[PPT] - Parts of biframes and a categorical approach to BiFrm Imanol Mozo PowerPoint Presentation

SLIDE 1

Parts of biframes and a categorical approach to BiFrm

Imanol Mozo Carollo1 imanol.mozo@ehu.eus

WorkALT in honour of Aleš Pultr, on the occasion of his 80th birthday

1Joint work with Andrew Moshier and Joanne Walters-Wayland

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 1 / 10

SLIDE 2

Parts of pointfree spaces: Sublocales

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

SLIDE 3

Parts of pointfree spaces: Sublocales

Subobjects of Frmop = Loc
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

SLIDE 4

Parts of pointfree spaces: Sublocales

Subobjects of Frmop = Loc
They are like subspaces. . .
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

SLIDE 5

Parts of pointfree spaces: Sublocales

Subobjects of Frmop = Loc
They are like subspaces. . .
. . . but way more fun!
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

SLIDE 6

Parts of pointfree spaces: Sublocales

Subobjects of Frmop = Loc
They are like subspaces. . .
. . . but way more fun!

Theorem (Isbell’s density theorem)

Each locale L contains a least dense sublocale, namely the Booleanization B(L) of L.

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

SLIDE 7

Parts of pointfree spaces: Sublocales

Subobjects of Frmop = Loc
They are like subspaces. . .
. . . but way more fun!

Theorem (Isbell’s density theorem)

Each locale L contains a least dense sublocale, namely the Booleanization B(L) of L. There are several ways to represent them:

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

SLIDE 8

Parts of pointfree spaces: Sublocales

Subobjects of Frmop = Loc
They are like subspaces. . .
. . . but way more fun!

Theorem (Isbell’s density theorem)

Each locale L contains a least dense sublocale, namely the Booleanization B(L) of L. There are several ways to represent them:

1 frame congruences

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

SLIDE 9

Parts of pointfree spaces: Sublocales

Subobjects of Frmop = Loc
They are like subspaces. . .
. . . but way more fun!

Theorem (Isbell’s density theorem)

Each locale L contains a least dense sublocale, namely the Booleanization B(L) of L. There are several ways to represent them:

1 frame congruences 2 nuclei

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

SLIDE 10

Parts of pointfree spaces: Sublocales

Subobjects of Frmop = Loc
They are like subspaces. . .
. . . but way more fun!

Theorem (Isbell’s density theorem)

Each locale L contains a least dense sublocale, namely the Booleanization B(L) of L. There are several ways to represent them:

1 frame congruences 2 nuclei 3 sublocale sets

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

SLIDE 11

Parts of pointfree spaces: Sublocales

Subobjects of Frmop = Loc
They are like subspaces. . .
. . . but way more fun!

Theorem (Isbell’s density theorem)

Each locale L contains a least dense sublocale, namely the Booleanization B(L) of L. There are several ways to represent them:

1 frame congruences 2 nuclei 3 sublocale sets 4 onto frame homomorphisms = extremal epimorphisms

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

SLIDE 12

Parts of pointfree spaces: Sublocales

Subobjects of Frmop = Loc
They are like subspaces. . .
. . . but way more fun!

Theorem (Isbell’s density theorem)

Each locale L contains a least dense sublocale, namely the Booleanization B(L) of L. There are several ways to represent them:

1 frame congruences 2 nuclei 3 sublocale sets 4 onto frame homomorphisms = extremal epimorphisms

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

SLIDE 13

Parts of pointfree spaces: Sublocales

Subobjects of Frmop = Loc
They are like subspaces. . .
. . . but way more fun!

Theorem (Isbell’s density theorem)

Each locale L contains a least dense sublocale, namely the Booleanization B(L) of L. There are several ways to represent them:

1 frame congruences 2 nuclei 3 sublocale sets 4 onto frame homomorphisms = extremal epimorphisms

Extremal epimorphisms

An extremal epimorphism e in a category C is an epimorphism such that if e = m ◦ g where m is a monomorphism = ⇒ m is an isomorphism.

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

SLIDE 14

Classical bispaces

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

SLIDE 15

Classical bispaces

Bispaces

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

SLIDE 16

Classical bispaces

Bispaces

A set of points X endowed with two topologies: (X, τ1, τ2)

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

SLIDE 17

Classical bispaces

Bispaces

A set of points X endowed with two topologies: (X, τ1, τ2) Bicontinuous functions: continuous w.r.t. both topologies.

