Bicategories with lax units and Morita theory lo Reimaa University - PowerPoint PPT Presentation

Bicategories with lax units and Morita theory Ülo Reimaa University of Tartu 10.07.2018 Ülo Reimaa Bicategories with lax units and Morita theory

For rings with identity and similar structures, several basic properties of the notion of Morita equivalence are a consequence of rings and bimodules forming a bicategory. and a consequence of the notion being essentially the same as the equivalence of objects of that bicategory. 𝑆 𝑈 𝑇 . 𝑁 𝑂 𝑁 ⊗ 𝑂 Note the direction of compositional fmow. Ülo Reimaa Bicategories with lax units and Morita theory

For rings with identity and similar structures, several basic properties of the notion of Morita equivalence are a consequence of rings and bimodules forming a bicategory and a consequence of the notion being essentially the same as the equivalence of objects of that bicategory. 𝑆 𝑈 𝑇 . 𝑁 𝑂 𝑁 ⊗ 𝑂 Note the direction of compositional fmow. Ülo Reimaa Bicategories with lax units and Morita theory

For rings with identity and similar structures, several basic properties of the notion of Morita equivalence are a consequence of rings and bimodules forming a bicategory and a consequence of the notion being essentially the same as the equivalence of objects of that bicategory. 𝑆 𝑈 𝑇 . 𝑁 𝑂 𝑁 ⊗ 𝑂 Note the direction of compositional fmow. Ülo Reimaa Bicategories with lax units and Morita theory

For rings with identity and similar structures, several basic properties of the notion of Morita equivalence are a consequence of rings and bimodules forming a bicategory and a consequence of the notion being essentially the same as the equivalence of objects of that bicategory. 𝑆 𝑈 𝑇 . 𝑁 𝑂 𝑁 ⊗ 𝑂 Note the direction of compositional fmow. Ülo Reimaa Bicategories with lax units and Morita theory

We have the invertible associator maps 𝑏∶ (𝑁 ⊗ 𝑂) ⊗ 𝑀 → 𝑁 ⊗ (𝑂 ⊗ 𝑀) and the invertible unitors 𝑚∶ 𝑆 ⊗ 𝑁 → 𝑁 , 𝑠 ⊗ 𝑛 ↦ 𝑠𝑛 , 𝑠∶ 𝑁 ⊗ 𝑆 → 𝑁 , 𝑠 ⊗ 𝑛 ↦ 𝑠𝑛 . Ülo Reimaa Bicategories with lax units and Morita theory

From this point forth, ring will mean one that structurally does not have an identity element. (Unless we specify that we mean a ring with identity .) Ülo Reimaa Bicategories with lax units and Morita theory

From this point forth, ring will mean one that structurally does not have an identity element. (Unless we specify that we mean a ring with identity .) Ülo Reimaa Bicategories with lax units and Morita theory

Rings (without structural identity) and bimodules do not form a bicategory solely because of the fact that the unitors 𝑚∶ 𝑆 ⊗ 𝑁 → 𝑁 , 𝑠 ⊗ 𝑛 ↦ 𝑠𝑛 , 𝑠∶ 𝑁 ⊗ 𝑆 → 𝑁 , 𝑛 ⊗ 𝑠 ↦ 𝑛𝑠 need not be invertible. Problem: How much Morita theory can we still do in a 2 -categorical setting? Ülo Reimaa Bicategories with lax units and Morita theory

Rings (without structural identity) and bimodules do not form a bicategory solely because of the fact that the unitors 𝑚∶ 𝑆 ⊗ 𝑁 → 𝑁 , 𝑠 ⊗ 𝑛 ↦ 𝑠𝑛 , 𝑠∶ 𝑁 ⊗ 𝑆 → 𝑁 , 𝑛 ⊗ 𝑠 ↦ 𝑛𝑠 need not be invertible. Problem: How much Morita theory can we still do in a 2 -categorical setting? Ülo Reimaa Bicategories with lax units and Morita theory

Let 𝑆 be a ring and let 𝑁 be a right 𝑆 -module. In the monoidal category Ab we have the coequalizer 𝑁 ⊙ 𝑆 ⊙ 𝑆 𝑁 ⊙ 𝑆 𝑁 ⊗ 𝑆 𝑁 1 ⊙ 𝜈 𝑆 𝜍 𝑁 𝑠 𝑁 𝑛 ↦ 𝑛 ⊙ 1 Using the map 𝑛 ↦ 𝑛 ⊙ 1 we can show that 𝑁 ⊙ 𝑆 → 𝑁 also coequalizes the pair. Ülo Reimaa Bicategories with lax units and Morita theory 𝜍 𝑁 ⊙ 1

