Tutorial on Localic Topology

Jorge Picado

Department of Mathematics University of Coimbra PORTUGAL

January 2015 Tutorial on localic topology BLAST 2015 – 0 – p. 0

OUTLINE

AIM: cover the basics of point-free topology

Slides give motivation, deﬁnitions and results, few proofs

January 2015 Tutorial on localic topology BLAST 2015 – 1 – p. 1 OUTLINE

AIM: cover the basics of point-free topology

Slides give motivation, deﬁnitions and results, few proofs

Part I. Frames: the algebraic facet of spaces

Part II. Categorical aspects of Frm

Part III. Locales: the geometric facet of frames

Part IV. Doing topology in Loc

January 2015 Tutorial on localic topology BLAST 2015 – 1 – p. 1

WHAT IS POINT-FREE TOPOLOGY?

It is an approach to topology taking the lattices of open sets as the primitive notion.

January 2015 Tutorial on localic topology BLAST 2015 – 2 – p. 2 WHAT IS POINT-FREE TOPOLOGY?

It is an approach to topology taking the lattices of open sets as the primitive notion.

The techniques may hide some geometrical intuition, but often offers powerful algebraic tools and opens new perspectives.

January 2015 Tutorial on localic topology BLAST 2015 – 2 – p. 2

WHAT IS POINT-FREE TOPOLOGY? is developed in the categories

frames

Frm frame homomorphisms

January 2015 Tutorial on localic topology BLAST 2015 – 2 – p. 2

WHAT IS POINT-FREE TOPOLOGY? is developed in the categories

frames

Frm frame homomorphisms

‘lattice theory applied to topology’

January 2015 Tutorial on localic topology BLAST 2015 – 2 – p. 2

WHAT IS POINT-FREE TOPOLOGY? is developed in the categories

frames

Frm frame homomorphisms

‘lattice theory applied to topology’ ‘topology itself’

January 2015 Tutorial on localic topology BLAST 2015 – 2 – p. 2 !

WHAT IS POINT-FREE TOPOLOGY? is developed in the categories

frames locales op

Frm Loc Frm frame homomorphisms localic maps

‘lattice theory applied to topology’ ‘topology itself’

January 2015 Tutorial on localic topology BLAST 2015 – 2 – p. 2 WHAT IS POINT-FREE TOPOLOGY? is developed in the categories

frames locales op

Frm Loc Frm frame homomorphisms localic maps

‘lattice theory applied to topology’ ‘topology itself’

! The topological structure of a locale cannot live in its points: the points,

if any, live on the open sets rather than the other way about." P. T. JOHNSTONE [The art of pointless thinking, Category Theory at Work (1991)]

January 2015 Tutorial on localic topology BLAST 2015 – 2 – p. 2 WHAT IS POINT-FREE TOPOLOGY? is developed in the categories

frames locales op

Frm Loc Frm frame homomorphisms localic maps

‘lattice theory applied to topology’ ‘topology itself’

! (...) what the pointfree formulation adds to the classical theory is a remarkable combination of elegance of statement, simplicity of proof, and

increase of extent." R.BALL &J.WALTERS-WAYLAND

[C- and C¦ -quotients in pointfree topology, Dissert. Math. (2002)]

January 2015 Tutorial on localic topology BLAST 2015 – 2 – p. 2 WHAT IS POINT-FREE TOPOLOGY? is developed in the categories

frames locales op

Frm Loc Frm frame homomorphisms localic maps

‘lattice theory applied to topology’ ‘topology itself’

! (...) what the pointfree formulation adds to the classical theory is a remarkable combination of elegance of statement, simplicity of proof, and

increase of extent." R.BALL &J.WALTERS-WAYLAND

[C- and C¦ -quotients in pointfree topology, Dissert. Math. (2002)]

MORE: different categorical properties with advantage to the point-free side.

January 2015 Tutorial on localic topology BLAST 2015 – 2 – p. 2

SOME HISTORY

The idea of approaching topology via algebra (lattice theory) goes back to the ’30s-40’s: Stone, McKinsey and Tarski, Wallman, ...

January 2015 Tutorial on localic topology BLAST 2015 – 3 – p. 3

SOME HISTORY

The idea of approaching topology via algebra (lattice theory) goes back to the ’30s-40’s: Stone, McKinsey and Tarski, Wallman, ...

ORIGINS: Seminar C. Ehresmann (1958) “local lattices” 1st talk (H. Dowker, Prague Top. Symp. 1966) groundbreaking paper (J. Isbell, Atomless parts of spaces, 1972) 1st book (P.T. Johnstone, Stone Spaces, CUP 1982)

January 2015 Tutorial on localic topology BLAST 2015 – 3 – p. 3

SOME HISTORY

The idea of approaching topology via algebra (lattice theory) goes back to the ’30s-40’s: Stone, McKinsey and Tarski, Wallman, ...

