coalgebras for enriched hausdorff (and vietoris)...

Coalgebras for enriched Hausdorff (and Vietoris) functors

Dirk Hofmann (collaboration with Pedro Nora and Renato Neves)July 12, 2019

CIDMA, Department of Mathematics, University of Aveiro, [email protected], http://sweet.ua.pt/dirk/

[email protected]

http://sweet.ua.pt/dirk/

Introduction

A quick reminder

DefinitionFor a functor F : C→ C, one defines coalgebra

homomorphism:

X

FX

Y

FYc d

f

Ff

The corresponding category we denote as CoAlg(F).

TheoremThe forgetful functor CoAlg(F)→ C creates all colimits and those limits which arepreserved by F.

A quick reminder

DefinitionFor a functor F : C→ C, one defines coalgebra homomorphism:

X

FX

Y

FYc d

f

Ff

The corresponding category we denote as CoAlg(F).

TheoremThe forgetful functor CoAlg(F)→ C creates all colimits and those limits which arepreserved by F.

Motivation

Recall• The final coalgebra for F : C→ C is a fix-point of F. a

• The power-set functor P : Set→ Set does not have a fix-point; hence P doesnot admit a final coalgebra.

• The finite power-set functor Pfin : Set→ Set admits a final coalgebra (forinstance, because Pfin is finitary).

• Somehow more general: the Vietoris functor V : CompHaus→ CompHausadmits a final coalgebra

(and the same is true for V : PosComp→ PosComp).

• A bit more general: the compact Vietoris functor Vc : Top→ Top admits a finalcoalgebra.

aJoachim Lambek. “A fixpoint theorem for complete categories”. In: Mathematische Zeitschrift103.(2) (1968), pp. 151–161.

Motivation

Recall• The final coalgebra for F : C→ C is a fix-point of F.• The power-set functor P : Set→ Set does not have a fix-point; hence P does

not admit a final coalgebra. a

• The finite power-set functor Pfin : Set→ Set admits a final coalgebra (forinstance, because Pfin is finitary).




aGeorg Cantor. “Über eine elementare Frage der Mannigfaltigkeitslehre”. In: Jahresbericht derDeutschen Mathematiker-Vereinigung 1 (1891), pp. 75–78.

Motivation


not admit a final coalgebra.• The finite power-set functor Pfin : Set→ Set admits a final coalgebra (for

instance, because Pfin is finitary). a




aMichael Barr. “Terminal coalgebras in well-founded set theory”. In: Theoretical Computer Science114.(2) (1993), pp. 299–315.

Motivation



instance, because Pfin is finitary).• Somehow more general: the Vietoris functor V : CompHaus→ CompHaus

admits a final coalgebraa

(and the same is true for V : PosComp→ PosComp).• A bit more general: the compact Vietoris functor Vc : Top→ Top admits a final

coalgebra.

aHere: V preserves codirected limits. This result appears as an exercise in Ryszard Engelking.General topology. 2nd ed. Vol. 6. Sigma Series in Pure Mathematics. Berlin: Heldermann Verlag, 1989.viii + 529.

Motivation




admits a final coalgebra (and the same is true for V : PosComp→ PosComp).ab


aLeopoldo Nachbin. Topologia e Ordem. University of Chicago Press, 1950.bDirk Hofmann, Renato Neves, and Pedro Nora. “Limits in categories of Vietoris coalgebras”. In:

Mathematical Structures in Computer Science 29.(4) (2019), pp. 552–587.

Motivation




admits a final coalgebra (and the same is true for V : PosComp→ PosComp).• A bit more general: the compact Vietoris functor Vc : Top→ Top admits a final

coalgebra.

Motivation




admits a final coalgebra (and the same is true for V : PosComp→ PosComp).• A bit more general: the compact Vietoris functor Vc : Top→ Top admits a final

coalgebra.• A bit surprising(?): Also the lower Vietoris functor V : Top→ Top admits a final

coalgebra.

Where to go from there?

QuestionsWhat about “power functors” on other (topological) base categories?

For instance,• the upset functor Up : Ord→ Ord?• li�ings of Set-functors to Met (or, more general, to V-Cat)?


QuestionsWhat about “power functors” on other (topological) base categories? For instance,• the upset functor Up : Ord→ Ord?

• li�ings of Set-functors to Met (or, more general, to V-Cat)?