Let 𝑆 be a ring and let 𝑁 be a right 𝑆 -module. In the monoidal category Ab we have the coequalizer 𝑁 ⊙ 𝑆 ⊙ 𝑆 𝑁 ⊙ 𝑆 𝑁 ⊗ 𝑆 𝑁 1 ⊙ 𝜈 𝑆 𝜍 𝑁 𝑠 𝑁 𝑛 ↦ 𝑛 ⊙ 1 Using the map 𝑛 ↦ 𝑛 ⊙ 1 we can show that 𝑁 ⊙ 𝑆 → 𝑁 also coequalizes the pair. Ülo Reimaa Bicategories with lax units and Morita theory 𝜍 𝑁 ⊙ 1

Let 𝑆 be a ring and let 𝑁 be a right 𝑆 -module. In the monoidal category Ab we have the coequalizer 𝑁 ⊙ 𝑆 ⊙ 𝑆 𝑁 ⊙ 𝑆 𝑁 ⊗ 𝑆 𝑁 1 ⊙ 𝜈 𝑆 𝜍 𝑁 𝑠 𝑁 𝑛 ↦ 𝑛 ⊙ 1 Using the map 𝑛 ↦ 𝑛 ⊙ 1 we can show that 𝑁 ⊙ 𝑆 → 𝑁 also coequalizes the pair. Ülo Reimaa Bicategories with lax units and Morita theory 𝜍 𝑁 ⊙ 1

Let 𝑆 be a ring and let 𝑁 be a right 𝑆 -module. In the monoidal category Ab we have the coequalizer 𝑁 ⊙ 𝑆 ⊙ 𝑆 𝑁 ⊙ 𝑆 𝑁 ⊗ 𝑆 𝑁 1 ⊙ 𝜈 𝑆 𝜍 𝑁 𝑠 𝑁 𝑛 ↦ 𝑛 ⊙ 1 Using the map 𝑛 ↦ 𝑛 ⊙ 1 we can show that 𝑁 ⊙ 𝑆 → 𝑁 also coequalizes the pair. Ülo Reimaa Bicategories with lax units and Morita theory 𝜍 𝑁 ⊙ 1

The Morita theory of rings without identity and various other internal semigroups objects has been studied by several people and some aspects do fjt in a 2 -categorical setting. The general themes are the following: Modules for which the maps 𝑚∶ 𝑆 ⊗ 𝑁 → 𝑁 , 𝑠 ⊗ 𝑛 ↦ 𝑠𝑛 , 𝑠∶ 𝑁 ⊗ 𝑆 → 𝑁 , 𝑛 ⊗ 𝑠 ↦ 𝑛𝑠 are . Ülo Reimaa Bicategories with lax units and Morita theory

The Morita theory of rings without identity and various other internal semigroups objects has been studied by several people and some aspects do fjt in a 2 -categorical setting. The general themes are the following: Modules for which the maps 𝑚∶ 𝑆 ⊗ 𝑁 → 𝑁 , 𝑠 ⊗ 𝑛 ↦ 𝑠𝑛 , 𝑠∶ 𝑁 ⊗ 𝑆 → 𝑁 , 𝑛 ⊗ 𝑠 ↦ 𝑛𝑠 are isomorphisms. Ülo Reimaa Bicategories with lax units and Morita theory

The Morita theory of rings without identity and various other internal semigroups objects has been studied by several people and some aspects do fjt in a 2 -categorical setting. The general themes are the following: Modules for which the maps 𝑚∶ 𝑆 ⊗ 𝑁 → 𝑁 , 𝑠 ⊗ 𝑛 ↦ 𝑠𝑛 , 𝑠∶ 𝑁 ⊗ 𝑆 → 𝑁 , 𝑛 ⊗ 𝑠 ↦ 𝑛𝑠 are surjections. Ülo Reimaa Bicategories with lax units and Morita theory

Notation: Objects of a bicategory: 𝐵 , 𝐶 , … 1 -cells of a bicategory: 𝑁 , 𝑂 , … 2 -cells of a bicategory: 𝑔 , 𝑕 , … unit 1 -cells: 𝐽 𝐵 Ülo Reimaa Bicategories with lax units and Morita theory