Later: autonomous subject with

January 2015 Tutorial on localic topology BLAST 2015 – 3 – p. 3 SOME HISTORY

The idea of approaching topology via algebra (lattice theory) goes back to the ’30s-40’s: Stone, McKinsey and Tarski, Wallman, ...

Later: autonomous subject with

RAMIFICATIONS: category theory, topos theory, logic and computer science.

January 2015 Tutorial on localic topology BLAST 2015 – 3 – p. 3 MAIN BASIC REFERENCES

P.T. Johnstone, Stone Spaces, CUP 1982.

S. Vickers, Topology via Logic, CUP 1989.

S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic - A ﬁrst introduction to topos theory, Springer 1992.

B. Banaschewski, The real numbers in pointfree topology, Textos de Matemática, vol. 12, Univ. Coimbra 1997. R. N. Ball and J. Walters-Wayland, C- and C*-quotients in pointfree topology, Dissert. Math, vol. 412, 2002.

JP, A. Pultr and A. Tozzi, Locales, Chapter II in “Categorical Foundations”, CUP 2004. JP and A. Pultr, Locales treated mostly in a covariant way, Textos de Matemática, vol. 41, Univ. Coimbra 2008.

January 2015 Tutorial on localic topology BLAST 2015 – 4 – p. 4 MAIN BASIC REFERENCES

January 2015 Tutorial on localic topology BLAST 2015 – 4 – p. 4 PART I. Frames:

the algebraic facet of spaces

January 2015 Tutorial on localic topology BLAST 2015 – 5 – p. 5

ê ä

é ã

Ö ¡ ×

Ö Ö¡ ¡× ×

ê ê

FROM SPACES TO FRAMES

Top

X, OX

January 2015 Tutorial on localic topology BLAST 2015 – 6 – p. 6

ê ä

é ã

Ö ¡ ×

Ö Ö¡ ¡× ×

ê ê

FROM SPACES TO FRAMES

Top

O X, OX X,

January 2015 Tutorial on localic topology BLAST 2015 – 6 – p. 6 ¡

Ö ¡ ×

Ö Ö¡ ¡× ×

ê ê

FROM SPACES TO FRAMES

Top complete lattice:

ê ä

Ui Ui, 0

O X, OX X,

U V U V , 1 X

é ã

Ui int Ui

January 2015 Tutorial on localic topology BLAST 2015 – 6 – p. 6 ¡

Ö ¡ ×

Ö Ö¡ ¡× ×

FROM SPACES TO FRAMES

Top complete lattice:

ê ä

Ui Ui, 0

O X, OX X,

U V U V , 1 X

é ã

Ui int Ui

ê ê

U I Vi I U Vi

January 2015 Tutorial on localic topology BLAST 2015 – 6 – p. 6 ¡

Ö ¡ ×

Ö Ö¡ ¡× ×

FROM SPACES TO FRAMES

Top complete lattice:

ê ä

Ui Ui, 0

O X, OX X,

U V U V , 1 X

é ã

Ui int Ui f more:

ê ê

U Vi U Vi O I I Y, Y OY,

January 2015 Tutorial on localic topology BLAST 2015 – 6 – p. 6 ê

FROM SPACES TO FRAMES

Top complete lattice:

ê ä

Ui Ui, 0

O X, OX X,

U V U V , 1 X

é ã

Ui int Ui

¡ 1

Ö ¡ × f f Ö Ö¡ ¡× × more:

ê ê

U Vi U Vi O I I Y, Y OY,

January 2015 Tutorial on localic topology BLAST 2015 – 6 – p. 6 FROM SPACES TO FRAMES

Top complete lattice:

ê ä

Ui Ui, 0

O X, OX X,

U V U V , 1 X

é ã

Ui int Ui

¡ 1

Ö ¡ × f f Ö Ö¡ ¡× × more:

ê ê

U Vi U Vi O I I Y, Y OY,

¡ 1 f preserves and

January 2015 Tutorial on localic topology BLAST 2015 – 6 – p. 6 FROM SPACES TO FRAMES

Top complete lattice L

ê ä

Ui Ui, 0

O X, OX X,

U V U V , 1 X

é ã

Ui int Ui

¡ 1

Ö ¡ × f f Ö Ö¡ ¡× × frame:

ê ê

a b a b O i i Y, Y OY, I I

frame homomorphisms: h: M L preserves and

January 2015 Tutorial on localic topology BLAST 2015 – 6 – p. 6 FROM SPACES TO FRAMES

Top Frm complete lattice L

ê ä

Ui Ui, 0

O X, OX X,

U V U V , 1 X

é ã

Ui int Ui

¡ 1

Ö ¡ × f f Ö Ö¡ ¡× × frame:

ê ê

a b a b O i i Y, Y OY, I I

frame homomorphisms: h: M L preserves and

January 2015 Tutorial on localic topology BLAST 2015 – 6 – p. 6 FROM SPACES TO FRAMES

Top Frm complete lattice L

ê ä

Ui Ui, 0

O X, OX X,

U V U V , 1 X

é ã

Ui int Ui

¡ 1

Ö ¡ × f f Ö Ö¡ ¡× × frame:

ê ê

a b a b O i i Y, Y OY, I I

frame homomorphisms: h: M L preserves and

The algebraic nature of the objects of Frm is obvious. More about that later on...

