Theory and Applications of Categories, Vol. 14, No. 10, 2005, pp. 221–243.

CANONICAL AND OP-CANONICAL LAX ALGEBRAS

GAVIN J. SEAL

Abstract. The definition of a category of (T, V)-algebras, where V is a unital commutative quantale and T is a Set-monad, requires the existence of a certain lax extension of T. In this article, we present a general construction of such an extension. This leads to the formation of two categories of (T, V)-algebras: the category Alg(T, V) of canonical (T, V)-algebras, and the category Alg(T′, V) of op-canonical (T, V)-algebras. The usual topological-like examples of categories of (T, V)-algebras (preordered sets, topological, metric and approach spaces) are obtained in this way, and the category of closure spaces appears as a category of canonical (P, V)-algebras, where P is the powerset monad. This unified presentation allows us to study how these categories are related, and it is shown that under suitable hypotheses both Alg(T, V) and Alg(T′, V) embed coreflectively into Alg(P, V).

1. Introduction

Following the description by Manes [11] of the category of compact Hausdorff spaces as the Eilenberg-Moore category of the ultrafilter monad U, Barr [1] showed that by weakening the axioms used to define a monad and its algebras, the resulting Eilenberg-Moore category could be seen to be isomorphic to the category Top of topological spaces. The category Met of premetric spaces benefitted from a similar treatment in Lawvere’s fundamental paper [9], via the identity monad I this time. In recent years, Clementino, Hofmann, and Tholen [2, 6, 5] extended these results and provided a unified setting that presented each of these categories as a particular instance of the category Alg(T, V) of so-called (T, V)-algebras, where T is a Set-monad and V a unital commutative quantale. For example, if V is the two-element lattice 2, the category Alg(T, 2) is isomorphic to either the category Ord of preordered sets or to Top, by taking T to be the identity or the ultrafilter monad, respectively. In the same way, if V is the extended real half-line \overline{R}_{+}, then Alg(T, \overline{R}_{+}) is isomorphic to either Met or to the category App of approach spaces depending on whether T = I or U.

Although the scope of this unified setting is striking, closure spaces do not seem to appear as such (T, V)-algebras. This gap comes as a surprise, since all the mentioned structures are intimately linked to certain “closure-like” operators. Also, the powerset monad—which is a natural candidate for T—does not appear to provide any meaningful example. A similar situation seems to occur for the filter monad F, whose corresponding (F, V)-algebras lack a certain monotonicity condition to describe topological spaces conveniently. Note however that by lifting the theory to a larger setting (see [13] and [8]), it is possible to include the monotonicity condition in the definition of the algebras.

Partial financial assistance by NSERC through a grant to M. Barr is gratefully acknowledged.

Received by the editors 2005-02-03 and, in revised form, 2005-05-24.

Transmitted by Walter Tholen. Published on 2005-06-29.

2000 Mathematics Subject Classification: 18C20, 18B30, 54A05.

Key words and phrases: V-matrix, (T, V)-algebra, ordered set, metric space, topological space, approach space, closure space, closeness space, topological category.

\begin{matrix}c \\ ◯ \end{matrix} Gavin J. Seal, 2005. Permission to copy for private use granted.

The original intent of the present work was to close the “closure space/powerset” gap by showing that closure spaces could be described as (T, V)-algebras via the powerset monad, modulo a slight modification in the axioms used to define (T, V)-algebras. In the process, a crucial property of lax algebras appeared, namely that the monotonicity condition is a consequence of the reflexivity and transitivity of the structure matrices. This led to reconsider the case of the filter monad, and it resulted that Top could be shown to be isomorphic to a category of (F, 2)-algebras without the use of any additional construction. By investigating further the algebras related to this monad, it also followed that App could be described as a category of (F, \overline{R}_{+})-algebras.

In fact, an important aspect of the theory was beginning to emerge. Indeed, before considering the category Alg(T, V) itself, a certain extension of the monad T is required. One approach to the existence of such an extension is discussed in [3], and unicity is obtained for V = 2. With the weaker axioms introduced here however, it is possible to put forth two other extensions of T by assuming similar conditions on T and V, but with very different techniques. Because all the significant examples may be obtained in this manner, the extensions described here are called the canonical and op-canonical extensions of T, depending on whether the structures of the resulting (T, V)-algebras are monotone increasing or decreasing in their first variable. For example, the category Clos of closure spaces may be obtained as a (P, 2)-algebra via the canonical extension of P, and the category Top as a (F, 2)-algebra via the op-canonical extension of F. Although all the examples mentioned in this introduction are isomorphic to canonical (T, V)-algebras, where T is one of I, U or P, they may also be described as either canonical or op-canonical (T, V)-algebras, where T is one of F or P (in all cases, V is either 2 or \overline{R}_{+}). Moreover, a new category appears: the category Clsn of closeness spaces whose objects are the metric counterpart of closure spaces, in the same way that approach spaces are the metric counterpart of topological spaces.

Thus, denoting by Alg(T, V) the category of canonical (T, V)-algebras, and by Alg(T′, V) the category of op-canonical (T, V)-algebras, we obtain the following list of isomorphisms:

Ord ≅ Alg(I, 2) ≅ Alg(P′, 2)  Met ≅ Alg(I, \overline{R}_{+}) ≅ Alg(P′, \overline{R}_{+})
Top ≅ Alg(U, 2) ≅ Alg(F′, 2)  App ≅ Alg(U, \overline{R}_{+}) ≅ Alg(F′, \overline{R}_{+})
Clos ≅ Alg(P, 2) ≅ Alg(F, 2)  Clsn ≅ Alg(P, \overline{R}_{+}) ≅ Alg(F, \overline{R}_{+})

From a general point of view, it is possible to determine a certain number of adjunctions between these categories which result in either embeddings or isomorphisms. In particular, the category of op-canonical (P, V)-algebras—which is isomorphic to Alg(I, V)—embeds as a full coreflective subcategory into the category of op-canonical (T, V)-algebras, and under a suitable hypothesis, the category of canonical (T, V)-algebras embeds as a full coreflective subcategory into the category of canonical (P, V)-algebras. For T = I or U, the (T, V)-algebras are both canonical and op-canonical, so the mentioned results are illustrated by the two horizontal lines in the following commutative diagram of coreflective embeddings:

\xymatrix{Ord \ar[d] \ar[r] & Top \ar[d] \ar[r] & Clos \ar[d] \\ Met \ar[r] & App \ar[r] & Clsn}

where the vertical arrows are induced by the coreflective embedding E : 2 → \overline{R}_{+}, as described in [5].

The general theory pertaining to these results is presented in Sections 2 to 4. Sections 5 and 6 contain the applications of the theory to the powerset and filter monads, although these are also used throughout the previous sections to illustrate the different definitions introduced therein.

2. Lax algebras

There are two major differences between the definition of lax algebras given in [5] or [6], and the weaker one given here. First, it is sufficient for our purpose to work with a lax extension T_{M} of a Set-functor T : Set → Set rather than with a V-admissible monad; indeed, the close interplay occurring between Set and Mat(V) allows us to use the properties of the original monad, without reference to the op-laxness in Mat(V) of either its unit or multiplication (see for example the proof of the monotonicity of a lax algebra’s structure matrix in 2.6, or the proof of Proposition 2.7). Second, in order to include closure spaces as models of lax algebras, the lax functor T_{M} : Mat(V) → Mat(V) must not extend the functor T : Set → Set strictly, nor commute with the involution ^{∘}; the replacement conditions are given in (1) below. For the sake of completeness and in order to settle the notations, we begin by recalling the main definitions and results pertaining to (T, V)-algebras.

2.1. Quantales. Throughout this article, V will denote a unital commutative quantale, i.e. a complete lattice provided with an associative and commutative binary operation ⊗ which preserves suprema in each variable:

a ⊗ ⋁_{i∈I} b_{i} = ⋁_{i∈I}(a ⊗ b_{i}),

and for which there is a neutral element k. The bottom and top elements of V are denoted by ⊥ and ⊤ respectively.

For example, the two-element chain 2 = {⊥, ⊤} with x ⊗ y being the infimum of x and y, and k = ⊤ is a suitable candidate for V.

The extended half-line \overline{R}_{+} = [0, ∞], considered for our purpose with the order opposite to the natural order, with ⊗ being the addition (for which ∞ is an absorbing element) and k the top element 0, is another candidate for V.

2.2. V-matrices. The objects of the category Mat(V) are sets, and the morphisms r : X ↛ Y are functions r : X × Y → V; these morphisms will often be referred to as Vmatrices, (or simply as matrices) since they may be viewed as matrices (r(x, y))_{x∈X,y∈Y}. Composition is given by

(sr)(x, z) = ⋁_{y∈Y} r(x, y) ⊗ s(y, z) ,

where r : X ↛ Y and s : Y ↛ Z. The identity 1_{X} : X ↛ X is defined by 1_{X}(x, y) = k if x = y and 1_{X}(x, y) = ⊥ otherwise.