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

SLIDE 18

Classical bispaces

Bispaces

A set of points X endowed with two topologies: (X, τ1, τ2) Bicontinuous functions: continuous w.r.t. both topologies.

J. C. Kelly,

Bitopological spaces,

Proc. London Math. Soc (3) 13, 71–89 (1963).
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

SLIDE 19

Classical bispaces

Bispaces

A set of points X endowed with two topologies: (X, τ1, τ2) Bicontinuous functions: continuous w.r.t. both topologies.

J. C. Kelly,

Bitopological spaces,

Proc. London Math. Soc (3) 13, 71–89 (1963).

How to forget about the points?

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

SLIDE 20

Classical bispaces

Bispaces

A set of points X endowed with two topologies: (X, τ1, τ2) Bicontinuous functions: continuous w.r.t. both topologies.

J. C. Kelly,

Bitopological spaces,

Proc. London Math. Soc (3) 13, 71–89 (1963).

How to forget about the points? τ0 = τ1 ∨ τ2

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

SLIDE 21

Classical bispaces

Bispaces

A set of points X endowed with two topologies: (X, τ1, τ2) Bicontinuous functions: continuous w.r.t. both topologies.

J. C. Kelly,

Bitopological spaces,

Proc. London Math. Soc (3) 13, 71–89 (1963).

How to forget about the points? τ0 = τ1 ∨ τ2

τ1 and τ2 embed into τ0
τ1 ∪ τ2 forms a subbasis of τ0
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

SLIDE 22

Classical bispaces

Bispaces

A set of points X endowed with two topologies: (X, τ1, τ2) Bicontinuous functions: continuous w.r.t. both topologies.

J. C. Kelly,

Bitopological spaces,

Proc. London Math. Soc (3) 13, 71–89 (1963).

How to forget about the points? τ0 = τ1 ∨ τ2

τ1 and τ2 embed into τ0
τ1 ∪ τ2 forms a subbasis of τ0

Any bicontinuous function is continuous w.r.t. τ0.

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

SLIDE 23

Getting rid of points: biframes

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 4 / 10

SLIDE 24

Getting rid of points: biframes

Biframes

A biframe L is formed by three frames (L0, L1, L2) where

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 4 / 10

SLIDE 25

Getting rid of points: biframes

Biframes

A biframe L is formed by three frames (L0, L1, L2) where

L1 and L2 are subframes of the ambient frame L0
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 4 / 10

SLIDE 26

Getting rid of points: biframes

Biframes

A biframe L is formed by three frames (L0, L1, L2) where

L1 and L2 are subframes of the ambient frame L0
L1 ∪ L2 forms a subbasis of L0
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 4 / 10

SLIDE 27

Getting rid of points: biframes

Biframes

A biframe L is formed by three frames (L0, L1, L2) where

L1 and L2 are subframes of the ambient frame L0
L1 ∪ L2 forms a subbasis of L0
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 4 / 10

SLIDE 28

Getting rid of points: biframes

Biframes

A biframe L is formed by three frames (L0, L1, L2) where

L1 and L2 are subframes of the ambient frame L0
L1 ∪ L2 forms a subbasis of L0

Biframe homomorphisms f : L → M are given by frame homomorphisms f0 : L0 → M0 that restricts to frame homomorphisms fi : Li → Mi (i = 1, 2)

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 4 / 10

SLIDE 29

Getting rid of points: biframes

Biframes

A biframe L is formed by three frames (L0, L1, L2) where

L1 and L2 are subframes of the ambient frame L0
L1 ∪ L2 forms a subbasis of L0

Biframe homomorphisms f : L → M are given by frame homomorphisms f0 : L0 → M0 that restricts to frame homomorphisms fi : Li → Mi (i = 1, 2)

J. B. Banaschewski, G. C. L. Brummer, K. A. Hardie

Biframes and bispaces,

Quaest. Math. 6, 13–25 (1983).
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 4 / 10

SLIDE 30

Extremal epis in BiFrm

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 5 / 10

SLIDE 31

Extremal epis in BiFrm

Theorem

Extremal epis in BiFrm are those biframe maps f : L → M such that:

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 5 / 10

SLIDE 32

Extremal epis in BiFrm

Theorem

Extremal epis in BiFrm are those biframe maps f : L → M such that:

f1 and f2 are extremal epis in Frm and
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 5 / 10