January 2015 Tutorial on localic topology BLAST 2015 – 6 – p. 6

½ ¾

MORE EXAMPLES of frames

Finite distributive lattices, complete Boolean algebras, complete chains.

January 2015 Tutorial on localic topology BLAST 2015 – 7 – p. 7

MORE EXAMPLES of frames

Finite distributive lattices, complete Boolean algebras, complete chains.

0 1

½ : ¾ :

January 2015 Tutorial on localic topology BLAST 2015 – 7 – p. 7

MORE EXAMPLES of frames

Finite distributive lattices, complete Boolean algebras, complete chains.

0 1

½ : ¾ :

subframe of a frame L: S L closed under arbitrary joins

(in part. 0 S) and ﬁnite meets (in part. 1 S).

January 2015 Tutorial on localic topology BLAST 2015 – 7 – p. 7 MORE EXAMPLES of frames

Finite distributive lattices, complete Boolean algebras, complete chains.

0 1

½ : ¾ :

subframe of a frame L: S L closed under arbitrary joins

(in part. 0 S) and ﬁnite meets (in part. 1 S).

intervals of a frame L: a,b L, a b

a,b x L a x b , b 0,b , a a, 1 .

January 2015 Tutorial on localic topology BLAST 2015 – 7 – p. 7

MORE EXAMPLES of frames

For any -semilattice A, , 1 , D A down-sets of A is a frame:

é ã ê ä

, .

January 2015 Tutorial on localic topology BLAST 2015 – 8 – p. 8 MORE EXAMPLES of frames

For any -semilattice A, , 1 , D A down-sets of A is a frame:

é ã ê ä

, .

SLat Frm G (forgetful functor)

January 2015 Tutorial on localic topology BLAST 2015 – 8 – p. 8

MORE EXAMPLES of frames

For any -semilattice A, , 1 , D A down-sets of A is a frame:

é ã ê ä

, .

SLat Frm G (forgetful functor)

January 2015 Tutorial on localic topology BLAST 2015 – 8 – p. 8 MORE EXAMPLES of frames

For any -semilattice A, , 1 , D A down-sets of A is a frame:

é ã ê ä

, .

SLat Frm G (forgetful functor)

HomFrm D A , L HomSLat A, G L

h h: a h a

g : S g S g

January 2015 Tutorial on localic topology BLAST 2015 – 8 – p. 8

MORE EXAMPLES of frames

For any distributive lattice A, I A ideals of A is a frame:

é ã

,J K a b a J, b K .

January 2015 Tutorial on localic topology BLAST 2015 – 9 – p. 9 MORE EXAMPLES of frames

For any distributive lattice A, I A ideals of A is a frame:

é ã

,J K a b a J, b K .

DLat Frm E (inclusion as a non-full subcategory)

January 2015 Tutorial on localic topology BLAST 2015 – 9 – p. 9 MORE EXAMPLES of frames

For any distributive lattice A, I A ideals of A is a frame:

é ã

,J K a b a J, b K .

DLat Frm E (inclusion as a non-full subcategory)

January 2015 Tutorial on localic topology BLAST 2015 – 9 – p. 9

¾ ½

ê ê

EXAMPLES of frame homomorphisms

All homomorphisms of ﬁnite distributive lattices.

January 2015 Tutorial on localic topology BLAST 2015 – 10 – p. 10

¾ ½

ê ê

EXAMPLES of frame homomorphisms

All homomorphisms of ﬁnite distributive lattices.

Complete homomorphisms of complete Boolean algebras.

January 2015 Tutorial on localic topology BLAST 2015 – 10 – p. 10 ¾ ½

ê ê

EXAMPLES of frame homomorphisms

All homomorphisms of ﬁnite distributive lattices.

Complete homomorphisms of complete Boolean algebras.

For any frame L, a L:

∆a : L a ∇a : L a

x x a x x a

January 2015 Tutorial on localic topology BLAST 2015 – 10 – p. 10

ê ê

EXAMPLES of frame homomorphisms

All homomorphisms of ﬁnite distributive lattices.

Complete homomorphisms of complete Boolean algebras.

For any frame L, a L:

∆a : L a ∇a : L a

x x a x x a

For any frame L, there exist unique ¾ L, L ½ .

January 2015 Tutorial on localic topology BLAST 2015 – 10 – p. 10

ê ê

EXAMPLES of frame homomorphisms

All homomorphisms of ﬁnite distributive lattices.