QuestionsWhat about “power functors” on other (topological) base categories? For instance,• the upset functor Up : Ord→ Ord?• li�ings of Set-functors to Met (or, more general, to V-Cat)? ab

V-Cat V-Cat

Set Set.

T

T

aPaolo Baldan, Filippo Bonchi, Henning Kerstan, and Barbara König. “Coalgebraic BehavioralMetrics”. In: Logical Methods in Computer Science 14.(3) (2018), pp. 1860–5974.

bAdriana Balan, Alexander Kurz, and Jiří Velebil. “Extending set functors to generalised metricspaces”. In: Logical Methods in Computer Science 15.(1) (2019).


QuestionsWhat about “power functors” on other (topological) base categories? For instance,• the upset functor Up : Ord→ Ord?• li�ings of Set-functors to Met (or, more general, to V-Cat)?• (in particular) the Hausdor� functor? abc

H : Met −→ MetaFelix Hausdor�. Grundzüge der Mengenlehre. Leipzig: Veit & Comp, 1914. viii + 476.bDimitrie Pompeiu. “Sur la continuité des fonctions de variables complexes”. In: Annales de la

Faculté des Sciences de l’Université de Toulouse pour les Sciences Mathématiques et les SciencesPhysiques. 2ième Série 7.(3) (1905), pp. 265–315.

c T. Birsan and Dan Tiba. “One hundred years since the introduction of the set distance by DimitriePompeiu”. In: System Modeling and Optimization. Ed. by F. Pandolfi et al. Springer, 2006, pp. 35–39.


QuestionsWhat about “power functors” on other (topological) base categories? For instance,• the upset functor Up : Ord→ Ord?• li�ings of Set-functors to Met (or, more general, to V-Cat)?• (in particular) the Hausdor� functor? ab

H : V-Cat −→ V-Cat

Here Ha(A,B) =∧y∈B

∨x∈A

a(x, y), for a V-category (X,a).

aAndrei Akhvlediani, Maria Manuel Clementino, and Walter Tholen. “On the categorical meaning ofHausdor� and Gromov distances, I”. In: Topology and its Applications 157.(8) (2010), pp. 1275–1295.

bIsar Stubbe. ““Hausdor� distance” via conical cocompletion”. In: Cahiers de Topologie etGéométrie Di�érentielle Catégoriques 51.(1) (2010), pp. 51–76.

Some "powerful functors"

About the upset funtor

TheoremLet X be a partially ordered set. Then there is no embedding ϕ : Up(X)→ X. ab

aRobert P. Dilworth and Andrew M. Gleason. “A generalized Cantor theorem”. In: Proceedings of theAmerican Mathematical Society 13.(5) (1962), pp. 704–705.

bRobert Rosebrugh and Richard J. Wood. “The Cantor-Gleason-Dilworth Theorem”. 1994.

CorollaryThe upset functor Up : Ord→ Ord does not admit a final coalgebra.

RemarkThe category CoAlg(Up) has equalisers.

About the upset funtor

TheoremLet X be a partially ordered set. Then there is no embedding ϕ : Up(X)→ X. abc

aRobert P. Dilworth and Andrew M. Gleason. “A generalized Cantor theorem”. In: Proceedings of theAmerican Mathematical Society 13.(5) (1962), pp. 704–705.

bRobert Rosebrugh and Richard J. Wood. “The Cantor-Gleason-Dilworth Theorem”. 1994.c F. William Lawvere. “Diagonal arguments and cartesian closed categories”. In: Category theory,

homology theory and their applications II. Springer, 1969, pp. 134–145.

CorollaryThe upset functor Up : Ord→ Ord does not admit a final coalgebra.

RemarkThe category CoAlg(Up) has equalisers.

About strict functorial liftings

TheoremConsider the following commutative diagram of functors.

X X

A A

F

U U

F

1. If F has a fix-point, then so has F. Hence, if F does not have a fix-point, thenneither does F.

2. If U : X→ A is topological, then so is U : CoAlg(F)→ CoAlg(F).

In particular, the category CoAlg(F) has limits of shape I if and only if CoAlg(F)

has limits of shape I.

About strict functorial liftings

TheoremConsider the following commutative diagram of functors.

X X

A A

F

U U

F

1. If F has a fix-point, then so has F. Hence, if F does not have a fix-point, thenneither does F.

2. If U : X→ A is topological, then so is U : CoAlg(F)→ CoAlg(F).In particular, the category CoAlg(F) has limits of shape I if and only if CoAlg(F)

has limits of shape I.

