completely distributive lattices; thereby relaxing on the condition of projectivity to admit a larger class of objects. In the literature, projectives over the regular epimorphisms are also called regular projectives. In certain categories such as the category KHausSp of compact Hausdorff spaces and continuous maps, every epimorphism is regular. Then, for such categories, the problem of characterising projectives then amounts to that of characterising the regular projectives; and this latter problem, in certain situations, may turn out to be easier. One such instance is Gleason's theorem, i.e., the projective objects in KHausSp (which coincide with the regular projective objects) are precisely the extremally disconnected spaces (refer to the proof found in [13, pp. 98-103]).

Recent years have seen a continued interest in the both the areas of characterisation of the E-injectives and projectives for poset-enriched categories. Along the 'injective' line, one important advancement was made by M. H. Escardo. His technique involves the use of KZ-monads to characterise injectives over certain em-beddings in some poset-enriched categories. Escardo's method relies heavily on the following result:

Theorem 1.1 ([9, Theorem 4.2.2, p.32])

Let T be a KZ-monad on X. Then, the following are equivalent for any A <E X:

(i) A is right injective over right T-embeddings.

(ii) A is injective over right T-embeddings.

(iii) A is a T-algebra.

This method is particularly powerful since this monadic approach characterises the injectives over right T-embeddings (where T is the given KZ-monad on the category) as precisely the underlying algebras of the monad T. The KZ-monadic machinery then forces upon us the characterisation of these underlying algebras by

invoking certain reflective adjunctions (see condition (KZ1) on [9, Lemma 4.1.1, p.31]). This technique proves versatile in reproducing well-known characterisations of injectives in various poset-enriched categories (e.g., the injectives over subspace embeddings are the continuous lattices, see [8,9]), as well as new ones (e.g., the injective locales over perfect sublocale embeddings are those that satisfy some new kind of stable continuity, see [10]).

On the 'projective' side of the development, most contributions came from the Asian authors ([23,21,16]). Notably, Zhao in [23] set the path for the latter two to follow. Based only on adjunctions between categories (not necessarily poset-enriched ones), Zhao's method has a much lighter mathematical overhead as compared to Escardoo's. Crucially, the preceding three works hinge on the following lemma:

Lemma 1.2 ([23, Lemma 3.1, p.43])

Let G : C —> D and F : D —> C be a pair of functors such that F is left adjoint to G, with co-unit denoted by e. Further, let E denote the collection of all C-morphisms f : A —> B such that G(f) has a section, i.e., a right inverse in D. Then, for any A e C, the following are equivalent:

(i) A is E-projective.

(ii) e : FG(A) —> A has a right inverse.

(iii) A is a retract of some FX for some object X in D.

Using the above lemma, [23] showed that the E-projective frames are precisely the stably Z-continuous ones, while [21] proved that the E-projective Z-quantales are precisely the stably Z-continuous ones. In a similar vein, [16] characterised the E-projective normal semi-rings as those stably F-continuous ones. All these three works share a common strategy in their use of Lemma 1.2: their characterisation of projectives relies heavily on condition (ii). In order to use (ii), one inevitably needs to obtain certain structural properties of FG(A) and its stability under retracts in order to deduce that A share those properties as FG(A). These structural properties are derived independent of Lemma 1.2. In other words, for the more general adjunctions F h G, their approach would work only if some salient structural properties of canonical structure FX have been independently identified.

Owing to this particular constraint, Zhao's approach fails precisely in those cases where the structural properties of the underlying coalgebra FG(A) of the comonad FG are not easily available. For instance, with respect to the adjunction pair G : Frm —> PreFrm and F : PreFrm —> Frm between the categories of pre-frames and frames (where G is the forgetful functor and F is the Scott-closed lattice functor), the question of characterising the E-projectives in the sense of Zhao [23] still remains open. Even though some explicit lattice-theoretic properties of F(P) have been worked out in [12], it is still not known how E-projectives may be characterised (see [12, p.313]). This question has drawn the attention of domain theorists, especially in the Asian 3 region.