SLIDE 33

Extremal epis in BiFrm

Theorem

Extremal epis in BiFrm are those biframe maps f : L → M such that:

f1 and f2 are extremal epis in Frm and

L1

ιL1

f1
eL1
L2

f2

eL2
L0

M1

eM1

M2

eM2

M0
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 5 / 10

SLIDE 34

Extremal epis in BiFrm

Theorem

Extremal epis in BiFrm are those biframe maps f : L → M such that:

f1 and f2 are extremal epis in Frm and

L1

ιL1

ιL1
f1
eL1
L1 ⊕ L2

L2

ιL2

f2
eL2
L0

M1

eM1

M2

eM2

M0
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 5 / 10

SLIDE 35

Extremal epis in BiFrm

Theorem

Extremal epis in BiFrm are those biframe maps f : L → M such that:

f1 and f2 are extremal epis in Frm and

L1

ιL1

ιL1
f1
eL1
L1 ⊕ L2

qL

L2

ιL2

f2
eL2
L0

M1

eM1

M2

eM2

M0
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 5 / 10

SLIDE 36

Extremal epis in BiFrm

Theorem

Extremal epis in BiFrm are those biframe maps f : L → M such that:

f1 and f2 are extremal epis in Frm and

L1

ιL1

ιL1
f1
eL1
L1 ⊕ L2

qL

f1⊕f2
L2

ιL2

f2
eL2
L0

M1

eM1

M1 ⊕ M2

M2

eM2

M0
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 5 / 10

SLIDE 37

Extremal epis in BiFrm

Theorem

Extremal epis in BiFrm are those biframe maps f : L → M such that:

f1 and f2 are extremal epis in Frm and

L1

ιL1

ιL1
f1
eL1
L1 ⊕ L2

qL

f1⊕f2
L2

ιL2

f2
eL2
L0

M1

eM1

M1 ⊕ M2

M2

eM2

M0
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 5 / 10

SLIDE 38

Extremal epis in BiFrm

Theorem

Extremal epis in BiFrm are those biframe maps f : L → M such that:

f1 and f2 are extremal epis in Frm and

L1

ιL1

ιL1
f1
eL1
L1 ⊕ L2

qL

f1⊕f2
L2

ιL2

f2
eL2
L0

f

M1

eM1

M1 ⊕ M2

qL

M2

eM2

M0
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 5 / 10

SLIDE 39

Extremal epis in BiFrm

Theorem

Extremal epis in BiFrm are those biframe maps f : L → M such that:

f1 and f2 are extremal epis in Frm and
f0 is the pushout of f1 ⊕ f2 along qL,

L1

ιL1

ιL1
f1
eL1
L1 ⊕ L2

qL

f1⊕f2
L2

ιL2

f2
eL2
L0

f

M1

eM1

M1 ⊕ M2

qL

M2

eM2

M0
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 5 / 10

SLIDE 40

Extremal epis in BiFrm

Theorem

Extremal epis in BiFrm are those biframe maps f : L → M such that:

f1 and f2 are extremal epis in Frm and
f0 is the pushout of f1 ⊕ f2 along qL, that is f0 = f

L1

ιL1

ιL1
f1
eL1
L1 ⊕ L2

qL

f1⊕f2
L2

ιL2

f2
eL2
L0

f

M1

eM1

M1 ⊕ M2

qL

M2

eM2

M0
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 5 / 10

SLIDE 41

Extremal epis in BiFrm

The assignment L → S(L) is functorial S : Frm → coFrm

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 6 / 10

SLIDE 42

Extremal epis in BiFrm

The assignment L → S(L) is functorial S : Frm → coFrm Thus, for a biframe (L0, L1, L2), one has

eL1

eL2
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 6 / 10

SLIDE 43

Extremal epis in BiFrm

The assignment L → S(L) is functorial S : Frm → coFrm Thus, for a biframe (L0, L1, L2), one has L1

eL1

eL1

L0

L2

eL2

eL2
S(eL1 )
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 6 / 10

SLIDE 44

Extremal epis in BiFrm

The assignment L → S(L) is functorial S : Frm → coFrm Thus, for a biframe (L0, L1, L2), one has L1

eL1

eL1
L0
L2
eL2
eL2
S(L1)

S(eL1 ) S(eL1 ) S(L0)

S(L2)

S(eL2 )

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 6 / 10

SLIDE 45

Extremal epis in BiFrm

The assignment L → S(L) is functorial S : Frm → coFrm Thus, for a biframe (L0, L1, L2), one has L1

eL1

eL1
L0
L2
eL2
eL2
S(L1)

S(eL1 ) S(eL1 ) S(L0)

S(L2)

S(eL2 )

Proposition

Let f : L → S be an extremal epimorphism in BiFrm. Then [f0] = S(eL1)[f1] ∧ S(eL2)[f2].