The Hausdorff monad on V-Cat

DefinitionLet f : (X,a)→ (Y,b) be a V-functor.

1. For every A ⊆ X, put ↑a A = {y ∈ X | k ≤∨

x∈A a(x, y)}.

2. We call a subset A ⊆ X of (X,a) increasing whenever A = ↑a A.3. We consider the V-category HX = {A ⊆ X | A is increasing}, equipped with

Ha(A,B) =∧y∈B

∨x∈A

a(x, y), for all A,B ∈ HX.

4. The map Hf : H(X,a) −→ H(Y,b) sends an increasing subset A ⊆ X to ↑b f (A).5. The functor H is part of a Kock–Zöberlein monad H = (H,w , h) on V-Cat.6. H = (H,w , h) is a submonad of the covariant presheaf monad on V-Cat; in

fact, H is the monad of “conical limit weights”.




x∈A a(x, y)}.2. We call a subset A ⊆ X of (X,a) increasing whenever A = ↑a A.

3. We consider the V-category HX = {A ⊆ X | A is increasing}, equipped withHa(A,B) =

∧y∈B

∨x∈A







x∈A a(x, y)}.2. We call a subset A ⊆ X of (X,a) increasing whenever A = ↑a A.3. We consider the V-category HX = {A ⊆ X | A is increasing}, equipped with

Ha(A,B) =∧y∈B

∨x∈A








Ha(A,B) =∧y∈B

∨x∈A


4. The map Hf : H(X,a) −→ H(Y,b) sends an increasing subset A ⊆ X to ↑b f (A).

5. The functor H is part of a Kock–Zöberlein monad H = (H,w , h) on V-Cat.6. H = (H,w , h) is a submonad of the covariant presheaf monad on V-Cat; in






Ha(A,B) =∧y∈B

∨x∈A


4. The map Hf : H(X,a) −→ H(Y,b) sends an increasing subset A ⊆ X to ↑b f (A).5. The functor H is part of a Kock–Zöberlein monad H = (H,w , h) on V-Cat.

hX : X −→ HX, w X : HHX −→ HX.

x 7−→ ↑x A 7−→⋃A

6. H = (H,w , h) is a submonad of the covariant presheaf monad on V-Cat; infact, H is the monad of “conical limit weights”.





Ha(A,B) =∧y∈B

∨x∈A




Some classical results

For metric spaces1. For every compact metric space X, the Hausdor� metric induces the Vietoris

topology (of the compact Hausdor� space X).

2. Hence, the Hausdor� functor sends compact metric spaces to compact metricspaces.

3. Furthermore, the Hausdor� functor preserves Cauchy completeness.4. . . .



topology (of the compact Hausdor� space X).2. Hence, the Hausdor� functor sends compact metric spaces to compact metric

spaces.

3. Furthermore, the Hausdor� functor preserves Cauchy completeness.4. . . .




spaces.3. Furthermore, the Hausdor� functor preserves Cauchy completeness.

4. . . .




spaces.3. Furthermore, the Hausdor� functor preserves Cauchy completeness.4. . . .

Ernest Michael. “Topologies on spaces of subsets”. In: Transactions of the American MathematicalSociety 71.(1) (1951), pp. 152–182.

Sandro Levi, Roberto Lucchetti, and Jan Pelant. “On the infimum of the Hausdor� and Vietoristopologies”. In: Proceedings of the American Mathematical Society 118.(3) (1993), pp. 971–978.

Coalgebras for the Hausdorff functor

TheoremLet V be a non-trivial quantale and (X,a) be a V-category. There is no embeddingof type H(X,a)→ (X,a).

CorollaryLet V be a non-trivial quantale. The Hausdor� functor H : V-Cat→ V-Cat does notadmit a terminal coalgebra, neither does any possible restriction to a fullsubcategory of V-Cat.

RemarkIn particular, the (non-symmetric) Hausdor� functor on Met does not admit aterminal coalgebra, and the same applies to its restriction to the full subcategoryof compact metric spaces.

Passing to the symmetric version of the Hausdor�functor does not remedy the situation.


TheoremLet V be a non-trivial quantale and (X,a) be a V-category. There is no embeddingof type H(X,a)→ (X,a).

CorollaryLet V be a non-trivial quantale. The Hausdor� functor H : V-Cat→ V-Cat does notadmit a terminal coalgebra, neither does any possible restriction to a fullsubcategory of V-Cat.

