@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 4, No. 1, 2003

pp. 193–200

Injective locales over perfect embeddings and
algebras of the upper powerlocale monad

Mart́ın Escardó

Abstract. We show that the locales which are injective over per-
fect sublocale embeddings coincide with the underlying objects of the
algebras of the upper powerlocale monad, and we characterize them
as those whose frames of opens enjoy a property analogous to stable
supercontinuity.

2000 AMS Classification: 06D22, 54C20, 54D80, 06B35
Keywords: Injective locale, perfect embedding, powerlocale, free frame,

Kock–Zöberlein monad, stably supercontinuous lattice.

1. Introduction

An object D of a category is said to be injective over a map j : X → Y if for
every map f : X → D there is at least one map f̄ : Y → D with f̄ ◦j = f. The
injectives over subspace embeddings in the category of T0 topological spaces
are known to be precisely the continuous lattices under the Scott topology [24].
Although the extension f̄ of f is far from unique in general, in this case there is
a canonical choice, the greatest extension in the pointwise specialization order,
denoted by f/j. It is natural to ask whether, for X and Y exponentiable
topological spaces and for D injective over subspace embeddings, the greatest-
extension operator f 7→ f/j : DX → DY is continuous [24]. It was shown in [6]
that, excluding the trivial situation in which D is the one-point space, this is
the case if and only if the embedding j : X → Y is a perfect map.

Here a continuous map g : X → Y of topological spaces is called perfect if the
right adjoint of the frame homomorphism g−1 : OY →OX preserves directed
joins, where OX and OY are the frames of open sets of X and Y . Hofmann
and Lawson [12] showed that, for X and Y sober, this is equivalent to saying
that (1) for every closed set C ⊆ X, the lower set of the direct image g[C] in
the specialization order of Y is closed and (2) the inverse image g−1(Q) of every
compact saturated set Q ⊆ Y is compact (saturated). Hofmann [11] showed
that, under the additional assumption of local compactness, the first condition



194 M. Escardó

follows automatically from the second. Notice that the perfect maps are the
continuous maps satisfying half of the definition of proper map of locales [27],
where the missing half is the Frobenius condition. Topologically, the proper
maps are known to be precisely the closed continuous maps that reflect compact
saturated sets.

Perfect maps, under various names and guises, arise frequently in topology
and locale theory. It is a folkloric fact that the category of perfect maps of stably
locally compact spaces is equivalent to Nachbin’s category [20] of monotone
continuous maps of compact order-Hausdorff spaces – see e.g. [9]. This is an
extension of the equivalence of the category of perfect maps of spectral spaces
with that of monotone continuous maps of ordered Stone spaces, previously
established by Priestley [21]. Localic versions and variations of this can be
found in [2, 26, 7] – see also Section 4 below.

Coming back to the subject of the first paragraph, the second natural ques-
tion is what are the injective spaces over perfect embeddings. It was also shown
in [6] that they coincide with the algebras of the upper powerspace monad, but
an intrinsic characterization of the algebras was left as an open problem.

This paper solves this problem in the localic case and shows that the char-
acterization of the injectives as algebras also holds in this setting. Notice that
the notion of perfect map as defined for topological spaces makes sense for
locales: A continuous map of locales is called perfect if the right adjoint of
its defining frame homomorphism preserves directed joins. For a definition of
the upper powerlocale monad and its basic properties, see Section 2 below. If
u,v ∈ OX are opens of a locale X, we write u ≺ v to mean that every Scott
closed subset C of OX with v ≤

∨
C has u as a member.

Theorem 1.1. The following are equivalent for any locale X.

(1) X is injective over perfect sublocale embeddings.
(2) X is the underlying locale of an algebra of the upper powerlocale monad.
(3) (a) Every open v ∈OX is the join of the set {u ∈OX | u ≺ v}, and

(b) 1 ≺ 1, and u ≺ v and u ≺ w together imply u ≺ v ∧w.

Schalk [22, 23] characterized the algebras of the restriction of the monad to
the category of locally compact locales using a different criterion. As already
pointed out by her, the monad on the whole category is of the Kock–Zöberlein
type [18]. Convenient references for our purposes are [5] or [8]. By a general
result established in [6, Theorem 4.2.2], the underlying objects of the alge-
bras of this monad coincide with the locales which are injective over perfect
embeddings, using the characterization of perfect maps given in Vickers [28].
Moreover, it also follows from [6, Theorem 4.2.2] that the greatest-extension
property discussed above also holds for injectives over perfect embeddings. Our
main tool for identifying the algebras, and hence the injectives, is Kock’s crite-
rion: An object is the underlying object of an algebra if and only if its unit has
a right adjoint, which is then its unique structure map – see Section 2 below.



