Ordered Cubes Ed Morehouse HoTT/UF, Oxford July 8, 2018 Various - PowerPoint PPT Presentation

Ordered Cubes Ed Morehouse HoTT/UF, Oxford July 8, 2018

Various criteria for choosing a cubical theory, including: ▶ homotopy theory (strict test categories), ▶ computational behavior (canonical forms, 𝑦 -Reedy structure, distributive laws), ▶ model structure (judgemental vs typal equalities), ▶ etc. Context Like simplicial sets , cubical sets provide a combinatorial model of homotopy theory. However, there are several varieties of cubical sets to choose from. Maps include faces , degeneracies , diagonals , connections , etc.. Relations witness properties of geometric cubes . 2 / 24

Context Like simplicial sets , cubical sets provide a combinatorial model of homotopy theory. However, there are several varieties of cubical sets to choose from. Maps include faces , degeneracies , diagonals , connections , etc.. Relations witness properties of geometric cubes . Various criteria for choosing a cubical theory, including: ▶ homotopy theory (strict test categories), ▶ computational behavior (canonical forms, 𝑦 -Reedy structure, distributive laws), ▶ model structure (judgemental vs typal equalities), ▶ etc. 2 / 24

▶ has a strong equational theory, ▶ is a strict test category, ▶ is closely related to simplices. Overview Motivated by order-theoretic and monoidal structure, we present a simple cube category that: ▶ contains all the familiar maps, 3 / 24

▶ is a strict test category, ▶ is closely related to simplices. Overview Motivated by order-theoretic and monoidal structure, we present a simple cube category that: ▶ contains all the familiar maps, ▶ has a strong equational theory, 3 / 24

▶ is closely related to simplices. Overview Motivated by order-theoretic and monoidal structure, we present a simple cube category that: ▶ contains all the familiar maps, ▶ has a strong equational theory, ▶ is a strict test category, 3 / 24

Overview Motivated by order-theoretic and monoidal structure, we present a simple cube category that: ▶ contains all the familiar maps, ▶ has a strong equational theory, ▶ is a strict test category, ▶ is closely related to simplices. 3 / 24

Combinatorial Aspects 4 / 24

The simplex category , “ ∆ ”, can be presented as the (skeleton of the) full subcategory of Ord containing inhabited, finite, totally ordered sets: e.g. Its maps are generated by: faces (dimension-raising maps) injective monotone functions e.g. 𝑒 1 = [0, 2] = {0, 1} ⟼ {0, 2} ∶ ∆ (⟨1⟩ → ⟨2⟩) degeneracies (dimension-lowering maps) surjective monotone functions e.g. 𝑡 1 = [0, 1, 1] = {0, 1, 2} ⟼ {0, 1, 1} ∶ ∆ (⟨2⟩ → ⟨1⟩) ⟨𝑜⟩ ≔ fin (𝑜 + 1) ⟨2⟩ ≔ {0, 1, 2} Simplicies, Order-Theoretically An 𝑜 -simplex , “ ⟨𝑜⟩ ”, is the walking path of 𝑜 serially composable arrows. 5 / 24

Its maps are generated by: faces (dimension-raising maps) injective monotone functions e.g. 𝑒 1 = [0, 2] = {0, 1} ⟼ {0, 2} ∶ ∆ (⟨1⟩ → ⟨2⟩) degeneracies (dimension-lowering maps) surjective monotone functions e.g. 𝑡 1 = [0, 1, 1] = {0, 1, 2} ⟼ {0, 1, 1} ∶ ∆ (⟨2⟩ → ⟨1⟩) ⟨2⟩ ≔ {0, 1, 2} ⟨𝑜⟩ ≔ fin (𝑜 + 1) Simplicies, Order-Theoretically An 𝑜 -simplex , “ ⟨𝑜⟩ ”, is the walking path of 𝑜 serially composable arrows. The simplex category , “ ∆ ”, can be presented as the (skeleton of the) full subcategory of Ord containing inhabited, finite, totally ordered sets: e.g. 5 / 24