3 Private communications with some Asian domain-theorists revealed that they were unaware of Escardo's works [10,9,10].

Naively speaking, Escardo's method appears to be more superior: the monadic machinery forces out the innate properties of the canonical structures (i.e., the algebras of the monad) without having to derive them elsewhere outside the main development. But this kind of comparison may not be meaningful or fair since the two methods were intended for different purposes: one is to characterise injectives and the other projectives; for different E morphism classes; and for different categorical environments: one requires poset-enriched categories and the other does not. Therefore, a natural question to ask is whether Escardó's method which is powerful in dealing with the injectives can be modified to deal with the projectives. In view of Theorem 1.1, an affirmative answer to this question might allow us to characterise the underlying coalgebra (of those comonads we are interested) as precisely the E-projectives for some class E of morphisms.

In this paper, we play the category-dual game by turning Escardó's set-up in the opposite direction. This involves carefully replacing all categorical concepts in the formulation of KZ-monads on [9, pp.31-34] by their duals, e.g., monomorphisms by epimorphisms, injectives by projectives, and choosing the correct inequalities. The derived results are then applied to three different adjunctions between poset-enriched categories to yield new characterisations of projectives. To be do this, we introduce all these three examples in the preliminaries, and use them as running examples later in the paper. The preliminaries section also contains some essential definitions and results concerning comonads and Kan lifts. Our aim is to demonstrate that (the dual version of) Escardo's set-up is, in fact, just as powerful in characterising projectives as it is for injectives. In the ensuing sections, we assume the reader is familiar with basic category theory, order and lattice theory and domain theory. For category theory, we refer the reader to [15,17]; order and lattice theory [6]; and domain theory [1,11].

2 Preliminaries

2.1 Running examples

2.1.1 Ordered monoids and normal .semirings

Ordered algebraic structures not only yield a rich resource of poset-enriched categories but also numerous instances of categorical adjunctions. Here we focus on ordered monoids and normal semi-rings.

Recall that a monoid is a semi-group, i.e., almost a group except for the existence of inverses. The triple (M, ■, <) is an ordered monoid if M is a monoid with identity 1M, together with a partial order < on it which is compatible with the monoid operation, i.e., for any a, b and c £ M, a < b implies a ■ c < b ■ c, and c ■ a < c ■ b, and the multiplicative identity 1m is the top element of M. Every monoid is a trivial ordered monoid with the discrete order. The set of natural numbers has two different well-known ordered monoid structures, namely, (N, +, <) and (N, max, <).

Let (M, ■, <) and (K, <) be any two ordered monoids, with identities 1m and 1k respectively. A mapping f : M —> K is an ordered monoid morphism if the following conditions hold: (i) f (a ■ b) = f (a) ® f (b) for any a, b £ M. (ii)

f (Im) = 1k• (iii) a < b implies f (a) < f (b). Having defined the salient arrows, we now have OrdMon the category whose objects are the ordered monoids and whose morphisms the ordered monoid morphisms.

We now turn to look at semi-rings. Semi-rings are a notion intended to generalize rings. A semi-ring is a non-empty set R on which operations of addition and multiplication have been defined such that: (i) (R, +) is a commutative monoid with identity element 0. (ii) (R, ■) is a monoid with identity element 1r (or simply 1). (iii)For all a, b and c £ R, a ■ (b + c) = a ■ b + a ■ c and (b + c) ■ a = b ■ a + c ■ a. (iv) For all r £ R, 0 ■ r = 0 = r ■ 0. (v) 1 = 0.

Some examples of semi-rings are given below:

(i) The set of natural numbers, together with the usual addition and multiplication, (N, +, ■), is a commutative semi-ring.

(ii) A bounded distributive lattice (L, V, A) is a commutative idempotent semi-ring. Here, idempotence of a semi-ring refers to the idempotence of both addition and multiplication.

(iii) Let R be a ring. The set of ideals of R, denoted by Id(R), with the usual addition I + J := {i + j | i £ I, j £ J} and and multiplication of ideals I ■ J := {i ■ j I i £ I, j £ J}, is a semi-ring.