Injective locales over perfect embeddings 195

We emphasize that the proof of Theorem 1.1, given in Section 3 below, is
constructive in the sense of topos logic.

Reinhold Heckmann is gratefully acknowledged for a careful reading of a
previous version of this paper.

2. Preliminaries

We take the basic notions concerning locales and frames for granted [14]
– we just emphasize that the category Loc of continuous maps of locales is
defined to be the opposite of the category Frm of homomorphisms of frames.
The topology, or frame of opens, of a locale X is denoted by OX and is ranged
over by the letters u,v,w. The defining frame homomorphism of a continuous
map f : X → Y of locales is denoted by f∗ : OY →OX. As a map of posets,
this has a right adjoint, which is denoted by f∗ : OX →OY .

For the sake of completeness, we recall the definition of the upper powerlocale
monad and show that it is of the Kock–Zöberlein type.

A preframe is a poset with finite meets and directed joins in which the
former distribute over the latter, and a preframe homomorphism is a map
that preserves both operations. The forgetful functor G: Frm → PrFrm
into the category of preframes has a left adjoint F : PrFrm → Frm. By
Banaschewski [1], and independently Heckmann [10], for any preframe L, the
free frame FL can be constructed as the set of Scott closed subsets of L ordered
by inclusion, with insertion of generators given by principal ideals:

�: L → FL
u 7→ ↓u.

If A is any frame and h: L → A is any preframe homomorphism, the unique
frame homomorphism h̄: FL → A with h̄(�u) = h(u) is given by

h̄(C) =
∨
{h(u) | u ∈ C}.

This induces a monad on PrFrm and a comonad on Frm, and hence a monad
(U,η,µ) on Loc, with OUX = FGOX, known as the upper powerlocale
monad. (Notice that, by virtue of the freeness property, the global points
1 → UX are in bijection with the preframe homomorphisms OX → O 1 and
hence with the Scott open filters of OX. Thus, by the localic Hofmann–Mislove
Theorem [15, 29], they are in bijection with the compact saturated sublocales
of X).

Explicitly, the upper powerlocale monad is constructed as follows. For any
continuous map f : X → Y , the continuous map Uf : UX →UY is given by

(Uf)∗ : OUY → OUX
�v 7→ �f∗(v).

The unit ηX : X →UX of a locale X is given by
η∗X : OUX → OX

�u 7→ u.



196 M. Escardó

In both cases, we are using the fact that the opens of the form �u freely generate
the frame of opens of the upper powerlocale, as explained above. Explicitly,
the unit is given by

η∗X(C) =
∨
C,

a fact used in the proof of Lemma 3.1 below. Multiplication is given by

µ∗X : OUX → OUUX
�u 7→ � � u,

but this is not needed for our considerations.
The category of locales is poset-enriched (and, in fact, dcpo-enriched) with

f ≤ g in Loc(X,Y ) if and only if f∗(v) ≤ g∗(v) for all v ∈OY . The following
lemma implicitly refers to this enrichment, which is known as the specialization
ordering.

A monad (T,η,µ) on a poset-enriched category C is said to be of the Kock–
Zöberlein type if the functor T is order-preserving and, for every object X, the
inequality ηTX ≤ TηX holds (the original, official definition has the inequality
in the opposite direction, but the convention that we adopt is the one which
is convenient for our purposes). In the presence of order preservation, the
inequality is equivalent to saying that structure maps α: TX → X are right
adjoint to units ηX : X → TX (and hence uniquely determined by X when
they exist), a fact which is also used in the proof of Lemma 3.1 below. Here,
by an adjunction f a g in the hom-poset C(X,Y ) it is meant a pair of maps
f : X → Y (the left adjoint) and g : Y → X (the right adjoint) with f◦g ≤ idY
and g ◦ f ≥ idX. The adjunction f a g is said to be reflective if g ◦ f =
idX. Once again for use in Lemma 3.1 below, observe that, by definition of
structure map [19], a right adjoint to a unit has to be reflective in order to be
a structure map. Notice that, because Loc = Frmop, we have that f a g holds
in Loc(X,Y ) with respect to the specialization order if and only if g∗ a f∗
holds in Frm(OY,OX) with respect to the pointwise order.

Lemma 2.1. The upper powerlocale monad is of the Kock–Zöberlein type.