Complete homomorphisms of complete Boolean algebras.

For any frame L, a L:

∆a : L a ∇a : L a

x x a x x a

For any frame L, there exist unique ¾ L, L ½ .

initial object terminal object

January 2015 Tutorial on localic topology BLAST 2015 – 10 – p. 10 EXAMPLES of frame homomorphisms

All homomorphisms of ﬁnite distributive lattices.

Complete homomorphisms of complete Boolean algebras.

For any frame L, a L:

∆a : L a ∇a : L a

x x a x x a

For any frame L, there exist unique ¾ L, L ½ .

initial object terminal object

ê ê

: I L L : D L L

ê ê

J J S S

January 2015 Tutorial on localic topology BLAST 2015 – 10 – p. 10

BACKGROUND: POSETS AS CATEGORIES poset A,

A, as a thin category

January 2015 Tutorial on localic topology BLAST 2015 – 11 – p. 11 D

BACKGROUND: POSETS AS CATEGORIES poset A,

OBJECTS: a A

A, as a thin category

January 2015 Tutorial on localic topology BLAST 2015 – 11 – p. 11 ¥

BACKGROUND: POSETS AS CATEGORIES poset A,

OBJECTS: a A

A, as a thin category

D !

MORPHISMS: a b whenever a b

(there is at most one arrow between any pair of objects)

January 2015 Tutorial on localic topology BLAST 2015 – 11 – p. 11 BACKGROUND: POSETS AS CATEGORIES poset A,

OBJECTS: a A

A, as a thin category

D !

MORPHISMS: a b whenever a b

(there is at most one arrow between any pair of objects)

1 1a b f a b

In fact, a preorder sufﬁces: g ¥ f g

(1) reﬂexivity: provides the identity morphisms 1a. c

(2) transitivity: provides the composition of morphisms g f. 1c

January 2015 Tutorial on localic topology BLAST 2015 – 11 – p. 11

¤ ¤

BACKGROUND: PREORDERS AS CATEGORIES

OBJECTS: a A

A, as a thin category

D !

MORPHISMS: a b whenever a b

FUNCTORS: f : A B

January 2015 Tutorial on localic topology BLAST 2015 – 11 – p. 11 BACKGROUND: PREORDERS AS CATEGORIES

OBJECTS: a A

A, as a thin category

D !

MORPHISMS: a b whenever a b

FUNCTORS: f : A B

a f a order-preserving maps

¤ ¤

½ a f a

January 2015 Tutorial on localic topology BLAST 2015 – 11 – p. 11 BACKGROUND: PREORDERS AS CATEGORIES

OBJECTS: a A

A, as a thin category

D !

MORPHISMS: a b whenever a b

FUNCTORS: f : A B

a f a order-preserving maps

¤ ¤

½ a f a

(binary) PRODUCTS: a ? b meets

January 2015 Tutorial on localic topology BLAST 2015 – 11 – p. 11 BACKGROUND: PREORDERS AS CATEGORIES

OBJECTS: a A

A, as a thin category

D !

MORPHISMS: a b whenever a b

FUNCTORS: f : A B

a f a order-preserving maps

¤ ¤

½ a f a

(binary) PRODUCTS: a ? b meets

January 2015 Tutorial on localic topology BLAST 2015 – 11 – p. 11 BACKGROUND: PREORDERS AS CATEGORIES

OBJECTS: a A

A, as a thin category

D !

MORPHISMS: a b whenever a b

FUNCTORS: f : A B

a f a order-preserving maps

¤ ¤

½ a f a

(binary) PRODUCTS: a a b b meets

January 2015 Tutorial on localic topology BLAST 2015 – 11 – p. 11 BACKGROUND: PREORDERS AS CATEGORIES

OBJECTS: a A

A, as a thin category

D !

MORPHISMS: a b whenever a b

FUNCTORS: f : A B

a f a order-preserving maps

¤ ¤

½ a f a

(binary) COPRODUCTS: a a b b joins

January 2015 Tutorial on localic topology BLAST 2015 – 11 – p. 11 BACKGROUND: PREORDERS AS THIN CATEGORIES

“Existence of limits” means “existence of products” (because equalizers exist trivially in thin categories)

January 2015 Tutorial on localic topology BLAST 2015 – 12 – p. 12 BACKGROUND: PREORDERS AS THIN CATEGORIES

“Existence of limits” means “existence of products” (because equalizers exist trivially in thin categories) so “existence of limits” (i.e. “complete category”) means “complete lattice”.

January 2015 Tutorial on localic topology BLAST 2015 – 12 – p. 12 BACKGROUND: PREORDERS AS THIN CATEGORIES

From this point of view: category theory is an extension of lattice th.