(iv) Let R be a commutative ring and A be the set of all elements of R which are not zero-divisors. Let S-1R be the ring of quotients of R. A fractional ideal K of R is an R-submodule of S satisfying the condition that aK C R for some a £ A. The set fract(R) of all fractional ideals of R is closed under taking intersections, sums and products. Moreover, (fract(R), +, ■) is a commutative semi-ring with additive identity (0) and multiplicative identity R.

(v) A commutative integral domain R is a Prüfer domain if every finitely generated fractional ideal of R has a multiplicative inverse in fract(R). This condition is equivalent to the condition that, in Id(R), the intersection distributes over addition, i.e., (Id(R), +,H) is a semi-ring.

If R and S are semi-rings, then a function 7 : R —> S is a semi-ring morphism if the following conditions hold: (i) 7(0r) = 0s. (ii) 7(1r) = 1s. (iii) 7(r + r') = Y(r) + 7(r') and 7(r ■ r') = 7(r) ■ Y(r') for all r, r' £ R.

Here, we deal with a special kind of semi-rings. A semi-ring R is normal if x ■ y + x = x = y ■ x + x for all x and y £ R. It is immediate that normal semirings are additively idempotent. Denote by SRng the category of semi-rings and semiring morphisms. The full subcategory, NSRng, of SRng consists of all normal semi-rings as objects.

For a poset (P, <) and X C P, we denote the set {p £ P I3x £ X. p < x} by IX. In case X = {x}, a singleton, we unambiguously write |x for IX. Dually, one defines f. If X is a finite subset of Y , we use the notation X £ Pfin(Y) or X Cfin Y.

Let (M, ■, <) be an ordered monoid. Define Do(M) := U A I A £ Pfin(M)}. Because M is an ordered monoid, M =| 1m, and so M £ D0(M). On D0(M), define the semi-ring addition as binary union. As for the semi-ring multiplication

g, define as follows: for any ],A, IB e Do(M), ^Ag IB :=|{a • b | a e A, b e B}. Crucially, for an ordered monoid (M, [_J, g), the triple (D0(M),[_J, g) is a normal semi-ring.

By virtue of this result, in order to extend the assignment (M, •, <) M (D0(M),[J, () (where M e obj(OrdMon)) to a functor F : OrdMon —> NSRng, it remains to assign to an arbitrarily given OrdMon-morphism f : M —> N a NSRng-morphism F(f) : Do(M) —> Do(N), defined by Ff(iA) =|f(A), for any A e Do(M). It is routine to verify that F is indeed a functor. In the opposite direction, it is easy to see that any normal semi-ring (S, +, •) can be given an ordered monoid structure, namely, (S, •, <), where a < b ^^ a + b = b for any a, b e S. Thus, we have the forgetful functor G : NSRng —> OrdMon. Regarding the order structure of a semi-ring, more can be said.

Proposition 2.1 Let (R, +, •) be a normal semi-ring and A Cfin R. Then

V A = £ a.

Proof. Follows from the additive idempotence of R. □

Corollary 2.2 Let R be a normal semi-ring and A, B C R with \.A =^B. Then

£ a = £ b.

aeA beB

Corollary 2.3 Let R be a normal semi-ring, x e R and A C R. Then

x • \J a = \J x • a

ae A ae A

Using these preceding properties, it is easy to establish the following:

Theorem 2.4 Let F : OrdMon —► NSRng and G : NSRng —► OrdMon be

the aforementioned functors. Then F is left adjoint to G with unit given by

nM : M —► GF(M), x (M e obj(OrdMon))

and co-unit given by

en : FG(R) —> R, |A m£ a (R e obj(NSRng)).

In the ensuing sections, whenever an example mentions the adjunction between OrdMon and NSRng, we are implicitly referring to the above adjunction, and the symbols F, G then refer to the above pair.

2.1.2 Z-frames and frames

By a semilattice, we mean a (finite) meet semilattice. Thus, every semilattice has a top element. For semilattices S and T, a semilattice homomorphism f : S —> T is a mapping that preserves finite meets (and hence the top element). Denote by SLat the category of semilattices and semilattice homomorphisms. A subset D of a poset is a lower set if D =\.D := {p e P | 3d e D. p < d}. We denote the collection of lower sets by D(P). For any semilattice S, D(S) is a complete lattice with respect

to inclusion, and hence a semilattice. Following the subset-systems approach first proposed by [20] and used later by [23,7], we define a subset system Z on SLat to be a function that assigns to each semilattice S a collection Z(S) of subsets of S, such that the following are satisfied:

(Z1) Z(S) is a subsemilattice of D(S) containing all ix for x e S. (Z2) For any D e Z(Z(S)), UDe Z(S).