Lax-unital bicategories Defjnition A lax-unital bicategory ℬ is given by the same data as a bicategory. The properties the data is required to satisfy are also the same, with the following exceptions: the unitors 𝑚∶ 𝐽𝑁 → 𝑁 , 𝑠∶ 𝑁𝐽 → 𝑁 need not be invertible, coherence follows from the diagrams below: Ülo Reimaa Bicategories with lax units and Morita theory

Lax-unital bicategories Defjnition A lax-unital bicategory ℬ is given by the same data as a bicategory. The properties the data is required to satisfy are also the same, with the following exceptions: the unitors 𝑚∶ 𝐽𝑁 → 𝑁 , 𝑠∶ 𝑁𝐽 → 𝑁 need not be invertible, coherence follows from the diagrams below: Ülo Reimaa Bicategories with lax units and Morita theory

Lax-unital bicategories Defjnition A lax-unital bicategory ℬ is given by the same data as a bicategory. The properties the data is required to satisfy are also the same, with the following exceptions: the unitors 𝑚∶ 𝐽𝑁 → 𝑁 , 𝑠∶ 𝑁𝐽 → 𝑁 need not be invertible, coherence follows from the diagrams below: Ülo Reimaa Bicategories with lax units and Morita theory

((𝑁𝑂)𝑀)𝐿 (𝑁𝑂)(𝑀𝐿) 𝑁(𝑂(𝑀𝐿)), (𝑁(𝑂𝑀))𝐿 𝑁((𝑂𝑀)𝐿) 𝑏 𝑏 𝑏1 𝑏 1𝑏 Ülo Reimaa Bicategories with lax units and Morita theory

(𝑁𝐽)𝑂 (𝐽𝑁)𝑂 Ülo Reimaa 𝑠 𝑚 𝐽 . 𝐽𝐽 𝑚 𝑚1 𝑏 𝑁𝑂, 𝐽(𝑁𝑂) 1𝑠 𝑁(𝐽𝑂) 𝑠 𝑏 𝑁𝑂, 𝑁(𝑂𝐽) (𝑁𝑂)𝐽 1𝑚 𝑠1 𝑏 , 𝑁𝑂 Bicategories with lax units and Morita theory

(𝑁𝐽)𝑂 𝐽(𝑁𝑂) Ülo Reimaa 𝑠 𝑚 𝐽 . 𝐽𝐽 𝑚 𝑚1 𝑏 𝑁𝑂, (𝐽𝑁)𝑂 𝑁(𝐽𝑂) 1𝑠 𝑠 𝑏 𝑁𝑂, 𝑁(𝑂𝐽) (𝑁𝑂)𝐽 1𝑚 𝑠1 𝑏 𝑁𝑂 , Bicategories with lax units and Morita theory

Defjnition 𝑠 Ülo Reimaa 𝑚 𝑠 𝜄1 1𝜚 𝑏 𝐽𝑄 (𝑄𝑅)𝑄 𝑄. 𝑄𝐽 𝑄(𝑅𝑄) 𝑚 𝜚1 A Morita context Γ∶ 𝐵 → 𝐶 consists of 1 -cells 1𝜄 𝑏 𝐽𝑅 (𝑅𝑄)𝑅 𝑅, 𝑅𝐽 𝑅(𝑄𝑅) such that the following diagrams commute: and 2 -cells Bicategories with lax units and Morita theory 𝑄 Γ ∶ 𝐵 → 𝐶 𝑅 Γ ∶ 𝐶 → 𝐵 𝜄 Γ ∶ 𝑄𝑅 → 𝐽 𝜚 Γ ∶ 𝑅𝑄 → 𝐽 .

Lax functors natural comparison 2 -cells comparison 2 -cells Φ 0 𝐺(𝑁)𝐽 𝐺(𝐶) 𝐺(𝑁)𝐺(𝐽 𝐶 ) 𝐺(𝑁) 𝐺(𝑁𝐽 𝐶 ), 1Φ 0 𝐶 Φ 𝑁,𝐽 𝐺(𝑠) 𝑠 𝐺(𝑁) Ülo Reimaa Bicategories with lax units and Morita theory Φ 𝑁,𝑂 ∶ 𝐺(𝑁)𝐺(𝑂) → 𝐺(𝑁𝑂) , 𝐵 ∶ 𝐽 𝐺(𝐵) → 𝐺(𝐽 𝐵 ) .