There is a partial order on the hom-sets of Mat(V) induced by the partial order on V, and given by r ≤ r′ if and only if r(x, y) ≤ r′(x, y) for all x ∈ X, y ∈ Y; this order is compatible with composition. There is also an order-preserving involution sending a morphism r : X ↛ Y to its transpose r^{∘} : Y ↛ X defined by r^{∘}(y, x) = r(x, y). Note that (1_{X})^{∘} = 1_{X} and (sr)^{∘} = r^{∘}s^{∘} by commutativity of ⊗.

Finally, there is a functor M : Set → Mat(V) which maps objects identically and sends a map f : X → Y to the matrix f : X ↛ Y given by

f(x, y) = \{ \begin{matrix}k & if f(x) = y \\ ⊥ & otherwise. \end{matrix}

Naturally, the functor M sends the identity map to the identity matrix. Since it will always be possible to deduce from the context whether we are working with a Set-map f : X → Y or its image f : X ↛ Y, we will not use the notation Mf : X ↛ Y. Thus, by composing a map f : X → Y, a matrix s : Y ↛ Z, and the transpose of a map g : Y → Z, we get the convenient formula (g^{∘}sf)(x, y) = s(f(x), g(y)). Notice also that 1_{X} ≤ f^{∘}f and ff^{∘} ≤ 1_{X}, so for t : X ↛ Z we have

t ≤ sf ⟺ tf^{∘} ≤ s  and  gr ≤ t ⟺ r ≤ g^{∘}t .

These properties may be used to obtain the pointwise notation of the various conditions presented further on (see [5] for details).

2.3. Monads. A Set-monad T is a triple (T, e, m), where T : Set → Set is a functor, and the unit e : Id → T and multiplication m : T T → T of T are natural transformations satisfying

m(T e) = 1 = m(eT) and m(T m) = m(mT) .

The identity monad I is simply the triple (Id, 1, 1).

The powerset monad P = (P, e, m) is defined as follows. The powerset functor P assigns to a set X the set P X of subsets of X, and sends a map f : X → Y to P f :

P X → P Y defined by P f(A) = {f(x) | x ∈ A}, where A ⊆ X. For x ∈ X and A ∈ P P X, the maps e_{X} and m_{X} are given by

e_{X}(x) = {x}  and  m_{X}(A) = ⋃A .

The filter monad F = (F, e, m) is defined as follows. The filter functor F assigns to a set X the set F X of filters on X, and sends a map f : X → Y to F f : F X → F Y defined by A ∈ F f(𝔣) ⟺ f^{−1}(A) ∈ 𝔣, where 𝔣 ∈ F X. The maps e_{X} and m_{X} are given by

A ∈ e_{X}(x) ⟺ x ∈ A  and  A ∈ m_{X}(𝔉) ⟺ A^{♯} ∈ 𝔉 ,

where A^{♯} = {𝔣 ∈ F X | A ∈ 𝔣}, x ∈ X and 𝔉 ∈ F F X.

Finally, the ultrafilter monad U = (U, e, m) is defined similarly to the filter monad, by replacing the filter functor F by the ultrafilter functor U which assigns to a set X the set of ultrafilters on X.

2.4. Lax extensions of T. A lax extension of a Set-functor T is a map

T_{M} : Mat(V) → Mat(V)
(r : X ↛ Y) ↦ (T_{M} r : T X ↛ T Y)

which preserves the partial order on the hom-sets and satisfies

(1) T f ≤ T_{M} f and (T f)^{∘} ≤ T_{M} f^{∘} ,

(2) (T_{M} s)(T_{M} r) ≤ T_{M} (sr)

for all f : X → Y, r : X ↛ Y and s : Y ↛ Z. A Set-monad T = (T, e, m) equipped with a lax extension T_{M} of T will be called a lax extension of (T, e, m), and will be denoted slightly abusively by T = (T_{M} , e, m). It should be stressed however that (T_{M} , e, m) is not a lax monad in the sense of [3]; in particular, e and m need not be op-lax in Mat(V) (see however Proposition 3.5 below).

In the presence of a Set-map, (2) may become an equality and allow part of (1) to be treated as such. Indeed, if f : X ↛ Y and g : Y ↛ Z come from Set-maps, then

T_{M} (sf) = (T_{M} s)(T f) = (T_{M} s)(T_{M} f) and T_{M} (g^{∘}r) = (T g)^{∘}(T_{M} r) = (T_{M} g^{∘})(T_{M} r) .

The first set of equalities follows from

T_{M} (sf) ≤ T_{M} (sf)(T f)^{∘}(T f) ≤ T_{M} (sf)(T_{M} f^{∘})(T f)
≤ (T_{M} (sff^{∘}))(T f) ≤ (T_{M} s)(T f) ≤ (T_{M} s)(T_{M} f) ≤ T_{M} (sf) ,

and the second is obtained similarly.

Notice that if T_{M} : Mat(V) → Mat(V) is a lax extension of T, then T ′_{M} : Mat(V) → Mat(V) given by

T ′_{M} r := (T_{M} r^{∘})^{∘} ,

is also a lax extension of T. As we will show further on, these two extensions are not necessarily equal.

2.5. The induced preorder. A lax extension T_{M} of a Set-functor T induces the following preorder on the set T X:

𝔵 ≤ 𝔶 ⟺ k ≤ T_{M} 1_{X}(𝔶, 𝔵) ,

where 𝔵, 𝔶 ∈ T X. Indeed, the condition (1) yields reflexivity, while (2) yields transitivity. As a consequence, if 𝔵 ≤ 𝔵′ and 𝔶′ ≤ 𝔶, then T_{M} r(𝔵, 𝔶) ≤ T_{M} r(𝔵′, 𝔶′), which means that T_{M} r preserves this preorder in the first variable and reverses it in the second. Similarly, T ′_{M} r reverses it in the first variable and preserves it in the second.

If f : X → Y is a Set-map, and 𝔵, 𝔶 are elements of T X such that 𝔵 ≤ 𝔶, then

T_{M} 1_{Y} (T f(𝔶), T f(𝔵)) = (T f)^{∘}(T_{M} 1_{Y})(T f)(𝔶, 𝔵) = T_{M} (f^{∘}f)(𝔶, 𝔵) ≥ 1_{X}(𝔶, 𝔵) ≥ k ,

so that T f(𝔵) ≤ T f(𝔶). This shows that T f preserves the preorder on T X (and T may be seen as a functor T : Set → Ord).

2.6. Lax algebras. For a Set-monad T = (T, e, m) equipped with a lax extension T_{M} of T, the category Alg(T, V) of (T, V)-algebras, also called lax algebras, has as objects pairs (X, r) with X a set and r : T X ↛ X a structure matrix satisfying the reflexivity and transitivity laws:

(3) 1_{X} ≤ re_{X} ,

(4) r(T_{M} r) ≤ rm_{X} .

A morphism f : (X, r) → (Y, s) is a Set-map f : X → Y satisfying:

(5) r ≤ f^{∘}s(T f) ,

and composing as in Set.

A crucial property of the structure matrix of a lax algebra (X, r) is the preservation of the preorder on T X (in the first variable):

𝔵 ≤ 𝔶 ⟹ r(𝔵, z) ≤ r(𝔶, z) .

Indeed, reflexivity of r yields 1_{T X} ≤ T_{M} 1_{X} ≤ (T_{M} r)(T e_{X}), so that if 𝔵, 𝔶 ∈ T X are such that 𝔵 ≤ 𝔶, we have

r(𝔵, z) ≤ T_{M} 1_{X}(𝔶, 𝔵) ⊗ r(𝔵, z) ≤ T_{M} r(T e_{X}(𝔶), 𝔵) ⊗ r(𝔵, z) ≤ r(𝔶, z) ,

by transitivity of r. Moreover, if the previous monad T is replaced by the monad T′ = (T ′_{M} , e, m) (but the preorder on T X is still defined via T_{M}), then a similar argument yields that r reverses the preorder on T X:

𝔵 ≤ 𝔶 ⟹ r(𝔶, z) ≤ r(𝔵, z) .

If k is the top element of V, then a morphism f : (X, r) → (Y, s) of (T, V)-algebras may be defined by using T_{M} in place of T. Indeed, in this case s(T_{M} 1_{Y})(𝔵, z) = ⋁_{𝔶≤𝔵} s(𝔶, z) = s(𝔵, z), so that f^{∘}s(T f) = f^{∘}s(T_{M} 1_{Y})(T f) = f^{∘}s(T_{M} f). The same argument holds in the case where f : (X, r) → (Y, s) is a morphism of (T′, V)-algebras.

2.7. Proposition. The category of (T, V)-algebras is topological. In fact, for a family of (Y_{i}, s_{i})_{i∈I} of (T, V)-algebras and (f_{i} : X → Y_{i})_{i∈I} of Set-maps, the initial structure r on X can be described by r = ⋀_{i∈I} f_{i}^{∘}s_{i}(T f_{i}), or

r(𝔵, y) = ⋀_{i∈I} s_{i}(T f_{i}(𝔵), f_{i}(y))

in pointwise notation, where 𝔵 ∈ T X and y ∈ X.