January 2015 Tutorial on localic topology BLAST 2015 – 12 – p. 12

GALOIS ADJUNCTIONS

f a b iff a g b A, B, g

January 2015 Tutorial on localic topology BLAST 2015 – 13 – p. 13

GALOIS ADJUNCTIONS

HomB f a , b HomA a, g b f

f a b iff a g b A, B, f g g

January 2015 Tutorial on localic topology BLAST 2015 – 13 – p. 13

GALOIS ADJUNCTIONS

HomB f a , b HomA a, g b f

f a b iff a g b A, B, f g g

fg id and id gf

(“quasi-inverses”)

January 2015 Tutorial on localic topology BLAST 2015 – 13 – p. 13

GALOIS ADJUNCTIONS

HomB f a , b HomA a, g b f

f a b iff a g b A, B, f g g

fg id and id gf

(“quasi-inverses”) Properties

1 fgf f and gfg g.

January 2015 Tutorial on localic topology BLAST 2015 – 13 – p. 13

GALOIS ADJUNCTIONS

HomB f a , b HomA a, g b f

f a b iff a g b A, B, f g g

fg id and id gf

(“quasi-inverses”) Properties

1 fgf f and gfg g.

2 A, B, g

g B f A

January 2015 Tutorial on localic topology BLAST 2015 – 13 – p. 13 GALOIS ADJUNCTIONS

HomB f a , b HomA a, g b f

f a b iff a g b A, B, f g g

fg id and id gf

(“quasi-inverses”) Properties

1 fgf f and gfg g.

2 A, B,

g g B a A gf a a

f A b B fg b b

g B f A

January 2015 Tutorial on localic topology BLAST 2015 – 13 – p. 13 ê

ê ê ê

ê ê

GALOIS ADJUNCTIONS

ADJOINT FUNCTOR THEOREM.

Let f : A B be an order-preserving map between posets. Then:

January 2015 Tutorial on localic topology BLAST 2015 – 14 – p. 14 ê

ê ê ê

ê ê

GALOIS ADJUNCTIONS

ADJOINT FUNCTOR THEOREM.

Let f : A B be an order-preserving map between posets. Then: (1) If f has a right adjoint, then f preserves all joins that exist in A.

January 2015 Tutorial on localic topology BLAST 2015 – 14 – p. 14 ê

ê ê

GALOIS ADJUNCTIONS

ADJOINT FUNCTOR THEOREM.

Let f : A B be an order-preserving map between posets. Then: (1) If f has a right adjoint, then f preserves all joins that exist in A.

ê ê ? ê

PROOF: Let f g, S A, S exists. f S f s s S

January 2015 Tutorial on localic topology BLAST 2015 – 14 – p. 14 ê

ê ê

GALOIS ADJUNCTIONS

ADJOINT FUNCTOR THEOREM.

Let f : A B be an order-preserving map between posets. Then: (1) If f has a right adjoint, then f preserves all joins that exist in A.

ê ê ? ê

PROOF: Let f g, S A, S exists. f S f s s S

upper bound X

January 2015 Tutorial on localic topology BLAST 2015 – 14 – p. 14 ê

GALOIS ADJUNCTIONS

ADJOINT FUNCTOR THEOREM.

Let f : A B be an order-preserving map between posets. Then: (1) If f has a right adjoint, then f preserves all joins that exist in A.

ê ê ? ê

PROOF: Let f g, S A, S exists. f S f s s S

upper bound X

least upper bound: f s b s iff s g b s

ê ê

f S b iff S g b

January 2015 Tutorial on localic topology BLAST 2015 – 14 – p. 14 ê

GALOIS ADJUNCTIONS

ADJOINT FUNCTOR THEOREM.

Let f : A B be an order-preserving map between posets. Then: (1) If f has a right adjoint, then f preserves all joins that exist in A.

ê ê ? ê

PROOF: Let f g, S A, S exists. f S f s s S

upper bound X

least upper bound: f s b s iff s g b s

ê ê

f S b iff S g b

January 2015 Tutorial on localic topology BLAST 2015 – 14 – p. 14 ê

GALOIS ADJUNCTIONS

ADJOINT FUNCTOR THEOREM.

Let f : A B be an order-preserving map between posets. Then: (1) If f has a right adjoint, then f preserves all joins that exist in A.

ê ê ? ê

PROOF: Let f g, S A, S exists. f S f s s S

upper bound X

least upper bound: f s b s iff s g b s

ê ê

f S b iff S g b

January 2015 Tutorial on localic topology BLAST 2015 – 14 – p. 14 ê

GALOIS ADJUNCTIONS

ADJOINT FUNCTOR THEOREM.

Let f : A B be an order-preserving map between posets. Then: (1) If f has a right adjoint, then f preserves all joins that exist in A.

ê ê ? ê

PROOF: Let f g, S A, S exists. f S f s s S

upper bound X

least upper bound: f s b s iff s g b s X

ê ê

f S b iff S g b

January 2015 Tutorial on localic topology BLAST 2015 – 14 – p. 14 GALOIS ADJUNCTIONS

ADJOINT FUNCTOR THEOREM.