(Z3) For any SLat-morphism f : S —> T and any D e Z(S), if (D) e Z(T).

We call the elements of Z(S) the Z-ideals of S. A subset A of S is called a Z-set if iA e Z(S).

A semilattice S is said to be Z-complete if V D exists for all D e Z(S). Given two Z-complete semilattices S and T, a Z-complete homomorphism f : S —> T is a semilattice homomorphism that preserves the suprema of Z-ideals. A Z-complete semilattice A is called a Z-frame if for each a e A and D e Z(A) it holds that a A V D = \/(a A D) where a A D := {a A d | d e D}. It can be shown that a Z-complete semilattice A is a Z-frame if and only if \J : Z(A) —> A is a Z-complete semilattice homomorphism. The notion of Z-frame is to generalise the notion of a frame, and this can be seen by taking Z to be the selection P of all subsets. Also, if Z selects all the directed subsets, then one recovers the notion of preframe. A Z-complete homomorphism between two Z-frames is also called a Z-frame homomorphism. We denote by ZFrm the category of all Z-frames and Z-frame homomorphisms. It can be shown that SLat H ZFrm; more precisely:

Theorem 2.5 ([23, Lemma 1.4, p.40])

Suppose Z is a subset system on SLat. Let Fz : SLat —> ZFrm be the functor defined by Fz(S) = (Z(S), C) and

Fz(f : S -— T) = (Fz(f) : Fz(S) -— Fz(T), D —if (D))

and Gz : ZFrm —> SLat the forgetful functor. Then Fz is left adjoint to Gz with unit given by

ns : S —> Z(S), x ——^ix (S e obj(SLat))

and co-unit given by

es : Z(S) -— S, D — \J D (S e obj(SLat)).

A subset X of a Z-complete semilattice A is Z-closed if (i) X =iX, and (ii) for any Z-set D, D C X implies D e X. The collection of all Z-closed subsets of A is denoted by Tz(A). Observe that Tz(A) is closed under arbitrary intersection. It is intended that the notion of Z-closed sets generalises that of Scott closed sets; just ask Z to select all the directed subsets. We denote the Z-closure of a subset X of a Z-complete semilattice A by clz(X) := f]{C e rz(A) | X C C}. It can be shown that for a semilattice A, A is a Z-frame if and only if Tz(A) is a frame. More importantly, ZFrm H Frm given below:

Theorem 2.6 ([22])

Suppose Z is a subset system on SLat. Let FZ : ZFrm —> Frm be the functor

defined by FZ(A) = (Tz(A), C) and

FZ(f : A B) = (FZ(f) : FZ(A) FZ(B), X ^ clz(f (X)))

and GZ : Frm —> ZFrm the forgetful functor. Then FZ is left adjoint to GZ with unit given by

nA : A —► rz(A), x ^{x (A e obj(ZFrm))

and co-unit given by

£a : rz (A) A, X X (A e obj(ZFrm)).

2.2 Comonads

A comonad in a category D consists of a functor U : D —> D together with two natural transformations e : U —> ido (the counit) and v : U —> U2 (the comultiplication), subject to the following conditions:

(i) (Associativity) Uvx ◦ vx = vux ◦ vx, and

(ii) (Unit laws) eux ◦ vx = Uex ◦ vx = idux for any object X of D.

The naturality conditions mean that for all f : X

Y , the following Tf

ey and vx

commute.

Let U = (U,e,v) be a comonad. A U-coalgebra is an object A (the underlying object) together with an arrow (3 : A —> UA (the co-structure map) subject to the following conditions:

(i) (Associativity) va ◦ ( = U( o (, and

(ii) (Unit law) 6a o 3 = idA

Sometimes, we refer to the underlying object of a co-algebra as the co-algebra.