Proof. The functor is monotone: Assume that f ≤ g in Loc(X,Y ) and let
v ∈ OY . Then �f∗(v) ≤ �g∗(v) by monotonicity of �, which amounts to
(Uf)∗(�v) ≤ (Ug)∗(�v). This completes the proof of monotonicity, because
the opens of the form �v form a subbase of the topology of UY .

The inequality ηUX ≤ UηX holds in Loc(UX,UUX): For C ∈ OUX, we
have that η∗UX(�C) = C ⊆↓

∨
C = �η∗X(C) = (UηX)

∗(�C). The result then
follows again from the fact that the opens of the form �C with C ∈ OUX
form a subbase of the topology of UUX. �

3. Proof of the theorem

Recall the definition of the relation ≺ on the opens of a locale formulated in
the paragraph preceding Theorem 1.1.



Injective locales over perfect embeddings 197

Lemma 3.1. A locale is the underlying object of an algebra of the upper pow-
erlocale monad if and only if the following two conditions hold.

(1) Every open v is the join of the opens u ≺ v, and
(2) 1 ≺ 1, and u ≺ v and u ≺ w together imply u ≺ v ∧w.

Proof. By the Kock–Zöberlein property, a locale X is the underlying object of
an algebra if and only if its unit ηX : X → UX has a reflective right adjoint
α: UX → X. This amounts to saying that α∗◦η∗X ≤ id and η

∗
X ◦α

∗ = id hold
in the category of frames. Because η∗X(C) =

∨
C, the adjoint-functor theorem

shows that the inequalities α∗ ◦ η∗X ≤ id and η
∗
X ◦ α

∗ ≥ id are equivalent to
the equation α∗(v) =

⋂
{C ∈ OUX | v ≤

∨
C}. It follows that u ∈ α∗(v) if

and only if u belongs to every C ∈ OUX with v ≤
∨
C, which amounts to

saying that u ≺ v; that is, α∗(v) = {u ∈ OX | u ≺ v}. Hence the equation
η∗X ◦ α

∗ = id amounts to condition 1. Being a left adjoint, α∗ preserves all
joins. Preservation of finite meets amounts to condition 2. �

Vickers showed that a continuous map f : X → Y of locales is perfect if
and only if the continuous map Uf : UX → UY has a left adjoint [28, Propo-
sition 5.6]. A slight modification of his proof establishes the following.

Lemma 3.2. A continuous map j : X → Y of locales is a perfect sublocale
embedding if and only if Uj : UX →UY has a reflective left adjoint.
Proof. Assume that j : X → Y is a perfect map. Being a right adjoint,
j∗ : OX → OY preserves meets and hence is a preframe homomorphism by
perfectness of j. By freeness of the frame OUX, there is a unique continuous
map U∗j : UY → UX such that (U∗j)∗(�u) = �j∗(u) for all u ∈ OX. Then
(Uj)∗ ◦ (U∗j)∗(�u) = (Uj)∗(�j∗(u)) = �j∗j∗(u) ≤ �u, and a similar calcu-
lation shows that (U∗j)∗ ◦ (Uj)∗(�v) ≥ �v. Since each inequality holds for
all opens of a subbase, they hold for all opens, which establishes the desired
adjunction U∗j a Uj. Reflectivity holds if and only if (Uj)∗ ◦ (U∗j)∗ = id if
and only if �j∗j∗(u) = �u for all u ∈ OX if and only if j∗j∗(u) = u for all
u ∈OX if and only if j is an embedding.

Conversely, assume that Uj : UX →UY has a (reflective) left adjoint U∗j :
UY →UX and define

OX
r
- OY = OX

�
- OUX

(U∗j)∗
- OUY

η∗Y- OY.
Being a composition of maps that preserve directed joins, r itself preserves
directed joins. We show that j∗ a r (reflectively), so that j∗ = r and hence j
is a perfect map (embedding):

j∗ ◦r(u) = j∗ ◦η∗Y ◦ (U
∗j)∗(�u) by definition of r

= (ηY ◦ j)
∗ ◦ (U∗j)∗(�u) by contravariance of (−)∗

= (Uj ◦ηX)
∗ ◦ (U∗j)∗(�u) by naturality of η

= η∗X ◦ (Uj)
∗ ◦ (U∗j)∗(�u) by contravariance of (−)∗

= η∗X ◦ (U
∗j ◦Uj)∗(�u) by contravariance of (−)∗

≤ η∗X(�u) because U
∗j aUj,

= u,



198 M. Escardó

where the inequality is an equality if the adjunction U∗j aUj is reflective, and
r ◦ j∗(v) = η∗Y ◦ (U

∗j)∗(�j∗(v)) by definition of r
= η∗Y ◦ (U

∗j)∗ ◦ (Uj)∗(�v) because (Uj)∗(�v) = �j∗(v)
= η∗Y ◦ (Uj ◦U

∗j)∗(�v) by contravariance of (−)∗
⊇ η∗Y (�v) because U

∗j aUj
= v.

