Extending Stone-Priestley duality along full embeddings C´eliaBorlido based on joint work with A. L. Suarez Centre for Mathematics, University of Coimbra BLAST 2021 9 - 13 June Σ op Pervin > Frith Ω full full CPervin CFrithop not full not full ∼ = op =∼ Priestley ∼= DLat 1. The spatial-sober duality (and its restriction to duality for bounded distributive lattices) 2. The categories of Pervin spaces and of Frith frames 3. Extending Stone-Priestley duality along full embeddings 4. The bitopological point of view Σ op Top > Frm Ω Σ op Pervin > Frith Ω full full CPervin CFrithop =∼ =∼ 1. The spatial-sober duality (and its restriction to duality for bounded distributive lattices) 2. The categories of Pervin spaces and of Frith frames 3. Extending Stone-Priestley duality along full embeddings 4. The bitopological point of view Σ op Top > Frm Ω not full not full op Priestley ∼= DLat Σ > Ω full full CPervin CFrithop =∼ =∼ 1. The spatial-sober duality (and its restriction to duality for bounded distributive lattices) 2. The categories of Pervin spaces and of Frith frames 3. Extending Stone-Priestley duality along full embeddings 4. The bitopological point of view Σ op Pervin > Frith Ω Σ op Top > Frm Ω not full not full op Priestley ∼= DLat Σ > Ω 1. The spatial-sober duality (and its restriction to duality for bounded distributive lattices) 2. The categories of Pervin spaces and of Frith frames 3. Extending Stone-Priestley duality along full embeddings 4. The bitopological point of view Σ op Pervin > Frith Ω full full Σ op Top > Frm Ω CPervin CFrithop not full not full ∼ = op =∼ Priestley ∼= DLat Σ > Ω 1. The spatial-sober duality (and its restriction to duality for bounded distributive lattices) 2. The categories of Pervin spaces and of Frith frames 3. Extending Stone-Priestley duality along full embeddings 4. The bitopological point of view Σ op Pervin > Frith Ω full full Σ op Top > Frm Ω CPervin CFrithop not full not full ∼ = op =∼ Priestley ∼= DLat The spatial-sober duality Frames Topological spaces A frame is a complete lattice L satisfying _ _ a ^ b = (a ^ b ): i i and continuous functions. i2I Frame homomorphisms preserve finite meets and arbitrary joins. C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 2 op I Σ: Frm ! Top −1 Σ(L) := fc. p. filters of Lg Σ(L −!h M) := (Σ(M) −−!h Σ(L)) ab := fF j a 2 F g; a 2 L op We have an adjunction Ω : Top Frm : Σ which restricts and co-restricts to a duality between sober spaces and spatial frames. − Spatial frame: a frame of the form Ω(X ) for some topological space X . − Sober space: a space that is completely determined by its set of open subsets. The spatial-sober duality op I Ω: Top ! Frm −1 Ω(X ) := (fopen subsets of X g; ⊆) Ω(X −!f Y ) := (Ω(Y ) −−!f Ω(X )) C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 3 op We have an adjunction Ω : Top Frm : Σ which restricts and co-restricts to a duality between sober spaces and spatial frames. − Spatial frame: a frame of the form Ω(X ) for some topological space X . − Sober space: a space that is completely determined by its set of open subsets. The spatial-sober duality op I Ω: Top ! Frm −1 Ω(X ) := (fopen subsets of X g; ⊆) Ω(X −!f Y ) := (Ω(Y ) −−!f Ω(X )) op I Σ: Frm ! Top −1 Σ(L) := fc. p. filters of Lg Σ(L −!h M) := (Σ(M) −−!h Σ(L)) ab := fF j a 2 F g; a 2 L C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 3 The spatial-sober duality op I Ω: Top ! Frm −1 Ω(X ) := (fopen subsets of X g; ⊆) Ω(X −!f Y ) := (Ω(Y ) −−!f Ω(X )) op I Σ: Frm ! Top −1 Σ(L) := fc. p. filters of Lg Σ(L −!h M) := (Σ(M) −−!h Σ(L)) ab := fF j a 2 F g; a 2 L op We have an adjunction Ω : Top Frm : Σ which restricts and co-restricts to a duality between sober spaces and spatial frames. − Spatial frame: a frame of the form Ω(X ) for some topological space X . − Sober space: a space that is completely determined by its set of open subsets. C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 3 Coherent homomorphisms are frame homomorphisms that preserve compact elements. I If D is a bounded distributive lattice, (Idl(D); ⊆) is a coherent frame. If h : C ! D is a lattice homomorphism, Idl(h):(J 2 Idl(C)) 7! hh[J]iIdl is a coherent homomorphism. I If L is a coherent frame, K(L) is a bounded distributive lattice. If h : L ! M is a coherent homomorphism, the restriction and co-restriction K(h): K(L) ! K(M) of h is a lattice homomorphism. These assignments define