Proposition 2.7 For any object X, UX is a coalgebra (called the free U-coalgebraj with the co-structure map vx : UX —> U2X.

Let A and B be U-coalgebras with co-structure maps a : A —> UA and ( : B —> UB. A U-coalgebra homomorphism from (A, a) to (B,() is an arrow h : A —> B such that Uh o a = ( o h.

Every adjunction between categories gives rise to a comonad. More precisely, given that F : C —> D is left adjoint to G : D —> C, define U : D —> D, U = FG. Set the co-unit to be that of the adjunction f H G, i.e., e, and the comultiplication to be v = F-qo. Then it can be shown that (U, e, v) is a comonad in D.

Example 2.8 (i) Specializing the above result to the adjunction F H G between the categories OrdMon and NSRng, U : NSRng —> NSRng is defined by U = FG. The counit e is given by

en : UR —> R, iA — J2 a = V A (A Cfin R e obj(NSRng)),

while the comultiplication v = Fqo is explicitly given by

vr :iA — {iB e FGR | 3a e A. iB Cia}.

(ii) Applying the above result to the adjunction Fz H Gz between the categories SLat and ZFrm, UZ : SLat —> ZFrm is defined by UZ = FZGZ. The counit e is given by

es : Z(S) -— S, D — \J D, while the co-multiplication v = Fz(qoZ) is explicitly defined by va : D — {E e Z(A) I 3d e D. E Cid}.

(iii) For the adjunction FZ H GZ between the categories ZFrm and Frm, UZ : ZFrm —> Frm is defined by UZ = F Z GZ. The counit e is given by

eA :rz(A) -— A, X — \/ X,

while the comultiplication v = FZ (noz) is explicitly given by

va : D — {X e rz(A) I 3d e D. X Cid}.

2.3 Kan lifts of morphisms in poset enriched categories

In what follows, the categorical notion of Kan lift (which is the categorical dual of Kan extension, see [15, p.236] is applied to the special case of posets and monotone maps.

Recall that for posets P and Q, a monotone map f : P —> Q is a function which preserves order, i.e., x <p y implies f (x) <q f (y). If a pair of monotone maps f : P —> Q and g : Q —> P is such that f (p) < q ^^ p < g(q) for all p e P and q e Q, then we say that f is left adjoint to g, or equivalently g is right adjoint to f. An adjunction pair as described above is denoted by f H g. For monotone maps, f H g implies that f o g < idQ and idp < g o f. If in addition f o g = idQ, we say that f H g is reflective; and dually, coreflective if g o f = idp.

Let p : Y < X and f : A —> X be monotone maps between posets. A left Kan lift of f along p is a monotone map f/p : A —> Y such that

(K1) p o (f/p) < f

(K2) p o g < f for g : A —> Y implies that g < (f/p).

In other words, the left Kan lift of f along p, if it exists, is the greatest map g : A —> Y for which po g < f. Since this amounts to the condition po g < f ^^ g < (f/p) for all g : A —> Y, the composition map

p o- : [A, Y] [A, X]

is right adjoint to

-/p : [A,X] [A, Y].

Here [P, Q] denotes the set of all monotone maps from P to Q.

Let C be a poset-enriched category and E a subclass of C-morphisms. An object A of C is called a left Kan object over E if for every E-morphism p : Y —> X the composition map

p o- : [A, Y] [A, X]

is right adjoint to

-/p : [A, X] [A, Y].

If A is a left Kan object over E and also the left Kan lift f/p is an actual lift (i.e., p o (f/p) = f), then we call A a left projective object over E.

3 KZ-comonadic machinery

We now develop the KZ-comonadic machinery, with which we characterize certain E-projectives in poset-enriched categories. Because what we are doing here is merely dualizing Escardo's monadic set-up into a comonadic one, we choose to omit all proofs. Experts in the KZ-monadic technology may choose to skip this entire section based on the duality principle. Readers who encounter this material for the first time should verify all the results reported here, and look up for hints (if one has trouble) from the original (monadic) version [9, p.31].

3.1 Definitional matters

Recall that a poset-enriched category is a category whose hom-sets are posets and whose composition operation is monotone. A poset-functor between poset-enriched categories is a functor which is monotone on hom-posets. A poset functor U : C —> D is poset-faithful if all A and B in C and all C-morphisms f, g : A —> B,

f < g ^ Uf < Ug.