Proof. This result may be proved as in [5].

2.8. Examples of (T, V)-algebras. Since a (T, V)-algebra in the sense of [5] is a (T, V)-algebra in the above sense, we have the following examples.

The category Alg(I, 2) is the category Ord of preordered sets, and Alg(I, \overline{R}_{+}) is the category Met of premetric spaces.

For the ultrafilter monad U = (U, e, m), we can define the lax extension

U_{M} r(𝔵, 𝔶) := ⋀_{\begin{matrix}A∈𝔵 \\ B∈𝔶 \end{matrix}} ⋁_{\begin{matrix}x∈A \\ y∈B \end{matrix}} r(x, y) ,

where r : X ↛ Y, 𝔵 ∈ UX and 𝔶 ∈ UY. Then Alg(U, 2) is isomorphic to the category Top of topological spaces, and Alg(U, \overline{R}_{+}) to the category App of approach spaces.

2.9. Taut monads. A functor T : Set → Set is taut if it preserves inverse images, i.e. for any map f : X → Y and subset B of Y, the pullback (T f)^{−1}(T B) is isomorphic to T (f^{−1}(B)). As a consequence, if ι : A → X is an injection, then T ι : T A → T X is one too (this allows us to avoid certain technical difficulties related to injections of empty sets into non-empty sets). In order to simplify notations, if A is a subset of X we will consider T A as a subset of T X, and similarly (T f)^{−1}(T B) will be identified with T (f^{−1}(B)). In the same way, we will write T f(T A) ⊆ T (f(A)) for any A ⊆ X. Finally, note that a taut functor preserves finite intersections.

Let T = (T, e, m) be a Set-monad with taut T. If the unit e and multiplication m are taut:

e_{X}(y) ∈ T A ⟺ y ∈ A and m_{X}(𝔛) ∈ T A ⟺ 𝔛 ∈ T T A

for any set X and A ⊆ X, then the monad T itself is said to be taut. Of course, e is taut if and only if e_{X} is injective for all X. Moreover, if T is taut, then so is e (see [12], Proposition 2.3).

The identity, powerset, filter and ultrafilter monads are all taut, and the previous identification convention yields natural results for their functors. Indeed, in the case of the powerset functor, we have for A, B ∈ P X that A ∈ P B ⟺ A ⊆ B. In the case of the filter functor, we have for 𝔣 ∈ F X that 𝔣 ∈ F A ⟺ A ∈ 𝔣, so with the notations of 2.3 we may write A^{♯} = F A.

2.10. Remark. Since the Beck-Chevalley condition (BC) of [2] has several important consequences in the theory of lax algebras, it is worth mentioning that if a functor satisfies (BC), then it it is naturally taut.

2.11. Completely distributive lattices. Let V be a complete lattice, and a, b ∈ V. Define a ≺ b whenever the following condition holds:

for any subset S ⊆ V with b ≤ ⋁ S, there exists s ∈ S satisfying a ≤ s .

The lattice V is completely distributive (see [14]) if for any b ∈ V, we have

b = ⋁{a ∈ V | a ≺ b} .

It follows immediately from its definition that the relation ≺ has the following properties: i) a ≺ b implies a ≤ b;

ii) a ≤ a′ ≺ b′ ≤ b implies a ≺ b;

iii) a ≺ ⋁ S implies there exists s ∈ S with a ≺ s.

For elements u and v of a completely distributive lattice, if a ≺ v for any a ∈ V with a ≺ u, then we can conclude that u ≤ v by taking the join of all elements a ≺ u. This argument will be used systematically in the following without necessarily explicit mention.

Notice that the lattice 2 is completely distributive (in which case ≺ is ≤), and the extended real half-line \overline{R}_{+} is too (with ≺ being <).

3. Canonical constructions

3.1. Canonical extensions. For a V-matrix r : X ↛ Y, define

r_{a}[A] := {y ∈ Y | there exists x ∈ A with a ≤ r(x, y)}
and T_{M} r(𝔵, 𝔶) := ⋁{a ∈ V | 𝔶 ∈ T (r_{a}[A]) for all A with T A ∋ 𝔵} ,

where a ∈ V, A ⊆ X, 𝔵 ∈ T X and 𝔶 ∈ T Y. Since the lax extension T ′_{M} obtained from T_{M} (see 2.4) will be of importance in the rest of this article, we explicit the corresponding definitions:

r_{a}^{∘}[B] := {x ∈ X | there exists y ∈ B with a ≤ r(x, y)}
and T ′_{M} r(𝔵, 𝔶) := ⋁{a ∈ V | 𝔵 ∈ T (r_{a}^{∘}[B]) for all B with T B ∋ 𝔶} ,

where a ∈ V, B ⊆ Y, 𝔵 ∈ T X and 𝔶 ∈ T Y. In case r is a 2-matrix, we will prefer to write r[A] and r^{∘}[A] in place of r_{⊤}[A] and r_{⊤}^{∘}[A] respectively. It will be proved further on that both T_{M} and T ′_{M} form lax extensions of T. The T_{M} defined above is called the canonical extension of T, and T ′_{M} is the op-canonical extension of T. Lemma 3.3 shows that if V is completely distributive, then the sets r_{a}[A] used to define T_{M} can be replaced by the smaller sets r\overline{a}[A] defined therein.

Remark that a ≤ b implies r_{b}[A] ⊆ r_{a}[A], and that A ⊆ B implies r_{a}[A] ⊆ r_{a}[B].

3.2. The canonical induced preorder. In the case where T_{M} is the canonical extension of T, the preorder on T X described in 2.5 is given by

𝔵 ≤ 𝔶 ⟺ for any A ⊆ X, if 𝔶 ∈ T A then 𝔵 ∈ T A ,

where 𝔵, 𝔶 ∈ T X. From now on, this will be the preorder used on T X. Note that it is preserved by m_{X} whenever m is taut.

This preorder yields the natural order induced by the functors considered in this article. Indeed, if T is the powerset functor P and A, B are subsets of X, then A ≤ B ⟺ A ⊆ B. If T is the filter functor F and 𝔵, 𝔶 are filters on X, then 𝔵 ≤ 𝔶 ⟺ 𝔵 is finer than 𝔶, since 𝔵 ∈ T A may be interpreted as A ∈ 𝔵 (see 2.9). Finally, if T is the identity functor I or the ultrafilter functor U, then 𝔵 ≤ 𝔶 ⟺ 𝔵 = 𝔶.

3.3. Lemma. Let r : X ↛ Y be a V-matrix, and suppose that V is completely distributive. For a ∈ V, define

r\overline{a}[A] := \{y ∈ Y | there exists x ∈ A with a ≺ r(x, y)\} .

Then for 𝔵 ∈ T X and 𝔶 ∈ T Y, we have

T_{M} r(𝔵, 𝔶) = ⋁\{a ∈ V | 𝔶 ∈ T (r\overline{a}[A]) for all A with T A ∋ 𝔵\} .

Proof. Define

S := {a ∈ V | 𝔶 ∈ T (r_{a}[A]) for all A with T A ∋ 𝔵} and
\overline{S} := \{a ∈ V | 𝔶 ∈ T (r\overline{a}[A]) for all A with T A ∋ 𝔵\} .

Since r\overline{a}[A] ⊆ r_{a}[A], we naturally have ⋁ \overline{S} ≤ ⋁ S.

Let b ∈ S and a ≺ b. This yields r_{b}[A] ⊆ r\overline{a}[A], so that if 𝔶 ∈ T (r_{b}[A]), then 𝔶 ∈ T (r\overline{a}[A]). Therefore, a ≤ ⋁ \overline{S} and b ≤ ⋁ \overline{S} because V is completely distributive, and we can conclude that ⋁ S ≤ ⋁ \overline{S}.

3.4. Lemma. Let r : X ↛ Y be a V-matrix, and suppose that V is completely distributive. For any A ⊆ X and a ∈ V, we have

(T_{M} r)\overline{a}[T A] ⊆ T (r\overline{a}[A]) ,

Proof. If 𝔶 ∈ (T_{M} r)\overline{a}[T A], then there exists 𝔵 ∈ T A with a ≺ T_{M} r(𝔵, 𝔶). Thus, there is an element b ∈ V with a ≺ b such that for all B with T B ∋ 𝔵, we have 𝔶 ∈ T (r_{b}[B]). In particular, 𝔶 ∈ T (r_{b}[A]) ⊆ T (r\overline{a}[A]), which yields the conclusion.

3.5. Proposition. If T is a taut functor, then T_{M} and T ′_{M} defined in 3.1 are lax extensions of T. Moreover, if T = (T, e, m) is a taut monad and V is completely distributive, then e and m are both op-lax in Mat(V) with respect to T_{M}, i.e. for any V-matrix r : X ↛ Y, we have

r ≤ e_{Y}^{∘} (T_{M} r)e_{X}  and  T 2_{M}r ≤ m_{Y}^{∘} (T_{M} r)m_{X} ,

and it follows that e and m are also op-lax with respect to T ′_{M}.