RemarkIn particular, the (non-symmetric) Hausdor� functor on Met does not admit aterminal coalgebra, and the same applies to its restriction to the full subcategoryof compact metric spaces. Passing to the symmetric version of the Hausdor�functor does not remedy the situation.

Adding topology

Generalised Nachbin spaces

Extending the Ultrafilter monadWe assume that V is a completely distributive quantale, then

ξ : UV −→ V, v 7−→∧A∈v

∨A

is the structure of an U-algebra on V (the Lawson topology).

Therefore we obtaina lax extension of the ultrafilter monad to V-Rel that induces a monad on V-Cat.Its algebras are V-categories equipped with a compatible compact Hausdor�topology; we call them V-categorical compact Hausdor� spaces, and denote thecorresponding Eilenberg–Moore category by V-CatCH.

TheoremFor an ordered set (X,≤) and a U-algebra (X, α), the following are equivalent.

(i) α : (UX,U≤)→ (X,≤) is monotone.

(ii) G≤ ⊆ X × X is closed.



ξ : UV −→ V, v 7−→∧A∈v

∨A

is the structure of an U-algebra on V (the Lawson topology). Therefore we obtaina lax extension of the ultrafilter monad to V-Rel that induces a monad on V-Cat.

Here:Ua(x, y) =

∧A,B

∨x,y

a(x, y), (X,a) 7−→ (UX,Ua).

Its algebras are V-categories equipped with a compatible compact Hausdor�topology; we call them V-categorical compact Hausdor� spaces, and denote thecorresponding Eilenberg–Moore category by V-CatCH.






ξ : UV −→ V, v 7−→∧A∈v

∨A

is the structure of an U-algebra on V (the Lawson topology). Therefore we obtaina lax extension of the ultrafilter monad to V-Rel that induces a monad on V-Cat.Its algebras are V-categories equipped with a compatible compact Hausdor�topology ab; we call them V-categorical compact Hausdor� spaces, and denote thecorresponding Eilenberg–Moore category by V-CatCH.

aLeopoldo Nachbin. Topologia e Ordem. University of Chicago Press, 1950.bWalter Tholen. “Ordered topological structures”. In: Topology and its Applications 156.(12) (2009),

pp. 2148–2157.






ξ : UV −→ V, v 7−→∧A∈v

∨A

is the structure of an U-algebra on V (the Lawson topology). Therefore we obtaina lax extension of the ultrafilter monad to V-Rel that induces a monad on V-Cat.Its algebras are V-categories equipped with a compatible compact Hausdor�topology; we call them V-categorical compact Hausdor� spaces, and denote thecorresponding Eilenberg–Moore category by V-CatCH.






ξ : UV −→ V, v 7−→∧A∈v

∨A



(i) α : (UX,U≤)→ (X,≤) is monotone.(ii) G≤ ⊆ X × X is closed.



ξ : UV −→ V, v 7−→∧A∈v

∨A


TheoremFor a V-category (X,a) and a U-algebra (X, α), the following are equivalent.

(i) α : U(X,a)→ (X,a) is a V-functor.(ii) a : (X, α)× (X, α)→ (V, ξ≤) is continuous.

Open “(”

TheoremFor an ordered compact Hausdor� space X, the ordered set X is directed complete.

Proof.

OrdCH Top K(X) is sober, . . .

Ord |X| is directed complete

K

|−|S

CorollaryEvery compact metric space is Cauchy complete.

Example(UX,Ud) is Cauchy complete.

Consider (UX,Ud,mX) . . . and close “)”

Open “(”

TheoremFor a metric compact Hausdor� space X, the metric space X is Cauchy complete.

Proof.

OrdCH Top K(X) is sober, . . .

Ord |X| is directed complete

K

|−|S




Open “(”


Proof.

MetCH Top K(X) is sober, . . .

Met |X| is directed complete

K

|−| S




Open “(”


Proof.

MetCH App K(X) is sober, . . .


K

|−| S

Approach space = “metric” topological space.Robert Lowen. “Approach spaces: a common supercategory of TOP and MET”. In: Mathematische

Nachrichten 141.(1) (1989), pp. 183–226.Bernhard Banaschewski, Robert Lowen, and Cristophe Van Olmen. “Sober approach spaces”. In:

Topology and its Applications 153.(16) (2006), pp. 3059–3070.




Open “(”


Proof.