Let f : A B be an order-preserving map between posets. Then: (1) If f has a right adjoint, then f preserves all joins that exist in A. (2) If A has all joins and f preserves them, then f has a right adjoint g, uniquely determined by f:

g b a A f a b .

ê ê ? ê

PROOF: Let f g, S A, S exists. f S f s s S

upper bound X

least upper bound: f s b s iff s g b s X

ê ê

f S b iff S g b

January 2015 Tutorial on localic topology BLAST 2015 – 14 – p. 14

6 6

FRAMES as HEYTING ALGEBRAS

Heyting algebra: lattice L with an extra satisfying

a b c iff b a c

January 2015 Tutorial on localic topology BLAST 2015 – 15 – p. 15

6 6

FRAMES as HEYTING ALGEBRAS

Heyting algebra: lattice L with an extra satisfying

a b c iff b a c i.e.

a a .

January 2015 Tutorial on localic topology BLAST 2015 – 15 – p. 15

6 6

FRAMES as HEYTING ALGEBRAS

Heyting algebra: lattice L with an extra satisfying

a b c iff b a c i.e.

a a .

A complete lattice is an Heyting algebra iff it is a frame.

January 2015 Tutorial on localic topology BLAST 2015 – 15 – p. 15

6 6

FRAMES as HEYTING ALGEBRAS

Heyting algebra: lattice L with an extra satisfying

a b c iff b a c i.e.

a a .

A complete lattice is an Heyting algebra iff it is a frame.

Proof: This is the ADJOINT FUNCTOR THEOREM!

January 2015 Tutorial on localic topology BLAST 2015 – 15 – p. 15

6 6

FRAMES as HEYTING ALGEBRAS

Heyting algebra: lattice L with an extra satisfying

a b c iff b a c i.e.

a a .

A complete lattice is an Heyting algebra iff it is a frame.

Proof: This is the ADJOINT FUNCTOR THEOREM!

: a is a left adjoint preserves joins.

January 2015 Tutorial on localic topology BLAST 2015 – 15 – p. 15 6

6 6

FRAMES as HEYTING ALGEBRAS

Heyting algebra: lattice L with an extra satisfying

a b c iff b a c i.e.

a a .

A complete lattice is an Heyting algebra iff it is a frame.

Proof: This is the ADJOINT FUNCTOR THEOREM!

: a is a left adjoint preserves joins.

: a preserves joins (=colimits) it has a right adjoint.

January 2015 Tutorial on localic topology BLAST 2015 – 15 – p. 15 FRAMES as HEYTING ALGEBRAS

Heyting algebra: lattice L with an extra satisfying

a b c iff b a c i.e.

a a .

A complete lattice is an Heyting algebra iff it is a frame.

Proof: This is the ADJOINT FUNCTOR THEOREM!

: a is a left adjoint preserves joins.

: a preserves joins (=colimits) it has a right adjoint.

6 6 6 frames = cHa. BUT different categories (morphisms).

January 2015 Tutorial on localic topology BLAST 2015 – 15 – p. 15

ê é

¦ ¦

¦¦ ¦¦¦ ¦

ê é é

¦ ¦

FRAMES as HEYTING ALGEBRAS

é é Properties

H1 a bi a bi .

January 2015 Tutorial on localic topology BLAST 2015 – 16 – p. 16 ê é

¦ ¦

¦¦ ¦¦¦ ¦

ê é é

¦ ¦

FRAMES as HEYTING ALGEBRAS

é é Properties

H1 a bi a bi .

H2 a b c iff b a c.

January 2015 Tutorial on localic topology BLAST 2015 – 16 – p. 16 ê

¦ ¦

¦¦ ¦¦¦ ¦

ê é é

¦ ¦

FRAMES as HEYTING ALGEBRAS

é é Properties

H1 a bi a bi .

H2 a b c iff b a c.

ê é

H3 ai b ai b . . .

January 2015 Tutorial on localic topology BLAST 2015 – 16 – p. 16 ¦

¦ ¦

¦¦ ¦¦¦ ¦

ê é é

¦ ¦

FRAMES as HEYTING ALGEBRAS

é é Properties

H1 a bi a bi .

H2 a b c iff b a c.

ê é

H3 ai b ai b . . .

Pseudocomplement: a a 0 b b a 0 .

January 2015 Tutorial on localic topology BLAST 2015 – 16 – p. 16 ¦ ¦

¦¦ ¦¦¦ ¦

ê é é

¦ ¦

FRAMES as HEYTING ALGEBRAS

é é Properties

H1 a bi a bi .

H2 a b c iff b a c.

ê é

H3 ai b ai b . . .

Pseudocomplement: a a 0 b b a 0 .

Example: U int X r U .