Proof. Let us first prove that T_{M} is a lax extension. If r, r′ : X ↛ Y are two Vmatrices such that r ≤ r′, then we have r_{a}[A] ⊆ r′_{a}[A] for any A ⊆ X and a ∈ V, so that T_{M} r ≤ T_{M} r′. Thus, T_{M} preserves the partial order on the hom-sets.

(1) Consider a map f : X → Y. To show that T f(𝔵, 𝔶) ≤ T_{M} f(𝔵, 𝔶), it is sufficient to consider the case 𝔶 = T f(𝔵). Let A ⊆ X be such that T A ∋ 𝔵, so T f(𝔵) ∈ T f(T A) ⊆ T (f(A)) and 𝔶 ∈ T (f(A)). Since f(A) ⊆ f_{a}[A] for any a ≤ k, we have k ≤ T_{M} f(𝔵, 𝔶) as required.

To verify that (T f)^{∘} ≤ T_{M} f^{∘}, suppose as before that T f(𝔵) = 𝔶. For any B ⊆ Y with 𝔶 ∈ T B, we have 𝔵 ∈ (T f)^{−1}(T B) = T (f^{−1}(B)). Remarking that f^{−1}(B) ⊆ f_{a}^{∘}[B] for all a ≤ k, we may conclude that k ≤ T_{M} f^{∘}(𝔶, 𝔵).

(2) Consider two V-matrices r : X ↛ Y and s : Y ↛ Z. Let a, b ∈ V be such that 𝔶 ∈ T (r_{a}[A]) for all A with T A ∋ 𝔵, and 𝔷 ∈ T (s_{b}[B]) for all B with T B ∋ 𝔶. For these A, we have 𝔷 ∈ T (s_{b}[r_{a}[A]]). Moreover,

s_{b}[r_{a}[A]] = {z ∈ Z | there exist x ∈ A, y ∈ Y with a ≤ r(x, y) and b ≤ s(y, z)}

⊆ {z ∈ Z | there exists x ∈ A with a ⊗ b ≤ (sr)(x, z)} = (sr)_{a⊗b}[A] .

Since ⊗ preserves suprema in each variable, we get (T_{M} s)(T_{M} r) ≤ T_{M} (sr) by taking the join of all a, b ∈ V chosen as above.

Therefore, T_{M} is a lax extension of T. But it also follows that T ′_{M} is a lax extension of T (see the concluding remark of 2.4).

To check that e is op-lax, consider a V-matrix r : X ↛ Y, x ∈ X, y ∈ Y and a = r(x, y). Thus, y ∈ r_{a}[A] for all A ∋ x. If e_{X}(x) ∈ T A, then x ∈ A by injectivity of e_{X}, so that y ∈ r_{a}[A]. This implies that a ≤ T_{M} r(e_{X}(x), e_{Y} (y)), and we can conclude that r ≤ e_{Y}^{∘} (T_{M} r)e_{X} as required. It also follows directly that r ≤ e_{Y}^{∘} (T ′_{M} r)e_{X}.

To check that m is op-lax, let 𝔛 ∈ T T X, 𝔜 ∈ T T Y and a ∈ V such that a ≺ T 2_{M}r(𝔛, 𝔜). Thus, if T T A ∋ 𝔛, then 𝔜 ∈ T ((T_{M} r)\overline{a}[T A]) ⊆ T T (r\overline{a}[A]) by Lemma 3.4. This implies that m_{Y} (𝔜) ∈ T (r\overline{a}[A]) for all A with T T A ∋ 𝔛, or equivalently for all A with T A ∋ m_{X}(𝔛) because m is taut. Lemma 3.3 then yields that a ≤ T_{M} r(m_{X}(𝔛), m_{Y} (𝔜)), so T 2_{M}r ≤ m_{Y}^{∘} (T_{M} r)m_{X} by complete distributivity of V. The corresponding inequality for T ′_{M} easily follows.

3.6. Canonical and op-canonical (T, V)-algebras. Consider the canonical and op-canonical extensions T_{M} and T ′_{M} of T. If T = (T_{M} , e, m) and T′ = (T ′_{M} , e, m), then a (T, V)-algebra will be called a canonical (T, V)-algebra, and a (T′, V)-algebra will be called an op-canonical (T, V)-algebra. Recall from 2.6 that the structure matrix of a canonical (T, V)-algebra preserves the preorder on T X in its first variable, and the structure matrix of an op-canonical (T, V)-algebra reverses it.

Examples of canonical and op-canonical (T, V)-algebras will be studied for the powerset and the filter monads in the last two sections. Notice that it may happen that a lax algebra is at the same time a canonical and op-canonical (T, V)-algebra: this is the case for the identity monad for which Id_{M} = Id′_{M} = Id, or the ultrafilter monad for which U ′_{M} r = U_{M} r (see Lemma 6.2); these situations are particular cases of the next proposition. Notice also that there exist (T, V)-algebras that are neither canonical nor op-canonical: this is the case for example for the lax algebras corresponding to the extension of Id considered in [5], Remark 3.2.

3.7. Proposition. Suppose that T ∅ = ∅ and for 𝔵 ∈ T X,

T B ∩ T A ≠ ∅ for all A with T A ∋ 𝔵 ⟹ 𝔵 ∈ T B ,

where B ⊆ X. Then the canonical and op-canonical extensions of T are equal.

Proof. Consider a V-matrix r : X ↛ Y, x ∈ T X, 𝔶 ∈ T Y, and a ∈ V. Suppose that a is such that 𝔶 ∈ T (r_{a}[A]) for all A with T A ∋ 𝔵. If moreover T B ∋ 𝔶, then 𝔶 ∈ T (r_{a}[A] ∩ B) because T preserves intersections, so that T ∅ = ∅ implies r_{a}[A] ∩ B ≠ ∅. Thus, for all A and B with T A ∋ 𝔵 and T B ∋ 𝔶, there exist x ∈ A and y ∈ B such that a ≤ r(x, y), or A ∩ r_{a}^{∘}[B] ≠ ∅. We then have T A ∩ T (r_{a}^{∘}[B]) ≠ ∅ because T preserves inclusions. Since this inequality holds for all A with T A ∋ 𝔵, we have 𝔵 ∈ T (r_{a}^{∘}[B]) by hypothesis. But this is true for all B with T B ∋ 𝔶, so that a ≤ T ′_{M} r(𝔵, 𝔶). Therefore, we have T_{M} r(𝔵, 𝔶) ≤ T ′_{M} r(𝔵, 𝔶). The same argument applied to r^{∘} : Y ↛ X shows that T ′_{M} r(𝔵, 𝔶) ≤ T_{M} r(𝔵, 𝔶), and we are done.

3.8. Corollary. If T preserves finite coproducts, then the canonical and op-canonical extensions of T are equal.

Proof. Let us verify the hypotheses of the proposition. Since T preserves finite coproducts, we have T ∅ = ∅. Suppose now that 𝔵 ∈ T X and B ⊆ X satisfy T B ∩ T A ≠ ∅ for all A with T A ∋ 𝔵, and set B′ = X \ B. If 𝔵 ∉ T B, then 𝔵 ∈ T X \ T B = T B′ because T preserves finite coproducts. But then T B ∩ T B′ = ∅, a contradiction. The proposition then yields the desired result.

To conclude this section, we exhibit an interesting property of the op-canonical (T, V)- algebras, which does not seem to have a counterpart in the canonical case.

3.9. Proposition. Let r : T X ↛ X be the structure matrix of an op-canonical (T, V)-algebra, and A a subset of T X. Then for 𝔛 ∈ T A and z ∈ X, we have

⋀_{𝔵∈A}r(𝔵, z) ≤ r(m_{X}(𝔛), z) .

Moreover, if all 𝔵 ∈ A satisfy 𝔵 ≤ m_{X}(𝔛), then ⋀_{𝔵∈A} r(𝔵, z) = r(m_{X}(𝔛), z).

Proof. Let a ∈ V be such that a ≤ ⋀_{𝔵∈A} r(𝔵, z). Thus, a ≤ r(𝔵, z) for all 𝔵 ∈ A, which implies that 𝔵 ∈ r_{a}^{∘}[{z}] for all 𝔵 ∈ A, and T A ⊆ T (r_{a}^{∘}[{z}]). So 𝔛 ∈ T (r_{a}^{∘}[{z}]) by hypothesis, and we can conclude that a ≤ T ′_{M} r(𝔛, e_{X}(z)) ≤ r(m_{X}(𝔛), z) because e_{X} is injective and r transitive.

The last equality naturally follows because r is preorder-reversing in its first variable.

