Frames and topological algebras for a double­power monad

: We study the algebras for the double power monad on the Sierpi´nski space in the Cartesian closed category of equilogical spaces and produce a connection of the algebras with frames. The results hint at a possible synthetic, constructive approach to frames via algebras, in line with that considered in Abstract Stone Duality by Paul Taylor and others.


Introduction
The category Equ of equilogical spaces introduced by Dana Scott in [24] offers a very nice extension of the category Top 0 of T 0 -spaces and continuous maps, as it is a locally Cartesian closed quasitopos and the embedding of T 0 -spaces is full and preserves all products and existing exponentials.In other words, one may work with T 0 -spaces as if they formed a Cartesian closed category, just that sometimes the necessary space need not be topological, but it is just equilogical; see also Bauer, Birkedal and Scott [1].
For instance, the Sierpiński space S is the open-subset classifier, in the sense that given a T 0 -space T , for every T 0 -space X a continuous map f : X −→ S T determines precisely an open subset of the space X × T .But there is a problem in reading the previous sentence: the object S T need not exist as a topological space.The immediate solution is to read that sentence in the category of equilogical spaces where S T always exists; it is just that it may be a true equilogical space (ie which is not topological).
Such an extension of the language of Cartesian closed categories (and of the λ-calculus) was tested in various guises in many papers, see for instance Taylor [28,32] and Vickers and Townsend [34].In particular, in Bucalo and Rosolini [5] that extension is used to prove an intrinsic description of the soberification of a T 0 -space which involved As shown in [24] the category Equ is equivalent to the category PEqu , where an object is a pair P = (L P , ≈ P ) consisting of an algebraic lattice L P and a symmetric and transitive relation ≈ , where two such continuous functions g, g : for all a ≈ P a.
The composition of two arrows [g] : To describe the equivalence of categories, for an object P in PEqu write D P for the domain {x ∈ |L P | | x ≈ P x} of the relation ≈ P and note that ≈ P ⊆ D P × D P .Also write τ Sc for the Scott topology on the algebraic lattice L P and τ sub for the subspace topology induced by the inclusion D P ⊆ |L P |.We can now recall the two results from [24] crucial for the developments in the paper.extends to a functor which is an equivalence of categories.
Proposition 2.2 Let P = (L P , ≈ P ) and Q = (L Q , ≈ Q ) be objects in PEqu .Then (i) their product can be chosen as where a, b ≈ P×Q a , b if a ≈ P a and b ≈ Q b , and (ii) their exponential can be chosen as Note that via the equivalence in Proposition 2.1 an object P of PEqu gives rise to a diagram: (1) The category PEqu is (equivalent to) the quotient completion of the subset doctrine on the category AL of algebraic lattices and Scott-continuous functions; see Maietti and Rosolini [18,19,20] and Rosolini and Streicher [22].
3 The monad of the double power is a regular projective of Equ .
We shall concentrate on the Sierpiński space as an object of Equ as we intend to study the algebraic theory of S, and for that we need a Cartesian closed category.That theory played a crucial role in a synthetic presentation of the soberification of a topological space in Bucalo and Rosolini [5,4] as it showed that the notion of soberification is intrinsically related to the topology and to a monad derived from S. From now on we shall write the equilogical space Y(S) simply as S, dropping the Y .
The main object of our study fits very well within a paradigm which was studied in depth in general category-theoretical terms, in particular we refer the reader to a series of papers by Paul Taylor [27,28,29,30,31,32], to Bucalo and Rosolini [3] and to Dubuc [7].We develop the basic details in an ambient category which is Cartesian closed.Since typed λ-calculus is the internal language of Cartesian closed categories (see Lambek and Scott [15]) we shall use it extensively in the following.
Let C be a Cartesian closed category, eg the category Equ , and let O be a fixed object in C , eg S in Equ .
Since the functor -it gives rise to a monad on C .The functor part of the monad sends an arbitrary object C of C to the object O (O C ) .The unit of the monad has components η C : the exponential adjunct of the composite which, in λ-notation using • to denote application, is written: , which is: The climbing exponentials are unpleasant to read and we follow Taylor lowering the exponent of the functor-so we write O(C) in place of O C -and denoting iterations as . . .In particular, we write the monad as O 2 .
Examples 3.1 A well-known example of this kind of monads is obtained when one takes the category Set of sets and functions as C and the set D = {0, 1} as the object O.The Eilenberg-Moore category of algebras for the monad D 2 is that of complete Boolean algebras.
Another example is with C the category Pos of posets and monotone functions, the object O is the standard order P on the previous set D. The Eilenberg-Moore category of algebras for the monad P 2 is that of completely distributive lattices.
An example is also that where O is again the poset P, but in a different category from the previous one: C is the category DPos of posets with sups of directed subsets and functions preserving sups of directed subsets.The Eilenberg-Moore algebras for the monad P 2 on the category DPos is that of frames and frame homomorphisms.
In order to present some properties of the monad S 2 on Equ , it is useful to introduce auxiliary full subcategories of PEqu .We denote RPEqu the full subcategory of PEqu Similarly, SPEqu is the full subcategory of PEqu on those objects where ≈ K is contained in the diagonal relation on |L K |-one may say that the relation ≈ K is subreflexive.Then: The two statements follow easily: For (i), when Corollary 3.6 Let P be an object in PEqu .Then: Corollary 3.7 The functor S (−) : Equ → Equ op applies the subcategory SEqu into REqu op and viceversa, the subcategory REqu into SEqu op .