January 2015 Tutorial on localic topology BLAST 2015 – 16 – p. 16 ¦¦ ¦¦¦ ¦

ê é é

¦ ¦

FRAMES as HEYTING ALGEBRAS

é é Properties

H1 a bi a bi .

H2 a b c iff b a c.

ê é

H3 ai b ai b . . .

Pseudocomplement: a a 0 b b a 0 .

Example: U int X r U .

¦ ¦

P1 a b b a .

January 2015 Tutorial on localic topology BLAST 2015 – 16 – p. 16 ê é é

¦ ¦

FRAMES as HEYTING ALGEBRAS

é é Properties

H1 a bi a bi .

H2 a b c iff b a c.

ê é

H3 ai b ai b . . .

Pseudocomplement: a a 0 b b a 0 .

Example: U int X r U .

¦ ¦

P1 a b b a .

¦¦ ¦¦¦ ¦

P2 a a , a a .

January 2015 Tutorial on localic topology BLAST 2015 – 16 – p. 16 FRAMES as HEYTING ALGEBRAS

é é Properties

H1 a bi a bi .

H2 a b c iff b a c.

ê é

H3 ai b ai b . . .

Pseudocomplement: a a 0 b b a 0 .

Example: U int X r U .

¦ ¦

P1 a b b a .

¦¦ ¦¦¦ ¦

P2 a a , a a .

ê é é

¦ ¦

P3 ai ai . De Morgan law (Caution: not for )

January 2015 Tutorial on localic topology BLAST 2015 – 16 – p. 16 PART II.

Categorical aspects of Frm

January 2015 Tutorial on localic topology BLAST 2015 – 17 – p. 17

ê ê ê ê

ALGEBRAIC ASPECTS OF Frm

1 Frm is equationally presentable i.e.

January 2015 Tutorial on localic topology BLAST 2015 – 17 – p. 17

ê ê ê ê

ALGEBRAIC ASPECTS OF Frm

1 Frm is equationally presentable i.e.

Objects are described by a (proper class of) operations and equations:

OPERATIONS 0 0-ary: 0, 1: L L 2 binary: L L, a,b a b

κ ê κ-ary (any cardinal κ): L L, ai κ κ ai

January 2015 Tutorial on localic topology BLAST 2015 – 17 – p. 17 ALGEBRAIC ASPECTS OF Frm

1 Frm is equationally presentable i.e.

Objects are described by a (proper class of) operations and equations:

OPERATIONS 0 0-ary: 0, 1: L L 2 binary: L L, a,b a b

κ ê κ-ary (any cardinal κ): L L, ai κ κ ai

EQUATIONS

L, , 1 is an idempotent commutative monoid

with a zero 0 sat. the absorption law a 0 0 0 a a.

ê ê ê ê

0 0 ai , aj κ ai aj, a κ ai κ a ai .

January 2015 Tutorial on localic topology BLAST 2015 – 17 – p. 17

ALGEBRAIC ASPECTS OF Frm

Then, by general results of category theory [E. Manes, Algebraic Theories, Springer, 1976]:

January 2015 Tutorial on localic topology BLAST 2015 – 18 – p. 18 ALGEBRAIC ASPECTS OF Frm

Then, by general results of category theory [E. Manes, Algebraic Theories, Springer, 1976]:

COROLLARY. Frm has all (small) limits (i.e., it is a COMPLETE category) and they are constructed exactly as in Set (i.e., the forgetful func-

tor Frm Set preserves them).

January 2015 Tutorial on localic topology BLAST 2015 – 18 – p. 18 ê

Ô Õ

ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ¨

Ô Õ

ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ¨

Ô Õ

ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ¨

Ô Õ

ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ¨

ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

X F X

f F Ô f Õ

Y F Y

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

X F X S X S ﬁnite ,

f F Ô f Õ

Y F Y f S

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

ηX

THE UNIT: X F X

x x

D ! ϕ È SLat

f A

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

ηX

THE UNIT: X F X

x x

D ! ϕ È SLat

f A

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19

ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

ηX

THE UNIT: X F X

x x

D ! ϕ È SLat

f A

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

ηX

THE UNIT: X F X x1,x2,...,xn

x x

D ! ϕ È SLat

f é n

f x A i 1 i

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19

ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

A D A

h DÔ hÕ

B D B

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

A D A S A S S S

h DÔ hÕ

B D B h S

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

ηA

THE UNIT: A D A

a a

D ! ϕ È Frm

f L

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

ηA

THE UNIT: A D A

a a

D ! ϕ È Frm

f L

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ª

ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

ηA

THE UNIT: A D A

a a

D ! ϕ È Frm

f L

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19 ALGEBRAIC ASPECTS OF Frm

2 Frm has free objects: there is a free functor Set Frm (i.e., a

left adjoint of the forgetful functor Frm Set):

CONSTRUCTION (in two steps):