3.10. Remark. From now on, we will suppose that the lattice V is completely distributive.

4. Isomorphisms and embeddings

4.1. Extension conditions. Let T be a Set-functor and A a collection of subsets of X. The following condition will be called the extension condition for A:

(E) for every 2-matrix r : X ↛ Y, and 𝔶 ∈ T Y such that 𝔶 ∈ T (r[B]) for all B ∈ A, there exists 𝔵 ∈ ⋂_{B∈A} T B such that 𝔶 ∈ T (r[A]) for all A ⊆ X with T A ∋ 𝔵.

For example, I and P both satisfy the extension condition for any A = {B} with B ⊆ X. Similarly, U and F satisfy (E) for any filter or filter base A (this is true for U by the Extension Lemma, see for example [8], Corollary 2.3).

4.2. Restriction of a monad. Let T = (T, e, m) and S = (S, d, n) be Set-monads such that T and S are both taut, and suppose that the preorder induced on the sets SX make them into complete atomistic lattices. We say that T is a restriction of S to atoms, if there is a natural transformation ι : T → S, such that ι_{X} sends T X bijectively onto the set of atoms of SX, d = ιe, and nι^{2} = ιm (with ι^{2} = (Sι)(ιT) = (ιS)(T ι)). In particular, ι is a morphism of monads, so that T is a submonad of S.

For 𝔵 ∈ T X we have 𝔵 ∈ T A ⟺ ι_{X}(𝔵) ∈ SA so the preorder induced on the sets T X is also an order. Remark also that T ∅ = ∅ because S∅ contains a unique element.

For B ⊆ SX, define

A_{B} := {𝔵 ∈ T X | there exists 𝔣 ∈ B with ι_{X}(𝔵) ≤ 𝔣} .

A restriction T of S is convenient if there exists a natural transformation σ : T S → ST satisfying ιT = σ(T ι), as well as the following two conditions:

𝔛 ∈ T B ⟹ σ_{X}(𝔛) ∈ SA_{B}  and  σ_{X}(𝔛) ∈ ST A ⟹ 𝔛 ∈ T SA

for any 𝔛 ∈ T SX, B ⊆ SX, and A ⊆ X. Since A_{SA} = T A for any A ⊆ X, these conditions imply in particular that 𝔛 ∈ T SA ⟺ σ_{X}(𝔛) ∈ ST A, so n(ιS) = n(Sι)σ whenever S is taut.

For example, the identity monad I is a restriction of P to singletons, and the ultrafilter monad U is a restriction of F to ultrafilters. Moreover, both these restrictions are convenient; this is immediate in the identity case since we may set σ = 1; the ultrafilter case will be considered in Proposition 6.3.

4.3. Lemma. Let T be a taut monad satisfying the extension condition for all sets A = {T B ⊆ T X | 𝔶 ∈ T B}, where 𝔶 ∈ T X. If r : T X ↛ X is the structure matrix of a canonical (T, V)-algebra, then for any 𝔵 ∈ T X and z ∈ X we have

r(𝔵, y) = ⋀_{T B∋𝔵} ⋁_{𝔷∈T B} r(𝔷, z) .

Proof. Define s(𝔵, z) := ⋀_{T B∋𝔵} ⋁_{𝔷∈T B} r(𝔷, z), and observe that r(𝔵, z) ≤ s(𝔵, z) naturally holds. Let a ∈ V be such that a ≺ s(𝔵, z). Thus, for all B with T B ∋ 𝔵, there exists 𝔷 ∈ T B with a ≤ r(𝔷, z), or z ∈ r_{a}[T B]. This implies that e_{X}(z) ∈ T (r_{a}[T B]) for all B with T B ∋ 𝔵. By the extension condition, there exists 𝔛 ∈ T T X such that 𝔛 ∈ T T B for all B with T B ∋ 𝔵, and e_{X}(z) ∈ T (r_{a}[B]) for all B with T B ⊇ 𝔛. This allows us to conclude that a ≤ T_{M} r(𝔛, e_{X}(z)) ≤ r(m_{X}(𝔛), z) by transitivity of r. Finally, m_{X}(𝔛) ≤ 𝔵 implies a ≤ r(𝔵, z) because r is increasing in its first variable. Thus, s(𝔵, z) ≤ r(𝔵, z) as required.

4.4. Lemma. Let T be a taut monad and r : T X ↛ X the structure matrix of a canonical (T, V)-algebra. Suppose that T preserves finite coproducts, and verifies the extension condition for all sets {T B ⊆ T X | 𝔵 ∈ T B}, where 𝔵 ∈ T X. If z ∈ X, 𝔶 ∈ T X and A ⊆ T X are such that 𝔶 ∈ T A for all A with T A ⊇ A, then

⋀_{𝔵∈A}r(𝔵, z) ≤ r(𝔶, z).

Proof. Remark first that if B ⊆ X is such that T B ∩ T A ≠ ∅ for all A with T A ⊇ A, then there exists 𝔵 ∈ A with 𝔵 ∈ T B (see the proof of Corollary 3.8). Thus, if T B ∋ 𝔶, then T B ∩ T A ≠ ∅ for all A with T A ⊇ A by hypothesis, so there exists 𝔵 ∈ A with 𝔵 ∈ T B. The previous lemma then implies

⋀_{𝔵∈A}r(𝔵, z) = ⋀_{𝔵∈A} ⋀_{T B∋𝔵} ⋁_{𝔷∈T B} r(𝔷, z) ≤ ⋀_{T B∋𝔶} ⋁_{𝔷∈T B} r(𝔷, z) = r(𝔶, z) .

4.5. Proposition. Let T = (T, e, m) and S = (S, d, n) be taut monads such that T is a convenient restriction of S to its atoms. Suppose furthermore that T preserves finite coproducts and satisfies the extension condition for all sets

{B ⊆ T X | A ⊆ T B}  and  {T B ⊆ T X | 𝔵 ∈ T B}

where A ⊆ T X and 𝔵 ∈ T X. Then the category of op-canonical (S, V)-algebras is isomorphic to the category of op-canonical (T, V)-algebras.

Proof. In this proof, elements 𝔵 ∈ T X will be considered as elements of SX via ι_{X}, and elements 𝔛 ∈ T T X will be considered as elements of SSX via ι_{SX}(T ι_{X}) = (Sι_{X})ι_{T X}. The symbols 𝔵, 𝔶 and 𝔛 will be used to designate elements of T X and T T X respectively, whereas 𝔣, 𝔤 and 𝔉 will denote elements of SX and SSX that are not necessarily atoms; there will also be mention of elements 𝔛′ of T SX.

For a relation r : SX ↛ X, let ř : T X ↛ X be the restriction of r to elements of T X. If (X, r) is an op-canonical (S, V)-algebra, then a routine verification shows that S′_{M} r(𝔛, 𝔶) = T ′_{M} ř(𝔛, 𝔶) and (X, ř) is naturally an op-canonical (T, V)-algebra.

Suppose that f : (X, r) → (Y, s) is a morphism of op-canonical (S, V)-algebras. Then ř(𝔵, y) ≤ s(Sf(𝔵), f(y)), and Sf(𝔵) = T f(𝔵) yields that f : (X, ř) → (Y, š) is a morphism of Alg(T′, V). Thus, we can define a functor R : Alg(S′, V) → Alg(T′, V) commuting with the underlying set functor, and sending (X, r) to (X, ř).

We now proceed to verify that there is a functor L : Alg(T′, V) → Alg(S′, V) commuting with the underlying set functor, and sending (X, r) to (X, r̂), where r̂: SX ↛ X is defined by

r̂(𝔣, y) = ⋀_{𝔵≤𝔣}r(𝔵, y)

(with the symbols 𝔵 designating atoms of SX). To prove that (X, r̂) is a (S′, V)-algebra, we only need to verify the transitivity condition for r̂ (because d_{X}(x) ∈ T X by hypothesis). Let 𝔉 ∈ SSX, 𝔤 ∈ SX, z ∈ X, and denote by 𝔛′ an element of T SX with 𝔛′ ≤ 𝔉. Note that for a ∈ V we have

r_{a}^{∘}[B] = A_{r̂_{a}^{∘}[B]} .