The algebraic theory of an object
We have seen in the previous section that the description of the monad O 2 can be performed in the internal language of the Cartesian closed category C .In fact, those functor and natural transformations can be internalised in the sense of enriched categories, see Kock [14], Linton [16,17], Power [21] and Street [25].
We refer the reader to previous references and to Kelly [13] for the notions of enriched category theory, in particular of monads in the enriched situation, or as they are called strong monads.Recall that a Cartesian closed category C has a canonical C -enrichment given by the exponentials: In fact, this can be done for any symmetric monoidal closed category, but that kind of generality is not needed for the purposes of the present study.In the λ-notation, a composition arrow is given by the term and, for T the terminal object in C , identities are i A : T → C C given by the term λx : C.x in the empty context.From now on, we may drop the application dot in case doing so generates no confusion, eg write g(fx) in the term above.
We shall adopt a standard notation for the enriched homsets, and for objects A and A in is given by the λ-term where f and g are in context x : C and y : D respectively.With respect to the canonical enrichment, C has C -tensors and C -cotensors given by product and power, respectively, since: Note that every object of C is a tensor of the terminal object T .Similarly, the Recall also that, for a fixed object O in C , the monad O 2 is C -enriched: the action of the functor O 2 on arrows, for every pair of objects C and D in C , is the C -arrow given by the term and there are commutative diagrams involving the natural transformations η and µ as follows (2) which internalise the standard (Set -enriched) commutative diagrams.
Remark 4.1 There is a very elegant analysis of this kind of monads in Kock [14] with an elementary characterisation of the conditions for enrichment given by the notion of strong monad.
We are interested in giving a presentation of the category of algebras for the monad O 2 in terms of certain enriched functors along the lines of Dubuc [6,7] and Power [21]; see also Kelly [12].The intuition about that presentation is that a C -enriched monad is an abstract presentation of an algebraic theory (in a suitable internal sense) and it hinges on the parallel between the Kleisli category of a monad and the terms of a theory; see Streicher and Reus [26]. of the following three arrows: Taking advantage of the enriched monad, the Kleisli category has a C -enrichment given by C quite similarly to how it inherits colimits from C .Like in C , every object of C M is a tensor of the terminal object T .
There is an identity-on-object, C -enriched functor from C to C M which maps an arrow Let L(M) be the opposite C -enriched category C M .
Remark 4.3 The category L(M) has C -cotensors.Every object of L(M) is a cotensor of the object T by the natural iso V ∼ / / V ∩ T .
Remark 4.4 Note that, in the case of a (standard) monad on the category Set for an algebraic theory, the Kleisli category is the category of free algebras and homomorphisms of the theory.