an equivalence of categories DLat =∼ CohFrm Bounded distributive lattices seen as coherent frames A coherent frame is a frame L whose set of compact elements K(L) is closed under finite meets (thus, a sublattice) and join-dense in L. C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4 I If D is a bounded distributive lattice, (Idl(D); ⊆) is a coherent frame. If h : C ! D is a lattice homomorphism, Idl(h):(J 2 Idl(C)) 7! hh[J]iIdl is a coherent homomorphism. I If L is a coherent frame, K(L) is a bounded distributive lattice. If h : L ! M is a coherent homomorphism, the restriction and co-restriction K(h): K(L) ! K(M) of h is a lattice homomorphism. These assignments define an equivalence of categories DLat =∼ CohFrm Bounded distributive lattices seen as coherent frames A coherent frame is a frame L whose set of compact elements K(L) is closed under finite meets (thus, a sublattice) and join-dense in L. Coherent homomorphisms are frame homomorphisms that preserve compact elements. C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4 I If L is a coherent frame, K(L) is a bounded distributive lattice. If h : L ! M is a coherent homomorphism, the restriction and co-restriction K(h): K(L) ! K(M) of h is a lattice homomorphism. These assignments define an equivalence of categories DLat =∼ CohFrm Bounded distributive lattices seen as coherent frames A coherent frame is a frame L whose set of compact elements K(L) is closed under finite meets (thus, a sublattice) and join-dense in L. Coherent homomorphisms are frame homomorphisms that preserve compact elements. I If D is a bounded distributive lattice, (Idl(D); ⊆) is a coherent frame. If h : C ! D is a lattice homomorphism, Idl(h):(J 2 Idl(C)) 7! hh[J]iIdl is a coherent homomorphism. C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4 These assignments define an equivalence of categories DLat =∼ CohFrm Bounded distributive lattices seen as coherent frames A coherent frame is a frame L whose set of compact elements K(L) is closed under finite meets (thus, a sublattice) and join-dense in L. Coherent homomorphisms are frame homomorphisms that preserve compact elements. I If D is a bounded distributive lattice, (Idl(D); ⊆) is a coherent frame. If h : C ! D is a lattice homomorphism, Idl(h):(J 2 Idl(C)) 7! hh[J]iIdl is a coherent homomorphism. I If L is a coherent frame, K(L) is a bounded distributive lattice. If h : L ! M is a coherent homomorphism, the restriction and co-restriction K(h): K(L) ! K(M) of h is a lattice homomorphism. C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4 Bounded distributive lattices seen as coherent frames A coherent frame is a frame L whose set of compact elements K(L) is closed under finite meets (thus, a sublattice) and join-dense in L. Coherent homomorphisms are frame homomorphisms that preserve compact elements. I If D is a bounded distributive lattice, (Idl(D); ⊆) is a coherent frame. If h : C ! D is a lattice homomorphism, Idl(h):(J 2 Idl(C)) 7! hh[J]iIdl is a coherent homomorphism. I If L is a coherent frame, K(L) is a bounded distributive lattice. If h : L ! M is a coherent homomorphism, the restriction and co-restriction K(h): K(L) ! K(M) of h is a lattice homomorphism. These assignments define an equivalence of categories DLat =∼ CohFrm C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4 Spectral maps are continuous functions such that the preimage of compact open subsets is again compact (and open). op The adjunction Ω : Top Frm : Σ restricts and co-restricts to an equivalence Spec =∼ CohFrmop Spectral spaces A spectral space is a T0 compact sober space (X ; τ) whose set of compact open subsets K o(X ; τ) is closed under finite meets and is a basis for the topology. C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 5 op The adjunction Ω : Top Frm : Σ restricts and co-restricts to an equivalence Spec =∼ CohFrmop Spectral spaces A spectral space is a T0 compact sober space (X ; τ) whose set of compact open subsets K o(X ; τ) is closed under finite meets and is a basis for the topology. Spectral maps are continuous functions such that the preimage of compact open subsets is again compact (and open). C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 5 Spectral spaces A spectral space is a T0 compact sober space (X ; τ) whose set of compact open subsets K o(X ; τ) is closed under finite meets and is a basis for the topology.