Let a ∈ V be such that a ≺ S′_{M} r̂(𝔉, 𝔤), and 𝔛 ∈ T T X with 𝔛 ≤ σ_{X}(𝔛′). Thus, for all B with SB ∋ 𝔤 we have 𝔉 ∈ S(r̂_{a}^{∘}[B]), so that 𝔛 ∈ T (r_{a}^{∘}[B]). The extension condition for the set {B ⊆ T X | 𝔤 ∈ SB} yields an atom 𝔶 ≤ 𝔤 with 𝔛 ∈ T (r_{a}^{∘}[B]) for all B

with T B ∋ 𝔶, and we have a ≤ T ′_{M} r(𝔛, 𝔶). Thus, a ⊗ r̂(𝔤, z) ≤ T ′_{M} r(𝔛, 𝔶) ⊗ r(𝔶, z) ≤

r(m_{X}(𝔛), z), which allows us to conclude that (S′_{M} r̂)(𝔉, 𝔤) ⊗ r̂(𝔤, z) ≤ r(m_{X}(𝔛), z) for all 𝔛 ≤ σ_{X}(𝔛′) and 𝔛′ ≤ 𝔉 with 𝔛′ ∈ T SX. Writing A′ = {𝔛′ ∈ T SX | 𝔛′ ≤ 𝔉} and A = {𝔛 ∈ T T X | there exists 𝔛′ ∈ A′ with 𝔛 ≤ σ_{X}(𝔛′)}, we have for B ⊆ SX:

𝔉 ∈ SSB ⟺ A′ ⊆ T SB ⟺ σ_{X}(A′) ⊆ ST B ⟺ A ⊆ T T B ,

so n_{X}(𝔉) ∈ SB ⟺ m_{X}(A) ⊆ T B. Since r is also the structure matrix of a canonical (T, V)-algebra by Corollary 3.8, we can apply Lemma 4.4 to get for any 𝔶 ∈ T X with 𝔶 ≤ n_{X}(𝔉):

(S′_{M} r̂)(𝔉, 𝔤) ⊗ r̂(𝔤, z) ≤ ⋀_{𝔵∈m_{X}(A)}r(𝔵, z) ≤ r(𝔶, z) .

This allows us to conclude that (S′_{M} r̂)(𝔉, 𝔤) ⊗ r̂(𝔤, z) ≤ r̂(m_{X}(𝔉), z), as required.

If f : (X, r) → (Y, s) is a morphism of Alg(T′, V), then r̂(𝔣, y) ≤ ⋀_{𝔵≤𝔣} s(T f(𝔵), f(y)). Let 𝔶 ≤ Sf(𝔣). By the extension condition, there exists 𝔵 ∈ T X with 𝔵 ≤ 𝔣 and T f(𝔵) = 𝔶.

This yields that

⋀_{𝔵≤𝔣}s(T f(𝔵), f(y)) = ⋀_{𝔶≤Sf(𝔣)}s(𝔶, f(y)) = ŝ(Sf(𝔣), f(y)) ,

so f : (X, r̂) → (Y, ŝ) is a morphism of Alg(S′, V).

Thus, we have two functors R : Alg(S′, V) → Alg(T′, V) and L : Alg(T′, V) → Alg(S′, V) commuting with the underlying set functors, and sending (X, r) to (X, ř), and (X, s) to (X, ŝ) respectively. By noticing that σ_{X}d_{SX}(𝔣) ∈ T A_{{𝔣}} and 𝔣 = n_{X}d_{SX}(𝔣) = n_{X}σ_{X}d_{SX}(𝔣), Proposition 3.9 implies ˆ ř(𝔣, y) = ⋀_{𝔵≤𝔣} r(𝔵, y) = r(𝔣, y). Moreover, we naturally have ˇ r̂(𝔵, y) = r(𝔵, y), so the functors R and L define an isomorphism between Alg(S′, V) and Alg(T′, V).

4.6. Corollary. The category Alg(P′, V) is isomorphic to Alg(I, V).

Proof. As mentioned previously, the monad I is a convenient restriction of P to its atoms. Since it preserves finite coproducts, and also satisfies the extension condition for the sets given in the proposition, the desired result follows. Note however that in this simple case, the previous proof may be considerably simplified.

4.7. Corollary. The category Alg(P′, 2) is isomorphic to Ord, and Alg(P′, \overline{R}_{+}) is isomorphic to Met.

Proof. This is an immediate consequence of the previous corollary and the fact that Alg(I, 2) ≅ Ord and Alg(I, \overline{R}_{+}) ≅ Met (see for example [5]).

For the rest of this section, the morphisms of (P, V)-algebras will be denoted by f : (X, c) → (Y, d), while those of (T, V)-algebras, will be denoted by f : (X, r) → (Y, s). The unit e and multiplication m of the powerset monad P will be given by their explicit formulation, whereas the monad T will be denoted T = (T, e, m).

4.8. Proposition. Let T = (T, e, m) be a taut monad satisfying the extension condition for any A = {B} with B ⊆ X. The functors R : Alg(P, V) → Alg(T, V) and L : Alg(T, V) → Alg(P, V) commuting with the underlying set functor, and defined on objects by R(X, c) = (X, ĉ), L(X, r) = (X, ř), where

ĉ(𝔵, y) := ⋀_{T B∋𝔵}c(B, y)  and
ř(A, y) := ⋁_{𝔶∈T A}r(𝔶, y) ,

(with A ∈ P X, 𝔵 ∈ T X, and y ∈ X) yield an adjunction L ⊣ R. Moreover, if T satisfies the extension condition for all sets {T B ⊆ T X | 𝔶 ∈ T B} with 𝔶 ∈ T X, then ˆ ř = r and L is a full coreflective embedding.

Proof. We first notice that ˇ ĉ ≤ c and that r ≤ ˆ ř for any matrices r : T X ↛ X and c : P X ↛ X.

Suppose that r : T X ↛ X is the structure matrix of a (T, V)-algebra. The reflexivity of ř follows immediately from the definition: k ≤ r(e_{X}(x), y) ≤ ř({x}, y). For the transitivity, let A ∈ P P X, B ∈ P X, z ∈ X, and a, b ∈ V be two elements such that a ≺ P_{M} ř(A, B) and b ≺ ř(B, z). This last inequality yields an element 𝔶 ∈ T B such that b ≤ r(𝔶, z). The first inequality implies that B ⊆ ř\overline{a}[A]. Furthermore,

ř\overline{a}[A] ⊆ \{y ∈ X | there exist A ∈ A, 𝔵 ∈ T A with a ≤ r(𝔵, y)\} ⊆ r_{a}[T (⋃A)] ,

and we may write 𝔶 ∈ T (r_{a}[T (⋃ A)]). By the extension condition, there exists 𝔛 ∈ T T (⋃ A) such that 𝔶 ∈ T (r_{a}[B]) for all B with T B ∋ 𝔛. This implies that a ≤ T_{M} r(𝔛, 𝔶), and by using that b ≤ r(𝔶, z), we get a ⊗ b ≤ r(m_{X}(𝔛), z) by transitivity of r. Moreover, 𝔛 ∈ T T (⋃ A) implies m_{X}(𝔛) ∈ T (⋃ A), so that r(m_{X}(𝔛), z) ≤ ř(⋃ A, z). Since a ⊗ b ≤ ř(⋃ A, z) for all elements a, b ∈ V with a ≺ P_{M} ř(A, B) and b ≺ ř(B, z), we may conclude that P_{M} ř(A, B) ⊗ ř(B, z) ≤ ř(⋃ A, z) as required. Consider now a morphism of (T, V)- algebras f : (X, r) → (Y, s). The map f : (X, ř) → (Y, š) is a morphism of (P, V)-algebras, since

ř(A, y) ≤ ⋁_{𝔵∈T (f^{−1}f(A))}r(𝔵, y) ≤ ⋁_{𝔵∈(T f)^{−1}(T (f(A)))}s(T f(𝔵), f(y)) ≤ š(P f(A), f(y)) .

Suppose now that c : P X ↛ X is the structure matrix of a (P, V)-algebra. The reflexivity of ĉ follows from the monotonicity of c and the injectivity of e_{X}; indeed, k ≤ c({x}, y) = ĉ(e_{X}(x), y). To prove the transitivity of ĉ, let 𝔛 ∈ T T X, 𝔶 ∈ T X, z ∈ X, and a ∈ V such that a ≺ T_{M} ĉ(𝔛, 𝔶). This last condition implies that 𝔶 ∈ T (ĉ_{a}[T A]) for all A with T T A ∋ 𝔛. Furthermore, ĉ_{a}[T A] ⊆ c_{a}[{A}], so that by setting B = c_{a}[{A}], we naturally have 𝔶 ∈ T B and B ⊆ c_{a}[A] for all A ⊇ {A}. This implies that a ≤ P_{M} c({A}, B), so that T_{M} ĉ(𝔛, 𝔶) ⊗ ĉ(𝔶, z) ≤ c(A, z) for all A with T T A ∋ 𝔛. Since m is taut, we have ĉ(m_{X}(𝔛), z) = ⋀_{T T A∋𝔛} c(A, z), and the transitivity of ĉ follows. Consider now a morphism of (P, V)-algebras f : (X, c) → (Y, d). The map f : (X, ĉ) → (Y, d̂) is a morphism of (T, V)-algebras, since

ĉ(𝔵, y) ≤ ⋀_{T A∋𝔵}d(P f(A), f(y)) ≤ ⋀_{T B∋T f(𝔵)}d(B, f(y)) = d̂(T f(𝔵), f(y)) ,

by using the monotonicity of d and the fact that 𝔵 ∈ T (f^{−1}(B)) ⟺ T f(𝔵) ∈ T B. The last statement follows directly from Lemma 4.3.