G Frosoni, G Rosolini and A Santamaria
For a monad of the form O 2 there is an isomorphic presentation of L O 2 which will be useful for section 5.For O an object in C , let T (O) be the C -enriched category with the same object as C -so the same objects as C O 2 and L O 2 -and let Proof The functor H is the identity on the objects of L O 2 .To define the Cenriched action on the arrows where one sees clearly that the "two arguments"of t-y : W and U : O V -get swapped.
It is straightforward to check that the assignment H is a C -enriched functor.Moreover, since H V,W is iso, the functor is fully faithful.Hence H is iso.
The Eilenberg-Moore category C M has objects the M-algebras, ie pairs A = (A, α : M(A) → A) of an object A and an arrow α : which internalises the commutativity condition for M-homomorphisms; see Dubuc [7].
Such an enrichment inherits cotensors from C .But in the general case of a category C with finite products that enrichment need not be available.
from the terminal object T of C .Preservation of cotensors requires that F transform each universal family p I,X into another such, in other words that the B -arrow obtained by exponential adjunction from is iso, necessarily natural.In case B is C , one can use λ-notation to write the adjunct q F,I,X : F(I ∩ X) → I ∩ F(X) of ψ F,I,X as: Following Dubuc [6] and Power [21] a model of L(M) is a C -enriched functor F : L(M) −→ C which preserve cotensors.
Example 4.10 An M-algebra A = (A, α : M(A) → A) determines a standard example of a C -enriched, cotensor-preserving functor A (−) : L(M) −→ C , see [6,21].It is defined as follows: on an object V in L(M), the value is given by the λ-term The proof that the assignment is indeed a functor is direct, though laborious, as it involves the categorical structure of L(M) = (C M ) op and the conditions in (2).It is immediate to see that A (−) preserves cotensors as and it is easy to check that, given an M-homomorphism h : A → B , post-composition with h Proof For the sake of space saving, in the proof write the category L(M) simply as T .Fix a C -enriched, cotensor-preserving functor F : T −→ C .Hence, for every object I in C and every object V in T , the C -arrow ψ F,I,V : ] is natural iso.Since every object in T is a cotensor of the terminal object T , the value F(T) determines the functor F up to a natural isomorphism.By Remark 4.3 there is a natural iso V ∼ / / V ∩ T .So we can identify V and V ∩T in T , and F(V) and C [V, F(T)] in C .Also, to simplify notation, write F(T) as A, so that: where, like in Example 4.6, the arrow j C : C [C [A, A], C] → C denotes evaluation at the identity i A , ie the composite Remark 4.12 As already mentioned, there are results in the literature related to Theorem 4.11, for instance Theorem III in [6] and Theorem 3.4 in [21].The reader may find more details about this in Santamaria's masters thesis [23].In case C has equalisers, hence finite limits, by Remark 4.7 the Eilenberg-Moore category C O 2 is C -enriched with cotensors.In particular, the natural isomorphism of cotensors for objects I in C and A in C O 2 , gives a C -enriched adjunction: In our case of interest, when C is Equ and O is S, if T is a T 0 -space and T denotes its soberification, then T ∼ / / Equ S 2 [S T , S] ; see Bucalo and Rosolini [5].

Global sections of S 2 -algebras
By viewing an S 2 -algebra A = (A, α : S 2 (A) −→ A) as an Equ -enriched, cotensorpreserving functor from T (S) to Equ , it is possible to distinguish some of the operations induced on the object A by the S 2 -structure α : S 2 (A) −→ A and determine the identities these satisfy.This is a slight abuse of notation since it should be A (−) : L S 2 −→ Equ , but by Proposition 4.5 the category T (S) is isomorphic to L S 2 .
There is a useful, functorial way to analyse part of the structure given by a model Let T Fin (S) be the full subcategory of T (S) whose objects are the discrete (finite) numerals.
Proposition 5.1 The category T Fin (S) is the smallest subcategory of T (S) which contains the object 1, is closed under finite products of T (S), and contains the arrows: Proof Each object n in T Fin (S) is the product of n copies of 1 as n = Proof By Proposition 5.1, T Fin (S) has finite products, computed by cotensors; so a cotensor-preserving functor from T (S) to Equ preserves such limits.So A 0 is terminal in Equ and A 2 is a product A × A in Equ .Functoriality ensures that the operations in the statement satisfy all the commutative diagrams in which they appear in T (S).For instance, distributivity of ∨ over ∧ is the commutative diagram in T (S).Therefore, A (−) transforms it in the corresponding commutative diagram involving the operations on A.
In fact, in the following we strengthen Proposition 5.3 to show that every S 2 -algebra has a unique structure of a frame.In order to do that, we introduce two other full subcategories of T (S): the full subcategory T FinPos (S) of T (S) on the finite posets (each considered with its Scott topology) and the full subcategory T Set (S) on the discrete T 0 -spaces.
Proposition 5. 4 The full subcategory T FinPos (S) of T (S) is the smallest subcategory of T (S) which contains the object 1, is closed under finite products and retracts and contains the arrows: Proof Because of Proposition 5.1, it is enough to show that any finite poset P with the Scott topology is in T FinPos (S).Let n be the cardinality of P and let : n → P be a bijection.Consider the idempotent: It is immediate to see that S P is (isomorphic to) the distributive lattice of fixpoints of h.
The following result is reminiscent of the study in Hyland [11]; see also Fiore and Rosolini [8].
Corollary 5.5 The lattice structure on an S 2 -algebra A = (A, α : S 2 (A) −→ A) depends only on the underlying object A.
reference to the infinite is the existence of arbitrary products in the category T Set (S).
This suggests that it is possible to see the notion of frame as a finitary one from some appropriate, non-classical point of view.There is a similar approach in Hyland [11].