�

Corollary 3.3. A locale is the underlying object of an algebra of the upper
powerlocale monad if and only if it is injective over perfect sublocale embeddings.

Proof. By [6, Theorem 4.2.2], for any Kock–Zöberlein monad T on a poset-
enriched category, the objects which are injective over maps j : X → Y such
that Tj : TX → TY has a reflective left adjoint coincide with the underlying
objects of T-algebras. �

This concludes the proof of Theorem 1.1. Notice that Lemma 3.2 shows
that the unit ηX : X → UX of any locale X is a perfect sublocale embed-
ding, because the map TηX : TX → TTX has µX : TTX → TX as a reflective
left adjoint for any Kock–Zöberlein monad (T,η,µ) on a poset-enriched cate-
gory [5]. Hence, by Corollary 3.3, every locale can be perfectly embedded into
a perfectly injective locale, because UX is a free algebra.

4. Remarks

The (independent and unpublished) work of Ho Weng Kin and Zhao Dong-
sheng [17] characterizes the lattices of Scott closed sets of various kinds of
posets. When applied to frames, their results imply that the free algebras of
the upper powerlocale monad are precisely the locales such that every open v is
the join of the opens u ≤ v with u ≺ u and such that the relation ≺ satisfies the
stability condition (3b) of Theorem 1.1. They also consider lattices in which
every element v is the join of the elements u ≺ v, with a different motivation
in mind, and establish a number of interesting results for them.

As far as we know, the characterization of the injectives over perfect embed-
dings given here is new. However, as shown in [6, 5, 8], several known injectivity
results can be established by proofs following the above pattern, provided the
monads under consideration are of the Kock–Zöberlein type. An example men-
tioned in [8] is that of stably locally compact locales. These coincide with the
injectives over flat sublocale embeddings [13, 16], and with the algebras of the
monad on locales that arises from the forgetful functor from frames into dis-
tributive lattices, whose left adjoint is ideal completion. In connection with a
discussion started in the introduction, we observe that the algebra homomor-
phisms are precisely the perfect maps, so that there is a further equivalence
with Nachbin’s category of compact order-Hausdorff spaces. This is related to
the work of Simmons [25]. An example not mentioned in [8] is that of injec-
tives over arbitrary sublocale embeddings; these coincide with the algebras of



Injective locales over perfect embeddings 199

the monad on locales that arises from the forgetful functor from frames into
meet-semilattices.

Let’s develop this second example in more detail. As in the main example
discussed in this paper, an explicit construction of the free frame is known: it
is the set of down sets of the meet-semilattice, ordered by inclusion. Again, the
insertion of generators is given by principal ideals. Thus, one is led to consider
the relation u ≺ v redefined to mean that every cover of v which is a down
set has u as a member, or, equivalently, every cover of v has a member that
covers u, and a theorem analogous to the one proved in this paper holds with an
analogous proof. The frames which satisfy the third condition of the theorem,
with the relation redefined as above, are called stably supercontinuous and are
known to coincide with the Scott topologies of continuous lattices. Part of
what is described here is proved in [3].

By the freeness property, the global points of the locale whose topology is
the free frame over a frame qua semilattice coincide with the meet-semilattice
homomorphisms from the frame into the initial frame, which in turn coincide
with the filters of the frame. Thus, this monad is the localic version of the filter
monad on the category of T0 topological spaces discussed in [5] and previously
by A. Day [4] and Wyler [30].

Notice that, in the main example developed in this paper and the ones
discussed in this section, it is crucial to know an explicit construction of the
free frame and of the insertion of generators. For instance, we are unable
to apply our technique to the lower powerlocale monad, because we lack a
sufficiently explicit description of the left adjoint of the forgetful functor from
frames into sup-lattices.

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Received August 2002

Revised December 2002

Mart́ın H. Escardó

School of Computer Science,
University of Birmingham,
Birmingham B15 2TT,
England.

E-mail address : m.escardo@cs.bham.ac.uk
http://www.cs.bham.ac.uk/~mhe/

http://www.cs.bham.ac.uk/~mhe/

	Injective locales over perfect embeddings and algebras of the upper powerlocale monad. By M. Escardó