MetCH App K(X) is sober, . . .


K

|−| S

Theorem (Lawvere (1973))A metric space X is Cauchy-complete if and only if every left adjoint distributorsϕ : 1 X is representable (i.e. ϕ = x∗).




Open “(”


Proof.

MetCH App K(X) is Cauchy complete, . . .

Met |X| is Cauchy complete

K

|−| S

Theorem (Lawvere (1973))A metric space X is Cauchy-complete if and only if every left adjoint distributorsϕ : 1 X is representable (i.e. ϕ = x∗).




Open “(”


Proof.



K

|−| S




Open “(”


Proof.



K

|−| S


ExampleEvery discrete metric space is Cauchy complete (any compact Hausdor� topology).



Open “(”


Proof.



K

|−| S




Open “(”


Proof.



K

|−| S


Example(UX,Ud) is Cauchy complete. Consider (UX,Ud,mX) . . . and close “)”

Towards “Urysohn”

LemmaLet (X,a, α) be a V-categorical compact Hausdor� space and A,B ⊆ X so thatA ∩ B = ∅, A is increasing and compact in (X, α≤)op and B is compact in (X, α≤).Then there exists some u� k so that, for all x ∈ A and y ∈ B, u 6≤ a(x, y).

Corollary

For every compact subset A ⊆ X of (X, α≤)op, ↑a A = ↑≤ A. In particular, for everyclosed subset A ⊆ X of (X, α), ↑a A = ↑≤ A.

Theorem (Nachbin)Let A ⊆ X be closed and decreasing and B ⊆ X be closed and increasing withA ∩ B = ∅. Then there exist V ⊆ X open and co-increasing and W ⊆ X open andco-decreasing with

A ⊆ V, B ⊆ W, V ∩W = ∅.

The Hausdorff monad (again)

DefinitionFor a V-categorical compact Hausdor� space X = (X,a, α), we put

HX = {A ⊆ X | A is closed and increasing}with the restriction of the Hausdor� structure to HX and the hit-and-miss topology(Vietoris topology). That is, the topology generated by the sets

V♦ = {A ∈ HX | A ∩ V 6= ∅} (V open, co-increasing)

andW� = {A ∈ HX | A ⊆ W} (W open, co-decreasing).

PropositionFor every V-categorical compact Hausdor� space X, HX is a V-categorical compactHausdor� space.

TheoremThe construction above defines a functor H : V-CatCH −→ V-CatCH.

In fact, we obtain a Kock–Zöberlein monad.



HX = {A ⊆ X | A is closed and increasing}with the restriction of the Hausdor� structure to HX and the hit-and-miss topology(Vietoris topology).


Compare with:For a compact metric space, the Hausdor� metric induces the Vietoris topology.







TheoremThe construction above defines a functor H : V-CatCH −→ V-CatCH.In fact, we obtain a Kock–Zöberlein monad.

Compact V-categories

Theorem (Lawvere (1973))A metric space X is Cauchy-complete if and only if every left adjoint distributorϕ : 1 X is representable (i.e. ϕ = x∗).

DefinitionFor a V-category X, A ⊆ X and x ∈ X, we define x ∈ A whenever “x represents a le�adjoint distributor 1 A”.

Remark

• Under suitable conditions, this closure operator is topological.• Moreover, if X is separated, then this topology is Hausdor�.• With V-Catch denoting the full subcategory of V-Catsep defined by thoseV-categories with compact topology, we obtain a functor V-Catch → V-CatCH.




Remark• Under suitable conditions, this closure operator is topological.

• Moreover, if X is separated, then this topology is Hausdor�.• With V-Catch denoting the full subcategory of V-Catsep defined by thoseV-categories with compact topology, we obtain a functor V-Catch → V-CatCH.




Remark• Under suitable conditions, this closure operator is topological.• Moreover, if X is separated, then this topology is Hausdor�.

• With V-Catch denoting the full subcategory of V-Catsep defined by thoseV-categories with compact topology, we obtain a functor V-Catch → V-CatCH.




Remark• Under suitable conditions, this closure operator is topological.• Moreover, if X is separated, then this topology is Hausdor�.• With V-Catch denoting the full subcategory of V-Catsep defined by thoseV-categories with compact topology, we obtain a functor V-Catch → V-CatCH.


PropositionFor the V-categorical compact Hausdor� space induced by a compact separatedV-category X, the hit-and-miss topology on HX coincides with the topologyinduced by the Hausdor� structure on HX.