4.9. Corollary. Let T = (T, e, m) be a taut monad satisfying the extension condition for all sets {B} with B ⊆ X, and {T B ⊆ T X | 𝔶 ∈ T B} with 𝔶 ∈ T X. If for all A ⊆ X, there exists 𝔵_{A} ∈ T A with 𝔵_{A} ∈ T B ⟺ A ⊆ B, then Alg(T, V) is isomorphic to Alg(P, V).

Proof. With the notations of the previous proposition, it suffices to prove that ˇ ĉ = c for any structure matrix c : P X ↛ X. The definition of 𝔵_{A} implies that 𝔵 ≤ 𝔵_{A} for all 𝔵 ∈ T A, so by monotonicity of ĉ and c,

ˇĉ(A, y) = ⋁_{𝔵∈T A} ⋀_{T B∋𝔵}c(B, y) = ⋀_{T B∋𝔵_{A}} c(B, y) = c(A, y) ,

and we are done.

4.10. Proposition. The category of op-canonical (P, V)-algebras embeds as a full coreflective subcategory into the category of op-canonical (T, V)-algebras. More precisely, the functors R : Alg(T′, V) → Alg(P′, V) and E : Alg(P′, V) → Alg(T′, V) commuting with the underlying set functor, and defined on objects by R(X, r) = (X, r̂), E(X, c) = (X, č), where

r̂(A, y) := ⋀_{x∈A} r(e_{X}(x), y)  and
č(𝔵, y) := ⋁_{T B∋𝔵}c(B, y) ,

(with A ∈ P X, 𝔵 ∈ T X, and y ∈ X) yield an adjunction E ⊣ R such that ˆ č = c.

Proof. Proposition 3.9 implies that ˇ r̂ ≤ r(𝔵, y) and that ˆ č = c for any (T′, V)-algebra structure matrix r : T X ↛ X and (P′, V)-algebra structure matrix c : P X ↛ X.

Corollary 4.9 states that Alg(P′, V) is isomorphic to Alg(I, V) via the adjunction described in Proposition 4.5, so it is sufficient to prove that the matrix r̂(x, y) = r(e_{X}(x), y) is a structure of Alg(I, V). In this case, reflexivity is immediate, and transitivity follows from

r(e_{X}(x), y) ⊗ r(e_{X}(y), z) ≤ T ′_{M} r(e_{T X}(e_{X}(x)), e_{X}(y)) ⊗ r(e_{X}(y), z) ≤ r(e_{X}(x), z) .

If f : (X, r) → (Y, s) is a morphism of (T′, V)-algebras, we naturally have r(e_{X}(x), y) ≤ s(T f(e_{X}(x)), f(y)), and since T f(e_{X}(x)) = e_{Y} (f(x)), f is a morphism of the corresponding (P′, V)-algebras.

Let c : P X ↛ X be the structure matrix of a (P′, V)-algebra. The reflexivity of č follows from the fact that c is order-reversing and e_{X} injective. To prove the transitivity, let 𝔛 ∈ T T X, 𝔶 ∈ T X, z ∈ X, and a, b ∈ V two elements such that a ≺ T ′_{M} č(𝔛, 𝔶) and b ≺ č(𝔶, z). First note that by setting A_{B} := c_{a}^{∘}[B], we naturally have a ≤ P ′_{M} c(A_{B}, B). Furthermore, there exists B with T B ∋ 𝔶 and b ≤ c(B, z), so that 𝔛 ∈ T (č∘ ; \overline{a}[B]). Since

č∘ ; \overline{a}[B] ⊆ \{𝔵 ∈ T X | there exist y ∈ B, A ⊆ X with T A ∋ 𝔵 and a ≤ c(A, y)\} ⊆ T (⋃A_{B}) ,

we have m_{X}(𝔛) ∈ T (⋃ A_{B}). Therefore,

T ′_{M} č(𝔛, 𝔶) ⊗ č(𝔶, z) ≤ P ′_{M} c(A_{B}, B) ⊗ c(B, z) ≤ c(⋃A_{B}, z) ≤ č(m_{X}(𝔛), z)

by transitivity of c. Consider now a morphism of (P′, V)-algebras f : (X, c) → (Y, d). Then

č(𝔵, y) ≤ ⋁_{T A∋𝔵}d(P f(A), f(y)) ≤ ⋁_{T B∋T f(𝔵)}d(B, f(y)) = ď(T f(𝔵), f(y))

since {P f(A) ∈ P Y | T A ∋ 𝔵} ⊆ {B ∈ P Y | T B ∋ T f(𝔵)}.

5. The powerset monad and associated lax algebras

In this section, we give an alternate description of the categories of canonical and opcanonical (P, V)-algebras; in particular, we show that Alg(P, 2) is isomorphic to the category of closure spaces. The following result gives another description of the powerset’s canonical extension; of course, a similar formula may be obtained for its op-canonical extension (although it is of less interest here since the op-canonical algebras may be obtained via the identity monad by Corollary 4.6). This canonical extension also appears in [3], Example 6.3 as a lax functor H : Mat(2) → Mat(2), and as its lax extension to Mat(\overline{R}_{+}).

5.1. Proposition. The canonical extension of P is given by

P_{M} r(A, B) = ⋀_{y∈B} ⋁_{x∈A} r(x, y) ,

where A ∈ P X and B ∈ P Y.

Proof. Denote by T_{M} the canonical extension of P defined in 3.1. Let A, B ∈ P X and a ∈ V such that a ≺ P_{M} r(A, B). Thus, for every y ∈ B there exists x ∈ A with a ≤ r(x, y), so B ⊆ r_{a}[A] or equivalently B ∈ P (r_{a}[A]). Furthermore, if P C contains A, then we necessarily have A ⊆ C, so that B ∈ P (r_{a}[C]) and we can conclude that a ≤ T_{M} r(A, B).

Suppose now that a ∈ V is such that a ≺ T_{M} r(A, B). This implies in particular that B ∈ P (r_{a}[A]), i.e. for each y ∈ B, there exists x ∈ A with a ≤ r(x, y), and we may conclude that a ≤ P_{M} r(A, B).

Let us recall the definition of a closure space.

5.2. Closure spaces. Let X be a set. An operator c : P X → P X is a closure operator if it is extensive, monotone and idempotent:

(C_{1}) A ⊆ c(A);

(C_{2}) B ⊆ A ⟹ c(B) ⊆ c(A);

(C_{3}) c(c(A)) ⊆ c(A);

where A, B ∈ P X. A couple (X, c) is called a closure space. Closure spaces form the objects of the category Clos, whose morphisms f : (X, c) → (Y, d) are the Set-maps f : X → Y satisfying f(c(A)) ⊆ d(f(A)) for all A ∈ P X.

5.3. Proposition. The category Alg(P, 2) is isomorphic to Clos. In fact, a canonical (P, 2)-algebra (X, r) and a closure space (X, c) determine each other via

x ∈ c(A) ⟺ r(A, x) = ⊤ .

Proof. This is a particular case of Proposition 5.6 which is proved further on.

This result motivates the introduction of closeness operators, which might be seen as the metric counterpart of closure operators, since the former measure the distance between points and sets, rather than simply ascribing a true or false value to every such couple. As mentioned in the Introduction, closeness spaces are related to approach spaces in the same way that closure spaces are related to topological spaces.

5.4. Closeness spaces. The objects of the category Clsn(V) are couples (X, c), where X is a set and c : P X × X → V is a closeness operator, i.e. a map satisfying:

(C′_{1}) x ∈ A ⟹ k ≤ c(A, x);

(C′_{2}) B ⊆ A ⟹ c(B, x) ≤ c(A, x);

(C′_{3}) a ⊗ c(A^{(a)}, x) ≤ c(A, x);

where x ∈ X, A ∈ P X, a ∈ V and A^{(a)} = {x ∈ X | a ≤ c(A, x)}. The couple (X, c) is called a closeness space. A morphism of closeness spaces f : (X, c) → (Y, d) is a Setmap f : X → Y satisfying c(A, y) ≤ d(P f(A), f(y)). If V = 2, then we naturally have Clsn(2) = Clos. Moreover, if V = \overline{R}_{+}, we simply write Clsn instead of Clsn(V).

5.5. Remark. As in the context of approach spaces (see [10]), we observe that the conditions (C′_{1})–(C′_{3}) are equivalent to (C′_{1}), (C′_{2}) and

(C′′_{3}) ⋀_{y∈B} c(A, y) ⊗ c(B, x) ≤ c(A, x) for all A, B ⊆ X and x ∈ X.

Indeed, on one hand (C′′_{3}) implies (C′_{3}) by setting B = A^{(a)}. On the other hand, (C′_{2}) and (C′_{3}) imply (C′′_{3}) by setting a = ⋁{b ∈ V | B ⊆ A^{(b)}}. Notice also that the set A^{(a)} corresponds to the set c_{a}[{A}] in the notations of 3.1.

5.6. Proposition. The category Clsn(V) is isomorphic to Alg(P, V) via the following correspondence: a relation r : P X × X → V determines a closeness operator on X if and only if the associated matrix r : P X ↛ X is the structure of a canonical (P, V)-algebra.