Example 3.1 Both OrdMon and NSRng are poset-enriched categories. Furthermore, the functors F and G are both poset-functors which are poset-faithful. Indeed, for any ordered monoid morphisms f, g : M —> N with f < g pointwise, then for any finite set A C M, we have Ff({ A) ={ f (A) C{g(A) = Fg({ A). Conversely, if Ff < Fg then for every x e A, one has

Ff({x) C Fg({x) =^{f (x) C{g(x) f (x) < g(x),

and hence f < g.

Example 3.2 The categories SLat and ZFrm (and of course, Frm) are poset-enriched. Furthermore, it is easy to check that Fz, F Z and the forgetful functors are all poset-functors.

For the poset-enriched category NSRng, the following property holds:

Proposition 3.3 For any normal semi-ring R, it holds that

eur < Uer.

Proof. We perform the following calculations for a finite subset A C FGR: eur(IA) =4UM \iA e A} and Uer(1 A) =4{V A ||A e A}. Clearly, eur(IA) C UEr(IA). □

3.2 Main results

Comonads which satisfy the inequality in the preceding proposition are of a special kind, namely the KZ-comonads. The definition of KZ-comonad, given below, is based on a result of Anders Kock [14], specialized to poset-enriched categories.

Lemma 3.4 Let (U, ev) be a comonad in a poset-enriched category D, and assume that U is a poset-functor. Then the following conditions are equivalent:

(KZo) eux < Uex for all X e D.

(KZ\) For all X e D, an arrow ( : X —> UX is a co-structure map if and only if 3 H ex is a coreflective adjunction (i.e., ex ◦ 3 = idx)■

(KZ2) vx H eux for all X e D.

(KZ3) Uex H vx for all X e D.

Definition 3.5 Let D be a poset-enriched category. A left KZ-comonad in D is a comonad (U,e,v) in D with U a poset functor, subject to the equivalent conditions of Lemma 3.4. Poset-dually, one defines right KZ-comonads. Whenever there is no confusion, we just write 'KZ-comonad' for 'left KZ-comonad'. Here "KZ" abbreviates "Kock-Zoberlein".

Example 3.6 By Proposition 3.3, the comonad induced by the adjunction between OrdMon and NSRng is a left KZ-comonad.

Example 3.7 It is quite straightforward to check that the comonads Uz = FzGz (induced by the adjunction Fz H Gz) and UZ = FZGZ (induced by the adjunction FZ H GZ) are KZ-comonads, following the respective definitions of the functors and the co-multiplications explicitly given earlier in Example 2.8(2) and (3).

Remark 3.8 By the condition (KZ1), each object A has at most one co-structure map. We denote the unique co-structure map of a U-coalgebra A of a KZ-comonad U = (U,e,v) by mA.

Definition 3.9 Let F : D —> D be a poset-functor on a poset-enriched category D. A left F-arrow is a morphism f : X —> Y in D such that Ff : FX —> FY

has a right adjoint denoted by f : FY —> FX. If the adjunction is reflective (i.e., Ff H f and Ff o f = id^y), we say that f is a left F-quotient.

Proposition 3.10 Let (U,e,v) be a left KZ-comonad in a poset-enriched category D. Then ex : UX —> X is a left U-quotient with ex = vx.

Proof. The condition (KZ3) states that Uex H vx. By the unit law eux o vx = idux, it follows that this adjunction is reflective. □

We arrive at the main theorem of this section.

Theorem 3.11 The following statements are equivalent for a left KZ-comonad (U,e,v) in a poset-enriched category D and any object A e D;

(1) A is a left projective over left U-quotients.

(2) A is projective over left U-quotients.

(3) A is a U-coalgebra. These conditions imply

(4) A is a left Kan object over the left U-arrows.

Moreover, assuming that any one of the equivalent conditions (1) - (3) holds, if p : Y —> X is a left U-arrow and f : A —> X is any arrow in D, then

f/p = ey o p o Uf o mA,

where mA : U —> UA is the co-structure map of the coalgebra A.