TheoremThe functor H : V-Cat→ V-Cat restricts to the category V-Catch, moreover, thediagram

V-Catch V-Catch

V-CatCH V-CatCH

H

H

commutes.

Coalgebras for Hausdorff functors

PropositionThe diagrams of functors commutes.

OrdCH OrdCH

V-CatCH V-CatCH

H

H

PropositionThe Hausdor� functor on V-CatCH preserves codirected initial cones with respectto the forgetful functor V-CatCH→ CompHaus.

TheoremThe Hausdor� functor H : V-CatCH→ V-CatCH preserves codirected limits.


TheoremFor H : V-CatCH→ V-CatCH, the forgetful functor CoAlg(H)→ V-CatCH iscomonadic. Moreover, V-CatCH has equalisers and is therefore complete.

TheoremThe category of coalgebras of a Hausdor� polynomial functor on V-CatCH is(co)complete.

DefinitionWe call a functor Hausdor� polynomial whenever it belongs to the smallest classof endofunctors on V-Cat that contains the identity functor, all constant functorsand is closed under composition with H, products and sums of functors.

Priestley spaces

RecallAn ordered compact Hausdor� space is a Priestley space whenever the conePosComp(X, 2) is an initial monocone. ab

aHilary A. Priestley. “Representation of distributive lattices by means of ordered stone spaces”. In:Bulletin of the London Mathematical Society 2.(2) (1970), pp. 186–190.

bHilary A. Priestley. “Ordered topological spaces and the representation of distributive lattices”. In:Proceedings of the London Mathematical Society. Third Series 24.(3) (1972), pp. 507–530.

DefinitionWe call a V-categorical compact Hausdor� space X Priestley if the coneV-CatCH(X,Vop) is initial (and mono).

V-Priest denotes the full subcategory defined by all Priestley spaces.

PropositionLet X be a V-categorical compact Hausdor� space. Consider a V-subcategoryR ⊆ VX that is closed under finite weighted limits and such that (ψ : X → Vop)ψ∈R

is initial with respect to V-CatCH→ CompHaus.

Then the cone (ψ♦ : HX → Vop)ψ∈R is initial with respect to V-CatCH→ CompHaus.

CorollaryThe Hausdor� functor restricts to a functor H : V-Priest→ V-Priest, hence theHausdor� monad H restricts to V-Priest.

Priestley spaces

RecallAn ordered compact Hausdor� space is a Priestley space whenever the conePosComp(X, 2) is an initial monocone. ab


bHilary A. Priestley. “Ordered topological spaces and the representation of distributive lattices”. In:Proceedings of the London Mathematical Society. Third Series 24.(3) (1972), pp. 507–530.

AssumptionFrom now on we assume that the maps hom(u,−) : V → V are continuous.







Priestley spaces







Priestley spaces

DefinitionWe call a V-categorical compact Hausdor� space X Priestley if the coneV-CatCH(X,Vop) is initial (and mono).V-Priest denotes the full subcategory defined by all Priestley spaces.





Priestley spaces

DefinitionWe call a V-categorical compact Hausdor� space X Priestley if the coneV-CatCH(X,Vop) is initial (and mono).V-Priest denotes the full subcategory defined by all Priestley spaces.


is initial with respect to V-CatCH→ CompHaus.Then the cone (ψ♦ : HX → Vop)ψ∈R is initial with respect to V-CatCH→ CompHaus.


Duality theory


TheoremPriestH ' DLop

∨,⊥.

TheoremPriest ' DLop (induced by 2).a


TheoremCoAlg(H) ' DLOop (distributive lattices with operator).ab

aAlejandro Petrovich. “Distributive lattices with an operator”. In: Studia Logica 56.(1-2) (1996),pp. 205–224.

bRoberto Cignoli, S. Lafalce, and Alejandro Petrovich. “Remarks on Priestley duality for distributivelattices”. In: Order 8.(3) (1991), pp. 299–315.

From 2 to [0, 1]

TheoremC : StablyComp

V−→ LaxMon([0, 1]-FinSup)op is fully faithful.a

aDirk Hofmann and Pedro Nora. “Enriched Stone-type dualities”. In: Advances in Mathematics 330(2018), pp. 307–360.

Remark

• A ⊆ X closed (1 −7−→ X) ! Φ : CX −→ [0, 1].