Proof. Suppose first that (X, r) is a canonical (P, V)-algebra. Since r is orderpreserving in the first variable, reflexivity of r yields k ≤ r({x}, x) ≤ r(A, x) whenever x ∈ A. It also follows that B ⊆ A implies r(A, x) ≤ r(B, x). To prove (C′_{3}), set A = {A}. Then ⋀_{y∈A^{(a)}} ⋁_{B∈A} r(B, y) ⊗ r(A^{(a)}, z) = ⋀_{y∈A^{(a)}} r(A, y) ⊗ r(A^{(a)}, z) ≥ a ⊗ r(A^{(a)}, z). By transitivity, a ⊗ r(A^{(a)}, z) ≤ r(A, z) as required.

Suppose now that (X, c) is a closeness space. It is clear that k ≤ c({x}, x). If B ∈ P X and A ∈ P P X, then ⋁_{A∈A} c(A, y) ≤ c(⋃ A, y). Setting a = ⋀_{y∈B} c(⋃ A, y), we observe that B ⊆ (⋃ A)^{(a)}, so

⋀_{y∈B} ⋁_{A∈A} c(A, y) ⊗ c(B, z) ≤ a ⊗ c((⋃A)^{(a)}, z) ≤ c(⋃A, z) ,

and we are done.

The conclusion follows by noticing that the conditions for morphisms are equivalent.

6. The filter monad and associated lax algebras

As in the powerset case, the filter functor’s canonical and op-canonical extensions may be described without the use of the sets r_{a}[A] of Section 3. In this case however, we give the formula for the op-canonical extension.

6.1. Proposition. The op-canonical extension of F is given by

F ′_{M} r(𝔣, 𝔤) = ⋀_{B∈𝔤} ⋁_{A∈𝔣} ⋀_{x∈A} ⋁_{y∈B} r(x, y) ,

where 𝔣 ∈ F X and 𝔤 ∈ F Y.

Proof. Denote by T ′_{M} the op-canonical extension of F. Let 𝔣, 𝔤 ∈ F X and a ∈ V be such that a ≺ F ′_{M} r(𝔣, 𝔤). Thus, for every B ∈ 𝔤 there exists A ∈ 𝔣 satisfying A ⊆ r_{a}^{∘}[B]. As a consequence, for every B ∈ 𝔤, we have 𝔣 ∈ F (r_{a}^{∘}[B]) and a ≤ T ′_{M} r(𝔣, 𝔤). It follows

that F ′_{M} r(𝔣, 𝔤) ≤ T ′_{M} r(𝔣, 𝔤).

Suppose now that a ∈ V is such that a ≺ T ′_{M} r(A, B). This implies that for every B ∈ 𝔤, we have r_{a}^{∘}[B] ∈ 𝔣. Therefore, for every B ∈ 𝔤 there exists A ∈ 𝔣, namely A = r_{a}^{∘}[B], such that for every x ∈ A, there exists y ∈ B satisfying a ≤ r(x, y). Thus, we may conclude that a ≤ F ′_{M} r(𝔣, 𝔤).

6.2. Lemma. The lax extension U_{M} of the ultrafilter functor given in 2.8, is equal to both the canonical and op-canonical extensions of U. Moreover, U_{M} is equal to the restriction of F ′_{M} to ultrafilters, i.e. for all 𝔵, 𝔶 ∈ UX we have U_{M} r(𝔵, 𝔶) = F ′_{M} r(𝔵, 𝔶).

Proof. Corollary 3.8 yields that the canonical and op-canonical extensions of U are equal. The fact that the expression given in 2.8 describes these extensions may be seen as in the previous proposition. Again, the last claim may be proved with arguments similar to those in the proof of Proposition 3.7.

6.3. Proposition. The category of op-canonical (F, V)-algebras is isomorphic to the category of canonical (U, V)-algebras.

Proof. In order to apply Proposition 4.5, we only need to verify that U is a convenient restriction of F. Define σ_{X} : UF X → F UX by σ_{X}(𝔛) = {A_{B} | B ∈ 𝔛} for 𝔛 ∈ UF X, and consider a map f : X → Y. In order to verify that σ is a natural transformation, it is useful to first show that A_{(F f)^{−1}(B)} = (Uf)^{−1}(A_{B}). On one hand, if 𝔵 ∈ A_{(F f)^{−1}(B)}, then there exists 𝔣 ∈ (F f)^{−1}(B) such that 𝔵 ≤ 𝔣, so Uf(𝔵) = F f(𝔵) ≤ F f(𝔣) ∈ B and A_{(F f)^{−1}(B)} ⊆ (Uf)^{−1}(A_{B}). On the other hand, if 𝔵 ∈ (Uf)^{−1}(A_{B}), then there exists 𝔤 ∈ B with F f(𝔵) ≤ 𝔤. By definition of F f, we have f^{−1}(A) ∈ 𝔵 for all A ∈ 𝔤. Thus, the sets f^{−1}(A) for A ∈ 𝔤 form a filter 𝔣 ∈ (F f)^{−1}(B) satisfying 𝔵 ≤ 𝔣, and (Uf)^{−1}(A_{B}) ⊆ A_{(F f)^{−1}(B)}.

Let A_{B} ∈ σ_{Y} (UF f)(𝔛), where B ∈ UF f(𝔛). This means that (F f)^{−1}(B) ∈ 𝔛, so A_{(F f)^{−1}(B)} ∈ σ(𝔛). By the previous point, we have A_{B} ∈ (F Uf)σ_{X}(𝔛), and (F Uf)σ_{X}(𝔛) is finer than σ_{Y} (UF f)(𝔛).

We now show that Uf(A_{B}) = A_{F f(B)}. On one hand, if 𝔶 ∈ Uf(A_{B}), there exist a filter 𝔣 ∈ B and an ultrafilter 𝔵 ≤ 𝔣 such that Uf(𝔵) = 𝔶, so 𝔶 ∈ A_{F f(B)} because Uf(𝔵) = F f(𝔵) ≤ F f(𝔣). On the other hand, if 𝔶 ∈ A_{F f(B)}, there exists 𝔣 ∈ B with 𝔶 ≤ F f(𝔣). By the Extension Lemma, there exists an ultrafilter 𝔵 ≤ 𝔣 with Uf(𝔵) = 𝔶, and we have 𝔶 ∈ Uf(A_{B}).

A basis for the filter (F Uf)σ_{X}(𝔛) is given by the sets Uf(A_{B}) = A_{F f(B)} with B ∈ 𝔛. But then F f(B) ∈ UF f(𝔛), so naturally A_{F f(B)} ∈ σ_{Y} (UF f)(𝔛), and σ_{Y} (UF f)(𝔛) is finer than (F Uf)σ_{X}(𝔛). Therefore, we may conclude that σ is a natural transformation.

The other conditions that σ must verify follow immediately from its definition.

6.4. Corollary. The category Alg(F′, 2) is isomorphic to Top, and Alg(F′, \overline{R}_{+}) is isomorphic to App.

Proof. The first assertion follows from the fact that Alg(U, 2) is isomorphic to Top (see [1]). The second from the fact that Alg(U, \overline{R}_{+}) is isomorphic to App (see [2]).

6.5. Corollary. The category Ord embeds as a full coreflective subcategory into Top, and Met embeds as a full coreflective subcategory into App. Similarly, Top embeds as a full coreflective subcategory into Clos, and App embeds as a full coreflective subcategory into Clsn.

Proof. Since Ord ≅ Alg(P′, 2) and Met ≅ Alg(P′, \overline{R}_{+}), the first assertion is a consequence of Proposition 4.10. Moreover, the isomorphisms Top ≅ Alg(U, 2) and App ≅ Alg(U, \overline{R}_{+}) yield the second assertion via Proposition 4.8. Notice that the adjunction used in this last proposition is the one used in the original proofs of the isomorphisms Top ≅ Alg(U, 2) and App ≅ Alg(U, \overline{R}_{+}), where Top was described in terms of (additive) closure operators, and App in terms of (additive) closeness operators.

6.6. Proposition. The category of canonical (F, V)-algebras is isomorphic to the category of canonical (P, V)-algebras.

Proof. This follows from Corollary 4.9: each set A ⊆ X gives rise to the filter 𝔵_{A} := {B ⊆ X | A ⊆ B}, which satisfies the required hypothesis.

6.7. Corollary. The category Alg(F, 2) is isomorphic to Clos, and Alg(F, \overline{R}_{+}) is isomorphic to Clsn.

Proof. Again, this is immediate, since Alg(P, 2) ≅ Clos and Alg(P, \overline{R}_{+}) ≅ Clsn by Corollary 4.7.

Acknowledgements. The author wishes to thank McGill University and M. Barr for their hospitality during the writing of this article. He is also grateful to Cl.-A. Faure and D. Hofmann for the valuable comments they provided by way of electronic mail, to M. Barr again for useful discussions, and to the referee for helpful advice and references. Last but not least, he would like to thank W. Tholen for his support, and for introducing him to this beautiful subject.

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