⟨2⟩ ≔ {0, 1, 2} ⟨𝑜⟩ ≔ fin (𝑜 + 1) Simplicies, Order-Theoretically An 𝑜 -simplex , “ ⟨𝑜⟩ ”, is the walking path of 𝑜 serially composable arrows. The simplex category , “ ∆ ”, can be presented as the (skeleton of the) full subcategory of Ord containing inhabited, finite, totally ordered sets: e.g. Its maps are generated by: faces (dimension-raising maps) injective monotone functions e.g. 𝑒 1 = [0, 2] = {0, 1} ⟼ {0, 2} ∶ ∆ (⟨1⟩ → ⟨2⟩) degeneracies (dimension-lowering maps) surjective monotone functions e.g. 𝑡 1 = [0, 1, 1] = {0, 1, 2} ⟼ {0, 1, 1} ∶ ∆ (⟨2⟩ → ⟨1⟩) 5 / 24

Example: composing 𝑒 1 ∶ ∆ (⟨1⟩ → ⟨2⟩) with 𝑡 1 ∶ ∆ (⟨2⟩ → ⟨1⟩) : 0 0 𝑡 𝑒 1 1 2 1 Simplicies, Monoidally The simplex category can also be presented via the walking monoid , which is the category 𝕅 with: ▶ one generating object, V ∶ 𝕅 ▶ two generating morphisms, 𝑡 ∶ 𝕅 (V ⊗ V → V) and 𝑒 ∶ 𝕅 ( I → V) ▶ relations that make (V, 𝑒, 𝑡) a monoid in (𝕅, ⊗, I ) . Then ∆ is the full subcategory of 𝕅 excluding the object V ⊗0 with ⟨𝑜⟩ ≔ V ⊗(𝑜+1) . 6 / 24

0 0 𝑡 𝑒 1 1 2 1 Simplicies, Monoidally The simplex category can also be presented via the walking monoid , which is the category 𝕅 with: ▶ one generating object, V ∶ 𝕅 ▶ two generating morphisms, 𝑡 ∶ 𝕅 (V ⊗ V → V) and 𝑒 ∶ 𝕅 ( I → V) ▶ relations that make (V, 𝑒, 𝑡) a monoid in (𝕅, ⊗, I ) . Then ∆ is the full subcategory of 𝕅 excluding the object V ⊗0 with ⟨𝑜⟩ ≔ V ⊗(𝑜+1) . Example: composing 𝑒 1 ∶ ∆ (⟨1⟩ → ⟨2⟩) with 𝑡 1 ∶ ∆ (⟨2⟩ → ⟨1⟩) : 6 / 24

Goal: a vertex-based cube category with all familiar maps and relations that is related to the simplex category by their order-theoretic presentations. Ordered (Monoidal) Cubes? The well-studied cube categories also have order-theoretic [Jar06] and monoidal [GM03] presentations. But in the monoidal presentation there is a “dimension mismatch”: the generating object is an interval rather than a point . 7 / 24

Ordered (Monoidal) Cubes? The well-studied cube categories also have order-theoretic [Jar06] and monoidal [GM03] presentations. But in the monoidal presentation there is a “dimension mismatch”: the generating object is an interval rather than a point . Goal: a vertex-based cube category with all familiar maps and relations that is related to the simplex category by their order-theoretic presentations. 7 / 24

▶ [𝑜] is the walking product of 𝑜 arrows. ▶ Each [𝑜] is a complete and distributive lattice. ▶ [𝑜] is isomorphic to the subset lattice of fin (𝑜) where 𝑤 𝑗 = 1 ⇔ 𝑗 ∈ 𝑤 : Therefore we define: Definition An ordered 𝑜 -cube , “ [𝑜] ”, is the preorderd set {0 ≤ 1} 101 {0} ∅ ≅ 111 011 110 010 001 {1} 100 000 {0, 2} ×𝑜 {0, 1} {1, 2} ⏟⏟⏟⏟⏟ {0, 1, 2} {2} Ordered Cubes The standard geometric 𝑜 -cube is the convex subspace of ℝ 𝑜 bounded by the 2 𝑜 vertex points 𝑤 = (𝑤 0 , ⋯ , 𝑤 𝑜−1 ) where 𝑤 𝑗 ∈ {0, 1} . “ 𝑤 0 ⋯𝑤 𝑜−1 ” 8 / 24