4 Applications

4.1 E-projective normal semi-rings

A semi-ring morphism f : R —> S between normal semi-rings R and S is said to be perfect if Uf has a reflective right adjoint, i.e., a semi-ring morphism s : US —> UR such that Uf H s and Uf o s = idus1.

Remark 4.1 Every perfect semi-ring morphism has a section in NSRng.

Throughout this section, we denote by E the class of perfect semi-ring mor-phisms. For the remaining part of this section, we specialize the definitions of F, G, U, e and v to be those of Example 2.8.

Remark 4.2 For any normal semi-ring R, the co-unit map

e : UR R, {A A

is such that UeR : U2R —> UR has a reflective right adjoint vr : UR —> U2R given by

{A ^ {{B | 3a e A. {B C{a}. This provides a natural example of a perfect semi-ring morphism.

Before arriving at the main result of this section, we need to introduce two definitions new to the theory of semi-rings.

Definition 4.3 Let (R, +, •) be a semi-ring. Define an auxiliary relation (read as finitely way-below) on R as follows:

x ^f y ^^ VA fi R. a > y 3a E A.x > a I .

VaeA /

Viewing a normal semi-ring as an ordered monoid, the sum aeA a may be seen as A.

Definition 4.4 A normal semi-ring (R, +, •) is said to be F-continuous if for any

x E R,

(i) |x := {r E R | r ^f x} is the lower closure (w.r.t. the induced partial order) of a finite subset of R, and

(ii) x = V 4x.

A semi-ring R is said to be stably F-continuous if in addition to (i) and (ii) it satisfies the following condition:

(iii) x ^f y • z if and only if there exist y' and z' in R such that y' ^f y, z' ^f z and x < y' • z'.

Lemma 4.5 A normal semi-ring (R, +, •) is the underlying object of a U-coalgebra if and only if it is stably F-continuous.

Proof. By Lemma 3.4, R is the underlying object of a U-coalgebra if and only if its co-unit £r : UR —> R has a coreflective right adjoint 3 : R —> UR. This is equivalent to the condition that 3 o er < id^R and e o ( = idR. Since £r(| A) = aeA a = V A, the adjoint situation forces the preceding inequalities to be equivalent to

3(r) = H{|A E FR I r <\J A}.

Thus, s E 3(r) if and only if s belongs to the lower closure of every finite set A C R with r < V A, i.e., s ^f r. Hence R is a continuous normal semi-ring. Since 3 preserves the multiplication •, it follows that for any y and z E R, ((y • z) = 3(y) ® 3(z). Thus, this second condition is equivalent to:

x <f (y • z) ^^ x E 3(y) ® 3(z)

^^ (3y' <f y) A (3z' <f z). (x < y' • z'), which is just the condition that R is stably F-continuous. □

Example 4.6 By virtue of the Fundamental Theorem of Arithmetic, the semi-ring R = (N, lcm, gcd) with lowest common multiple as addition and greatest common divisor as multiplication is normal. Viewed as an ordered monoid, the partial order < on R is just the divisibility relation, i.e., a < b ^^ a | b. Clearly, 0 ^f 0 and 1 ^f 1 in R, and crucially, if y = 0,1, then x ^f y ^^ x = pk for some prime p

and x \ y. It follows immediately by Euclid's lemma that R is a stably continuous normal semi-ring.

Theorem 4.7 The following are equivalent for any normal semi-ring R:

(i) R is E-projective.

(ii) R is the underlying object of a U-coalgebra.

(iii) R is stably F-continuous.

Proof. A direct consequence of Lemmas 3.4 and 4.5, and Theorem 3.11. □

Remark 4.8 An independent work [16] of N. T. Nai has established results that are similar to ours but her class E of semi-ring morphisms f are those for which G(f) has a section. Not surprising, Nai has to separately prove that for a normal semiring S, the canonical structure (D0(S),U, <8>) is stably finitely continuous. (see [16, Proposition 5.2.10, p.67]) since she relies only on Lemma 1.2.

4.2 E-projective Z-frames and frames

In this subsection, we overload some of the definitions used in the preceding subsection.