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 6 / 10

SLIDE 46

Extremal epis in BiFrm

The assignment L → S(L) is functorial S : Frm → coFrm Thus, for a biframe (L0, L1, L2), one has L1

eL1

eL1
L0
L2
eL2
eL2
S(L1)

S(eL1 ) S(eL1 ) S(L0)

S(L2)

S(eL2 )

Proposition

Let f : L → S be an extremal epimorphism in BiFrm. Then [f0] = S(eL1)[f1] ∧ S(eL2)[f2].

Proposition

Let L be a biframe. For any [f1] ∈ S(L1) and [f2] ∈ S(L2) there exists a an extremal epimorphism f : L → M such that [f0] = S(eL1)[f1] ∧ S(eL2)[f2].

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 6 / 10

SLIDE 47

Isbell’s Density Theorem for biframes

Let f0 : L0 → M0 ∈ Frm and extremal epi. One has a biframe homomorphism f : L → (M,f0(L1), f2(L2)).

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 7 / 10

SLIDE 48

Isbell’s Density Theorem for biframes

Let f0 : L0 → M0 ∈ Frm and extremal epi. One has a biframe homomorphism f : L → (M,f0(L1), f2(L2)). Let f0 be the pushout of f1 ⊕ f2 along qL.

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 7 / 10

SLIDE 49

Isbell’s Density Theorem for biframes

Let f0 : L0 → M0 ∈ Frm and extremal epi. One has a biframe homomorphism f : L → (M,f0(L1), f2(L2)). Let f0 be the pushout of f1 ⊕ f2 along qL.

Proposition

The assignment [f0] → [f0] is a closure operator on S(L0).

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 7 / 10

SLIDE 50

Isbell’s Density Theorem for biframes

Let f0 : L0 → M0 ∈ Frm and extremal epi. One has a biframe homomorphism f : L → (M,f0(L1), f2(L2)). Let f0 be the pushout of f1 ⊕ f2 along qL.

Proposition

The assignment [f0] → [f0] is a closure operator on S(L0). There is a one-to-one correscondence between

Closed elements of S(L0) w.r.t. this closure operator
Equivalence classes of ext epis of L
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 7 / 10

SLIDE 51

Isbell’s Density Theorem for biframes

Let f0 : L0 → M0 ∈ Frm and extremal epi. One has a biframe homomorphism f : L → (M,f0(L1), f2(L2)). Let f0 be the pushout of f1 ⊕ f2 along qL.

Proposition

The assignment [f0] → [f0] is a closure operator on S(L0). There is a one-to-one correscondence between

Closed elements of S(L0) w.r.t. this closure operator
Equivalence classes of ext epis of L

Theorem

Any biframe L contains is a least dense “sublocale”,

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 7 / 10

SLIDE 52

Isbell’s Density Theorem for biframes

Let f0 : L0 → M0 ∈ Frm and extremal epi. One has a biframe homomorphism f : L → (M,f0(L1), f2(L2)). Let f0 be the pushout of f1 ⊕ f2 along qL.

Proposition

The assignment [f0] → [f0] is a closure operator on S(L0). There is a one-to-one correscondence between

Closed elements of S(L0) w.r.t. this closure operator
Equivalence classes of ext epis of L

Theorem

Any biframe L contains is a least dense “sublocale”, precisely the one given by βL0.

