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

Parts of biframes and a categorical approach to BiFrm Imanol Mozo Carollo 1 imanol.mozo@ehu.eus WorkALT in honour of Aleš Pultr, on the occasion of his 80th birthday 1 Joint 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

Parts of pointfree spaces: Sublocales I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

Parts of pointfree spaces: Sublocales • Subobjects of Frm op = Loc I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

Parts of pointfree spaces: Sublocales • Subobjects of Frm op = Loc • They are like subspaces. . . I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 2 / 10

Parts of pointfree spaces: Sublocales • Subobjects of Frm op = 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

Parts of pointfree spaces: Sublocales • Subobjects of Frm op = 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

Parts of pointfree spaces: Sublocales • Subobjects of Frm op = 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

Parts of pointfree spaces: Sublocales • Subobjects of Frm op = 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

Parts of pointfree spaces: Sublocales • Subobjects of Frm op = 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

Parts of pointfree spaces: Sublocales • Subobjects of Frm op = 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

Parts of pointfree spaces: Sublocales • Subobjects of Frm op = 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

Parts of pointfree spaces: Sublocales • Subobjects of Frm op = 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

Parts of pointfree spaces: Sublocales • Subobjects of Frm op = 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

Classical bispaces I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

Classical bispaces Bispaces I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

Classical bispaces Bispaces A set of points X endowed with two topologies: ( X , τ 1 , τ 2 ) I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

Classical bispaces Bispaces A set of points X endowed with two topologies: ( X , τ 1 , τ 2 ) 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

Classical bispaces Bispaces A set of points X endowed with two topologies: ( X , τ 1 , τ 2 ) 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

Classical bispaces Bispaces A set of points X endowed with two topologies: ( X , τ 1 , τ 2 ) 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

Classical bispaces Bispaces A set of points X endowed with two topologies: ( X , τ 1 , τ 2 ) 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? τ 0 = τ 1 ∨ τ 2 I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

Classical bispaces Bispaces A set of points X endowed with two topologies: ( X , τ 1 , τ 2 ) 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? τ 0 = τ 1 ∨ τ 2 • τ 1 and τ 2 embed into τ 0 • τ 1 ∪ τ 2 forms a subbasis of τ 0 I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

Classical bispaces Bispaces A set of points X endowed with two topologies: ( X , τ 1 , τ 2 ) 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? τ 0 = τ 1 ∨ τ 2 • τ 1 and τ 2 embed into τ 0 • τ 1 ∪ τ 2 forms a subbasis of τ 0 Any bicontinuous function is continuous w.r.t. τ 0 . I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 3 / 10

Getting rid of points: biframes I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 4 / 10

Getting rid of points: biframes Biframes A biframe L is formed by three frames ( L 0 , L 1 , L 2 ) where I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 4 / 10

Getting rid of points: biframes Biframes A biframe L is formed by three frames ( L 0 , L 1 , L 2 ) where • L 1 and L 2 are subframes of the ambient frame L 0 I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 4 / 10

Getting rid of points: biframes Biframes A biframe L is formed by three frames ( L 0 , L 1 , L 2 ) where • L 1 and L 2 are subframes of the ambient frame L 0 • L 1 ∪ L 2 forms a subbasis of L 0 I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 4 / 10

Getting rid of points: biframes Biframes A biframe L is formed by three frames ( L 0 , L 1 , L 2 ) where • L 1 and L 2 are subframes of the ambient frame L 0 • L 1 ∪ L 2 forms a subbasis of L 0 I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 4 / 10

Getting rid of points: biframes Biframes A biframe L is formed by three frames ( L 0 , L 1 , L 2 ) where • L 1 and L 2 are subframes of the ambient frame L 0 • L 1 ∪ L 2 forms a subbasis of L 0 Biframe homomorphisms f : L → M are given by frame homomorphisms f 0 : L 0 → M 0 that restricts to frame homomorphisms f i : L i → M i ( i = 1 , 2) I. Mozo Carollo (UPV/EHU) Parts of biframes and a categorical approach to BiFrm WorkALT 2018 4 / 10