• A is irreducible ⇐⇒ Φ is in Mon([0, 1]-FinSup).• Every X in StablyComp is sober.

TheoremC :([0, 1]-Priest

)V−→

([0, 1]-FinSup

)op is fully faithful.

Remark

• ϕ : X −→ [0, 1] (1 ϕ−◦−→ X ) ! Φ : CX −→ [0, 1].

• ϕ : X −→ [0, 1] is irreducible(?) ⇐⇒ Φ is ????

From 2 to [0, 1]


V−→ LaxMon([0, 1]-FinSup)op is fully faithful.

Remark• A ⊆ X closed (1 −7−→ X) ! Φ : CX −→ [0, 1].



)V−→

([0, 1]-FinSup


Remark

• ϕ : X −→ [0, 1] (1 ϕ−◦−→ X ) ! Φ : CX −→ [0, 1].


From 2 to [0, 1]




• A is irreducible ⇐⇒ Φ is in Mon([0, 1]-FinSup).

• Every X in StablyComp is sober.


)V−→

([0, 1]-FinSup


Remark

• ϕ : X −→ [0, 1] (1 ϕ−◦−→ X ) ! Φ : CX −→ [0, 1].


From 2 to [0, 1]






)V−→

([0, 1]-FinSup


Remark

• ϕ : X −→ [0, 1] (1 ϕ−◦−→ X ) ! Φ : CX −→ [0, 1].


“Irreducible” distributors

PropositionAn distributor ϕ : X → [0, 1] is left adjoint

⇐⇒ the [0, 1]-functor[ϕ,−] : Fun(X, [0, 1]) −→ [0, 1] preserves tensors and finite suprema.

For Łukasiewicz ⊗ = �

[0, 1] is a Girard quantale: for every u ∈ [0, 1], u = u⊥⊥, hom(u,⊥) = 1− u =: u⊥.Furthermore, the diagram

CX �� //

Φ &&

Fun(X, [0, 1]op)(−)⊥ //

(−·ϕ)��

Fun(X, [0, 1])op

[ϕ,−]op

��[0, 1]

(−)⊥// [0, 1]op

commutes in [0, 1]-Cat

and CX ↪→ Fun(X, [0, 1]op) is∨

-dense.Conclusion: ϕ : 1 −⇀◦ X is le� adjoint ⇐⇒ Φ preserves finite weighted limits.


PropositionAn distributor ϕ : X → [0, 1] is left adjoint ⇐⇒ the [0, 1]-functor[ϕ,−] : Fun(X, [0, 1]) −→ [0, 1] preserves tensors and finite suprema.ab

aMaria Manuel Clementino and Dirk Hofmann. “Lawvere completeness in topology”. In: AppliedCategorical Structures 17.(2) (2009), pp. 175–210.

bDirk Hofmann and Isar Stubbe. “Towards Stone duality for topological theories”. In: Topology andits Applications 158.(7) (2011), pp. 913–925.



CX �� //

Φ &&

Fun(X, [0, 1]op)(−)⊥ //

(−·ϕ)��

Fun(X, [0, 1])op

[ϕ,−]op

��[0, 1]

(−)⊥// [0, 1]op





PropositionAn distributor ϕ : X → [0, 1] is left adjoint ⇐⇒ the [0, 1]-functor[ϕ,−] : Fun(X, [0, 1]) −→ [0, 1] preserves tensors and finite suprema.



CX �� //

Φ &&

Fun(X, [0, 1]op)(−)⊥ //

(−·ϕ)��

Fun(X, [0, 1])op

[ϕ,−]op

��[0, 1]

(−)⊥// [0, 1]op








CX �� //

Φ &&

Fun(X, [0, 1]op)(−)⊥ //

(−·ϕ)��

Fun(X, [0, 1])op

[ϕ,−]op

��[0, 1]

(−)⊥// [0, 1]op

commutes in [0, 1]-Cat and CX ↪→ Fun(X, [0, 1]op) is∨

-dense.

Conclusion: ϕ : 1 −⇀◦ X is le� adjoint ⇐⇒ Φ preserves finite weighted limits.





CX �� //

Φ &&

Fun(X, [0, 1]op)(−)⊥ //

(−·ϕ)��

Fun(X, [0, 1])op

[ϕ,−]op

��[0, 1]

(−)⊥// [0, 1]op

commutes in [0, 1]-Cat and CX ↪→ Fun(X, [0, 1]op) is∨


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