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 7 / 10

SLIDE 53

An categorical approach to BiFrm

Motivation: For any f : L → M in BiFrm L1

f1

eL1
L2

eL2

f2
L0

f0

M1

eM1

M0

M2

eM2

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 8 / 10

SLIDE 54

An categorical approach to BiFrm

Motivation: For any f : L → M in BiFrm L1

f1

eL1
f1
L2

f2

eL2
f2
L0

f0

f1(L1)

m1

f2(L2)

m2

M1

eM1

M0

M2

eM2

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 8 / 10

SLIDE 55

An categorical approach to BiFrm

Motivation: For any f : L → M in BiFrm L1

f1

eL1
ιL1
f1
L1 ⊕ L2

qL

L2

ιL2

f2
eL2
f2
L0

f0

f1(L1)

m1

f2(L2)

m2

M1

eM1

M0

M2

eM2

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 8 / 10

SLIDE 56

An categorical approach to BiFrm

Motivation: For any f : L → M in BiFrm L1

f1

eL1
ιL1
f1
L1 ⊕ L2

qL

f1⊕f2
L2

ιL2

f2
eL2
f2
L0

f0

f1(L1)

m1

f(L1) ⊕ f(L2)

f2(L2)

m2

M1

eM1

M0

M2

eM2

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 8 / 10

SLIDE 57

An categorical approach to BiFrm

Motivation: For any f : L → M in BiFrm L1

f1

eL1
ιL1
f1
L1 ⊕ L2

qL

f1⊕f2
L2

ιL2

f2
eL2
f2
L0
f0
f0
f1(L1)

m1

f(L1) ⊕ f(L2)

qL

f2(L2)

m2

f(L0)

M1

eM1

M0

M2

eM2

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 8 / 10

SLIDE 58

An categorical approach to BiFrm

Motivation: For any f : L → M in BiFrm L1

f1

eL1
ιL1
f1
L1 ⊕ L2

qL

f1⊕f2
L2

ιL2

f2
eL2
f2
L0
f0
f0
f1(L1)

ιf1(L1)

m1
f(L1) ⊕ f(L2)

qL

f2(L2)

ιf2(L2)

m2
f(L0)

M1

eM1

M0

M2

eM2

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 8 / 10

SLIDE 59

An categorical approach to BiFrm

Motivation: For any f : L → M in BiFrm L1

f1

eL1
ιL1
f1
L1 ⊕ L2

qL

f1⊕f2
L2

ιL2

f2
eL2
f2
L0
f0
f0
f1(L1)

ιf1(L1)

m1
f(L1) ⊕ f(L2)

qL

g
f2(L2)

ιf2(L2)

m2
f(L0)

M1

eM1

M0

M2

eM2

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 8 / 10

SLIDE 60

An categorical approach to BiFrm

Motivation: For any f : L → M in BiFrm L1

f1

eL1
ιL1
f1
L1 ⊕ L2

qL

f1⊕f2
L2

ιL2

f2
eL2
f2
L0
f0
f0
f1(L1)

ιf1(L1)

m1
f(L1) ⊕ f(L2)

qL

g
f2(L2)

ιf2(L2)

m2
f(L0)

e0

M1

eM1

M0

M2

eM2

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 8 / 10

SLIDE 61

An categorical approach to BiFrm

Objects L ≡ a pair of jointly epic monomorphisms in C

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 9 / 10

SLIDE 62

An categorical approach to BiFrm

Objects L ≡ a pair of jointly epic monomorphisms in C Morphisms

f≡

f1

f0
f2
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 9 / 10

SLIDE 63

An categorical approach to BiFrm

Objects L ≡ a pair of jointly epic monomorphisms in C Morphisms

f≡

f1

f0
f2
Proposition

f is an extremal epi iff

f1 and f2 are extremal epis
f0 is the pushout of f1 ⊕ f2 along qL.
I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 9 / 10

SLIDE 64

An categorical approach to BiFrm

Objects L ≡ a pair of jointly epic monomorphisms in C Morphisms

f≡

f1

f0
f2
Proposition

f is an extremal epi iff

f1 and f2 are extremal epis
f0 is the pushout of f1 ⊕ f2 along qL.

Proposition

The assignment f0 → f0 determines a closure operator on the poset of quotients of C.

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 9 / 10

SLIDE 65

Dˇ ekuji za pozornost! Obrigado pela sua atenção! Mila esker arretarengatik! Thank you for your attention!

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 10 / 10

SLIDE 66

Dˇ ekuji za pozornost! (I hope this is proper Czech) Obrigado pela sua atenção! (I also hope this is proper Portuguese) Mila esker arretarengatik! (This is Basque) Thank you for your attention!

I. Mozo Carollo (UPV/EHU)

Parts of biframes and a categorical approach to BiFrm WorkALT 2018 10 / 10