A 1 Y
map of equilogical spaces [f ] : E −→ F is an equivalence class of continuous functions f : (|E|, τ E ) −→ (|F|, τ F ) preserving the equivalence relations, ie if x ≡ E x then f (x) ≡ F f (x ) for all x and x in |E|, where two such continuous functions f , f : (|E|, τ E ) −→ (|F|, τ F ) are equivalent if f (x) ≡ F f (x) for all x ∈ |E|.Composition of maps of equilogical spaces [f ] : E −→ F and [g] : F −→ G is given on (any of) their continuous representatives: [g] • [f ] := [g • f ].The data above determine the category Equ of equilogical spaces which is an extension of the category of T 0 -spaces: the embedding Top 0 full / / Equ maps a T 0 -space (T, τ ) to the equilogical space on (T, τ ) with the diagonal relation, ie the equilogical space (T, τ, =).
P ⊆ |L P | × |L P | on |L P |, ie a partial equivalence relation on |L P |.An arrow in PEqu [g] : P −→ Q is an equivalence class of Scott-continuous functions g : L P −→ L Q such that whenever a ≈ P b, also g(a) ≈ Q g(b) Consider the Sierpiński T 0 -space S which consists of two points { , ⊥} and one non-trivial open subset { }.In other words, one point is open, the other is closed.It is an algebraic lattice with the Scott topology.So the equilogical space Y(S) = (|S|, τ S , =)

Proposition 3 . 2 Notation 3 . 4
The restriction of the equivalence PEqu Z / / Equ to the subcategory SPEqu determines an equivalence between SPEqu and the image of the embedding Top 0 1 Y / / Equ .Proof It is enough to consider the diagram (1) and note that ≈ P is subreflexive if and only if the map (D P , τ sub , =) [id D P ] , 2 Z(P) is iso.Remark 3.3 A condition similar to that used in the proof of Proposition 3.2 characterises a full subcategory REqu of Equ equivalent to RPEqu : the objects of REqu are those equilogical spaces E for which there is an algebraic lattice L and a regular epi (L, =) , 2 E .In line with the notation used in Remark 3.3 we shall write SEqu for the closure under isos of the image of the embedding Top 0 1 Y / / Equ .Proposition 3.5 Let K be an object in SPEqu and let R be an object in RPEqu .
the enriched C -category A we write A[A, A ] for the C -object of A -arrows from A to A .So the C -enrichment of the Cartesian closed category C is C [C, D] = D C , and for G Frosoni, G Rosolini and A Santamaria arrows f : C → C and g

Remark 4 . 2
The precise sense in which to consider an "internal" algebraic theory requires the review of a few constructions on a C -enriched monad which appear in the references.Unfortunately we have not been able to single out the explicit result we need, see Theorem 4.11.So, although the interest of the present paper is for monads of the form O 2 , in the following we sketch that result for an arbitrary C -enriched monad M = (M, η, µ) on C .First we briefly recall the notions of enriched Kleisli category and enriched Eilenberg-Moore category of an enriched monad.The Kleisli category C M of the monad M consists of the same objects as C , but an arrow t : X → Y is an arrow t : X → M(Y) in C .The composition of t : X → Y and s : Y → Z is defined by the composition in C

Proposition 4 . 5
Composition and identities are as in C .Clearly, T (O) is equivalent to the full C -enriched subcategory of C on the objects of the form O V .Let O be an object in C .Then there is an isomorphism of C -enriched categories H : L O 2 −→ T (O).