Definition 4.9 A Z-frame homomorphism f : A —> B between Z-frames A and B is said to be perfect if Uzf has a reflective right adjoint, i.e., a Z-frame homomorphism s : Uz B —> Uz A such that Uz f H s and Uz f o s = iduz b .

A frame homomorphism f : A —> B between frames A and B is said to be perfect if Uzf has a reflective right adjoint, i.e., a frame homomorphism s : UzB —► Uz A such that Uz f H s and Uzf o s = idUzb.

In domain theory, the way-below relation, is the central relation that has been extensively studied [11, pp.49-78]. We modify it for the situation of Z-frame:

Definition 4.10 Let A be a Z-frame. Define an auxiliary relation ^z (read as Z-way-below) on A as follows:

x <Z y ^ VD e Z (A). (\J D > y =^3d e D. x < d).

Corresponding to the adjunction ZFrm H Frm, we have to define a new binary relation -<z for a frame:

Definition 4.11 Let A be a frame. Define an auxiliary relation —z (read as Z-beneath) on A as follows:

x —z y ^ VC e rz (A). fyc > y x e ü}.

Remark 4.12 A special instance of —z has appeared in both the work of Es-cardo([10]), and the joint work of the author and Zhao ([12]) independently. In their works, Z selects all the directed subsets.

Definition 4.13 A Z-frame A is said to be Z-continuous if for any x e A,

(i) |zx := {a e A \ a ^z x} e Z(A) and

(ii) x = V izx.

A Z-frame A is said to be stably Z-continuous if in addition to (i) and (ii) it satisfies the following conditions:

(iii) 1 ^z 1, and

(iv) x ^z y A z if and only if there exist y' and z' in A such that y' ^z y, z' ^z z

and x < y' A z'.

Definition 4.14 A frame A is said to be Tz-continuous if for any x e A, it holds that x = V 7(x), where 7(x) := {a e A \ a —z x}. A frame A is said to be .stably Fz-continuous if in addition it satisfies the following conditions:

(i) 1 —z 1 and

(ii) x —z y A z if and only if there exist y' and z' in A such that y' —z y, z' —z z

and x < y' A z'.

Remark 4.15 Notice that it always holds that j(x) e rz(A) for any x e A. This follows immediately from the definitions of —z and Z-closed sets.

Lemma 4.16 Let Z be a subset system on SLat. Then the following statements hold:

(i) A Z-frame A is the underlying object of a Uz-coalgebra if and only if A is stably Z-continuous.

(ii) A frame A is the underlying object of a Uz-coalgebra if and only if A is stably Fz-continuous.

Proof. Similar to the proof of Lemma 4.5. □

Theorem 4.17 (i) Let E be the set of perfect Z-frame homomorphisms. The following are equivalent for any Z-frame A:

(i) A is E-projective.

(ii) A is the underlying object of a Uz-coalgebra.

(iii) A is stably Z-continuous.

(ii) Let E be the set of perfect frame homomorphisms. The following are equivalent for any frame A:

(i) A is E-projective.

(ii) A is the underlying object of a Uz-coalgebra.

(iii) A is stably Fz-continuous.

Proof. A direct consequence of Lemmas 3.4 and 4.16, and Theorem 3.11. □

5 Conclusion

This paper makes a simple adaptation of Escardo's KZ-monadic machinery to deal with the characterisation of projectives. Instead of re-inventing the wheel, the

main idea here is to turn the wheel in the opposite direction. Our Theorem 3.11, which is the category-dual of Theorem 1.1, allows us to characterise the underlying coalgebras of the comonads in consideration as precisely the E-projectives with E as the class of those left monad-quotients. Such a technique is applied to the several comonadic situations arising from the theory of ordered monoids and normal semirings, and from the theory of frames, and in each case characterises the E-projectives as precisely those stably continuous structures (each respect to ^f, ^Z and <z). In summary, we demonstrated that it is easy to formulate and apply the dual version of Escardó's monadic machinery, and that his method is, in fact, just as powerful in characterising projectives as it is for injectives. One important piece of future work is to find explicit characterisations of the various perfect morphisms for each of the different categories in the running examples. Currently, we have not obtained such characterisations.

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