F D

Set SLat ê Frm

forgets ^ forgets

ηA

THE UNIT: A D A S

a a

D ! ϕ È Frm

f ª L f s

sÈ S

January 2015 Tutorial on localic topology BLAST 2015 – 19 – p. 19

ALGEBRAIC ASPECTS OF Frm

Then, by general results of category theory [E. Manes, Algebraic Theories, Springer, 1976]:

January 2015 Tutorial on localic topology BLAST 2015 – 20 – p. 20

ALGEBRAIC ASPECTS OF Frm

Then, by general results of category theory [E. Manes, Algebraic Theories, Springer, 1976]:

COROLLARY. Frm is an ALGEBRAIC category. In particular:

January 2015 Tutorial on localic topology BLAST 2015 – 20 – p. 20

ALGEBRAIC ASPECTS OF Frm

Then, by general results of category theory [E. Manes, Algebraic Theories, Springer, 1976]:

COROLLARY. Frm is an ALGEBRAIC category. In particular: (1) It has all (small) colimits (i.e., it is a COCOMPLETE category).

January 2015 Tutorial on localic topology BLAST 2015 – 20 – p. 20

ALGEBRAIC ASPECTS OF Frm

Then, by general results of category theory [E. Manes, Algebraic Theories, Springer, 1976]:

COROLLARY. Frm is an ALGEBRAIC category. In particular: (1) It has all (small) colimits (i.e., it is a COCOMPLETE category). (2) Monomorphisms = injective.

January 2015 Tutorial on localic topology BLAST 2015 – 20 – p. 20

ALGEBRAIC ASPECTS OF Frm

Then, by general results of category theory [E. Manes, Algebraic Theories, Springer, 1976]:

COROLLARY. Frm is an ALGEBRAIC category. In particular: (1) It has all (small) colimits (i.e., it is a COCOMPLETE category). (2) Monomorphisms = injective. (3) Epimorphisms need not be surjective; Regular epis = surjective.

January 2015 Tutorial on localic topology BLAST 2015 – 20 – p. 20 ALGEBRAIC ASPECTS OF Frm

Then, by general results of category theory [E. Manes, Algebraic Theories, Springer, 1976]:

(4) RegEpi,Mono is a factorization system.

January 2015 Tutorial on localic topology BLAST 2015 – 20 – p. 20 ALGEBRAIC ASPECTS OF Frm

Then, by general results of category theory [E. Manes, Algebraic Theories, Springer, 1976]:

(4) RegEpi,Mono is a factorization system. (5) Quotients are described by congruences; there exist presentations by generators and relations.

January 2015 Tutorial on localic topology BLAST 2015 – 20 – p. 20 EXAMPLE: PRESENTATIONS

PRESENTATIONS BY GENERATORS AND RELATIONS:

just take the quotient of the free frame on the given set of generators modulo the congruence generated by the pairs

u, v for the given relations u v.

January 2015 Tutorial on localic topology BLAST 2015 – 21 – p. 21 EXAMPLE: PRESENTATIONS Frame of reals L Ê

generated by all ordered pairs p, q , p, q É , subject to the relations

(R1) p, q r, s p r, q s ,

(R2) p, q r, s p, s whenever p r q s,

(R3) p, q r, s p r s q ,

(R4) p, q 1. p,q È É

p r q s

r s

p q

January 2015 Tutorial on localic topology BLAST 2015 – 21 – p. 21 EXAMPLE: PRESENTATIONS Frame of reals L Ê

generated by all ordered pairs p, q , p, q É , subject to the relations

(R1) p, q r, s p r, q s ,

(R2) p, q r, s p, s whenever p r q s,

(R3) p, q r, s p r s q ,

(R4) p, q 1. p,q È É

p s

r s

p q

January 2015 Tutorial on localic topology BLAST 2015 – 21 – p. 21 EXAMPLE: PRESENTATIONS Frame of reals L Ê

generated by all ordered pairs p, q , p, q É , subject to the relations

(R1) p, q r, s p r, q s ,

(R2) p, q r, s p, s whenever p r q s,

(R3) p, q r, s p r s q ,

(R4) p, q 1. p,q È É

p q

January 2015 Tutorial on localic topology BLAST 2015 – 21 – p. 21 EXAMPLE: PRESENTATIONS Frame of reals L Ê

generated by all ordered pairs p, q , p, q É , subject to the relations

(R1) p, q r, s p r, q s ,

(R2) p, q r, s p, s whenever p r q s,

(R3) p, q r, s p r s q ,

(R4) p, q 1. p,q È É

January 2015 Tutorial on localic topology BLAST 2015 – 21 – p. 21 EXAMPLE: PRESENTATIONS Frame of reals L Ê

Nice features: Continuous real functions,

semicontinuous real functions, ...

MORE, in next lectures.

January 2015 Tutorial on localic topology BLAST 2015 – 21 – p. 21