▶ [𝑜] is the walking product of 𝑜 arrows. ▶ Each [𝑜] is a complete and distributive lattice. ▶ [𝑜] is isomorphic to the subset lattice of fin (𝑜) where 𝑤 𝑗 = 1 ⇔ 𝑗 ∈ 𝑤 : 111 011 110 010 101 {0} 100 000 001 ∅ ×𝑜 {2} {0, 2} {1} {0, 1} {1, 2} ⏟⏟⏟⏟⏟ {0, 1, 2} ≅ Ordered Cubes The standard geometric 𝑜 -cube is the convex subspace of ℝ 𝑜 bounded by the 2 𝑜 vertex points 𝑤 = (𝑤 0 , ⋯ , 𝑤 𝑜−1 ) where 𝑤 𝑗 ∈ {0, 1} . “ 𝑤 0 ⋯𝑤 𝑜−1 ” Therefore we define: Definition An ordered 𝑜 -cube , “ [𝑜] ”, is the preorderd set {0 ≤ 1} 8 / 24

▶ Each [𝑜] is a complete and distributive lattice. ▶ [𝑜] is isomorphic to the subset lattice of fin (𝑜) where 𝑤 𝑗 = 1 ⇔ 𝑗 ∈ 𝑤 : 001 011 110 010 101 ∅ 100 000 {0} ≅ ×𝑜 {2} {0, 2} {1} {0, 1} {1, 2} ⏟⏟⏟⏟⏟ {0, 1, 2} 111 Ordered Cubes The standard geometric 𝑜 -cube is the convex subspace of ℝ 𝑜 bounded by the 2 𝑜 vertex points 𝑤 = (𝑤 0 , ⋯ , 𝑤 𝑜−1 ) where 𝑤 𝑗 ∈ {0, 1} . “ 𝑤 0 ⋯𝑤 𝑜−1 ” Therefore we define: Definition An ordered 𝑜 -cube , “ [𝑜] ”, is the preorderd set {0 ≤ 1} ▶ [𝑜] is the walking product of 𝑜 arrows. 8 / 24

▶ [𝑜] is isomorphic to the subset lattice of fin (𝑜) where 𝑤 𝑗 = 1 ⇔ 𝑗 ∈ 𝑤 : 001 ∅ 110 010 101 ≅ 100 000 {0} 111 ×𝑜 {2} {0, 2} {1} {0, 1} {1, 2} ⏟⏟⏟⏟⏟ {0, 1, 2} 011 Ordered Cubes The standard geometric 𝑜 -cube is the convex subspace of ℝ 𝑜 bounded by the 2 𝑜 vertex points 𝑤 = (𝑤 0 , ⋯ , 𝑤 𝑜−1 ) where 𝑤 𝑗 ∈ {0, 1} . “ 𝑤 0 ⋯𝑤 𝑜−1 ” Therefore we define: Definition An ordered 𝑜 -cube , “ [𝑜] ”, is the preorderd set {0 ≤ 1} ▶ [𝑜] is the walking product of 𝑜 arrows. ▶ Each [𝑜] is a complete and distributive lattice. 8 / 24

001 {0} 010 101 111 100 000 ≅ ∅ ×𝑜 011 {2} {0, 2} {1} {0, 1} {1, 2} ⏟⏟⏟⏟⏟ {0, 1, 2} 110 Ordered Cubes The standard geometric 𝑜 -cube is the convex subspace of ℝ 𝑜 bounded by the 2 𝑜 vertex points 𝑤 = (𝑤 0 , ⋯ , 𝑤 𝑜−1 ) where 𝑤 𝑗 ∈ {0, 1} . “ 𝑤 0 ⋯𝑤 𝑜−1 ” Therefore we define: Definition An ordered 𝑜 -cube , “ [𝑜] ”, is the preorderd set {0 ≤ 1} ▶ [𝑜] is the walking product of 𝑜 arrows. ▶ Each [𝑜] is a complete and distributive lattice. ▶ [𝑜] is isomorphic to the subset lattice of fin (𝑜) where 𝑤 𝑗 = 1 ⇔ 𝑗 ∈ 𝑤 : 8 / 24