Example 4 . 6
an arrow h : (A, α) → (B, β) is an arrow h : A → B in C which is a M-homomorphism, ie the following is a commutative diagram: For a fixed object O in C , an example of an O 2 -algebra is O = (O, j O ) where j O : O 2 (O) → O denotes evaluation at i O , ie the composite O 2 (O) id O 2 (O) , i O ! / / O 2 (O) × O ev / / O for !: O 2 (O) → T the unique arrow to the terminal object, which can be written in λ-notation as φ • (λx : O. x) in context φ : O 2 (O).

Remark 4 . 7
In case C has equalisers, the Eilenberg-Moore category C M has the C -enrichment which, for M-algebras A = (A, α) and B = (B, β), is given by (a choice of) the equaliser

Remark 4 . 8
In the case of a (standard) monad on the category Set for an algebraic theory considered in Remark 4.4, the category L(M), being the opposite of the category of free algebras and homomorphisms between them, can be considered equivalently as the sets V of the generators-think of variables.With that point of view, a L(M)-arrowt : V → W is a W -listof terms of the theory written in the variables (ie the elements) of V and the composition V t / / W s / / U in L(M) is substitution of the variables W in the terms of the U -list with the W -list of terms.Notation 4.9 We recall when a C -enriched functor F : A −→ B preserves cotensors since we shall need it to prove Theorem 4.11.For objects X in A and I in C , write I ∩ X for the cotensor X by I and consider the universal I -family of A -arrows G Frosoni, G Rosolini and A Santamaria p I,X : I → A[I ∩ X, X] of the cotensor I ∩ X , obtained by the exponential adjunction from the composite arrow is a natural transformation.Journal of Logic & Analysis 11:FT5 (2019) Frames and Topological Algebras for a Double-Power Monad 13 The result mentioned in Remark 4.2 is the statement that the examples in 4.10 are the most general.Theorem 4.11 Let C be a Cartesian closed category and let M = (M, η, µ) be C -enriched monad on C .Then the functor that assign to an M-algebra A the C -enriched, cotensor-preserving functor A (−) : L(M) −→ C is an equivalence of categories between C M and the full subcategory of the functor category [L(M) , C ] on the C -enriched, cotensor-preserving functors.

G
Frosoni, G Rosolini and A Santamaria In the particular case of the monad of an object O in C , one can see the models F : T (O) → C as an interpretation of all the operations O V → O on O available in C which satisfies all identities that the operations satisfy on O. Or, turning things around, one can see T (O) as the algebraic theory of all the operations on O.And Theorem 4.11 states that the models are precisely the O 2 -algebras.

A 1 I/
(−) : T (S) −→ Equ .Let D be a subcategory of T (S) and write I : D 1 / / T (S) the inclusion functor.Then the restriction functor D / T (S) A (−) / / Equ Journal of Logic & Analysis 11:FT5 (2019) Frames and Topological Algebras for a Double-Power Monad 15 is a (Set -enriched) model of D in Equ .If a syntactic presentation of the category D is available by means of a logical theory, then the functor A (−) • I is a model of D with underlying object A. As an instance of this procedure, we show that every S 2 -algebra A induces a distributive lattice structure on A.

n times 1 +Corollary 5 . 2 Proposition 5 . 3
. . .+ 1.An arrow f : n −→ m in T Fin (S) is a continuous function f : S n −→ S m between finite powers of S. As such it is monotone and the four arrows in the statement are enough to generate by composition and pairing all such monotone maps.Note that T Fin (S) has an enrichment on the category FinSet of finite sets and functions and it has FinSet -cotensors.The category T Fin (S) is the Lawvere algebraic theory of distributive lattices.ProofThe identities satisfied by the four arrows in Proposition 5.1 are precisely those given by commutative diagrams in T (S), in other words are precisely the identities satisfied by those operations in their interpretation as meet and join in S. Every S 2 -algebra A inherits a structure of a (bounded) distributive lattice in Equ as given by the maps A : A 0 −→ A 1 , A ⊥ : A 0 −→ A 1 , A ∧ : A 2 −→ A 1 and A ∨ : A 2 −→ A 1 .