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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

• f 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

• f 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

• f 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

• f 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

𝜍𝑁 ⊙ 1 1 ⊙ 𝜈𝑆 𝜍𝑁 𝑠𝑁 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

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

𝜍𝑁 ⊙ 1 1 ⊙ 𝜈𝑆 𝜍𝑁 𝑠𝑁 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

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

𝜍𝑁 ⊙ 1 1 ⊙ 𝜈𝑆 𝜍𝑁 𝑠𝑁 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

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

𝜍𝑁 ⊙ 1 1 ⊙ 𝜈𝑆 𝜍𝑁 𝑠𝑁 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

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: 𝐽𝐵

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 𝑁(𝐽𝑂) 𝑁𝑂 ,

𝑏 𝑠1 1𝑚

(𝑁𝑂)𝐽 𝑁(𝑂𝐽) 𝑁𝑂,

𝑏 𝑠 1𝑠

(𝐽𝑁)𝑂 𝐽(𝑁𝑂) 𝑁𝑂,

𝑏 𝑚1 𝑚

𝐽𝐽 𝐽 .

𝑚 𝑠

Ülo Reimaa Bicategories with lax units and Morita theory

(𝑁𝐽)𝑂 𝑁(𝐽𝑂) 𝑁𝑂 ,

𝑏 𝑠1 1𝑚

(𝑁𝑂)𝐽 𝑁(𝑂𝐽) 𝑁𝑂,

𝑏 𝑠 1𝑠

(𝐽𝑁)𝑂 𝐽(𝑁𝑂) 𝑁𝑂,

𝑏 𝑚1 𝑚

𝐽𝐽 𝐽 .

𝑚 𝑠

Ülo Reimaa Bicategories with lax units and Morita theory

Defjnition

A Morita context Γ∶ 𝐵 → 𝐶 consists of 1-cells 𝑄Γ ∶ 𝐵 → 𝐶 𝑅Γ ∶ 𝐶 → 𝐵 and 2-cells 𝜄Γ ∶ 𝑄𝑅 → 𝐽 𝜚Γ ∶ 𝑅𝑄 → 𝐽 . such that the following diagrams commute:

𝑅(𝑄𝑅) 𝑅𝐽 𝑅, (𝑅𝑄)𝑅 𝐽𝑅

𝑏 1𝜄 𝜚1 𝑠 𝑚

𝑄(𝑅𝑄) 𝑄𝐽 𝑄. (𝑄𝑅)𝑄 𝐽𝑄

𝑏 1𝜚 𝜄1 𝑠 𝑚

Ülo Reimaa Bicategories with lax units and Morita theory

Lax functors

natural comparison 2-cells Φ𝑁,𝑂 ∶ 𝐺(𝑁)𝐺(𝑂) → 𝐺(𝑁𝑂), comparison 2-cells Φ0

𝐵 ∶ 𝐽𝐺(𝐵) → 𝐺(𝐽𝐵).

𝐺(𝑁)𝐽𝐺(𝐶) 𝐺(𝑁)𝐺(𝐽𝐶) 𝐺(𝑁) 𝐺(𝑁𝐽𝐶),

1Φ0

𝐶

Φ𝑁,𝐽 𝐺(𝑠) 𝑠𝐺(𝑁)

Ülo Reimaa Bicategories with lax units and Morita theory

Lax functors

natural comparison 2-cells Φ𝑁,𝑂 ∶ 𝐺(𝑁)𝐺(𝑂) → 𝐺(𝑁𝑂), comparison 2-cells Φ0

𝐵 ∶ 𝐺(𝐽𝐵) → 𝐽𝐺(𝐵).

𝐺(𝑁)𝐽𝐺(𝐶) 𝐺(𝑁)𝐺(𝐽𝐶) 𝐺(𝑁) 𝐺(𝑁𝐽𝐶),

1Φ0

𝐶

Φ𝑁,𝐽 𝐺(𝑠) 𝑠𝐺(𝑁)

Ülo Reimaa Bicategories with lax units and Morita theory

Lax functors take Morita contexts to Morita contexts. 𝑄𝑅

𝜄

→ 𝐽

↦

𝐺(𝑄)𝐺(𝑅)

Φ

→ 𝐺(𝑄𝑅)

𝜄

→ 𝐺(𝐽)

Φ0

→ 𝐽 .

Ülo Reimaa Bicategories with lax units and Morita theory

From this point forth we will suppose that every hom-category ℬ(𝐵, 𝐶) carries an orthogonal factorization system (ℰ, ℳ), where ℰ = strongly epimorphic 2-cells and ℳ = monomorphic 2-cells. The composition functor of ℬ is required to map elements of ℰ into ℰ. 𝑔 ∈ ℰ ⇒ 𝑔1 ∈ ℰ ∧ 1𝑔 ∈ ℰ .

Ülo Reimaa Bicategories with lax units and Morita theory

From this point forth we will suppose that every hom-category ℬ(𝐵, 𝐶) carries an orthogonal factorization system (ℰ, ℳ), where ℰ = strongly epimorphic 2-cells and ℳ = monomorphic 2-cells. The composition functor of ℬ is required to map elements of ℰ into ℰ. 𝑔 ∈ ℰ ⇒ 𝑔1 ∈ ℰ ∧ 1𝑔 ∈ ℰ .

Ülo Reimaa Bicategories with lax units and Morita theory

From this point forth we will suppose that every hom-category ℬ(𝐵, 𝐶) carries an orthogonal factorization system (ℰ, ℳ), where ℰ = strongly epimorphic 2-cells and ℳ = monomorphic 2-cells. The composition functor of ℬ is required to map elements of ℰ into ℰ. 𝑔 ∈ ℰ ⇒ 𝑔1 ∈ ℰ ∧ 1𝑔 ∈ ℰ .

Ülo Reimaa Bicategories with lax units and Morita theory

Defjnition We defjne a 1-cell 𝑁∶ 𝐵 → 𝐶 to be right unitary when 𝑠∶ 𝑁𝐽 → 𝑁 lies in ℰ, right fjrm when 𝑠∶ 𝑁𝐽 → 𝑁 is invertible. Defjnition An object 𝐵 is called fjrm or unitary if the corresponding 1-cell 𝐽𝐵 is so.

Ülo Reimaa Bicategories with lax units and Morita theory

Defjnition Let Γ∶ 𝐵 → 𝐶 be a Morita context. When 𝑄Γ and 𝑅Γ are unitary 1-cells and the 2-cells 𝜄Γ and 𝜚Γ belong to ℰ, we will call Γ an ℰ-Morita context. Proposition The relation of ℰ-equivalence is a transitive and symmetric relation on the objects of a lax-unital bicategory. Proposition Let 𝐵 and 𝐶 be arbitrary objects of a lax-unital bicategory and suppose that there exists an ℰ-Morita context from 𝐵 to 𝐶. Then 𝐵 and 𝐶 are unitary.

Ülo Reimaa Bicategories with lax units and Morita theory

Defjnition Let Γ∶ 𝐵 → 𝐶 be a Morita context. When 𝑄Γ and 𝑅Γ are unitary 1-cells and the 2-cells 𝜄Γ and 𝜚Γ belong to ℰ, we will call Γ an ℰ-Morita context. Proposition The relation of ℰ-equivalence is a transitive and symmetric relation on the objects of a lax-unital bicategory. Proposition Let 𝐵 and 𝐶 be arbitrary objects of a lax-unital bicategory and suppose that there exists an ℰ-Morita context from 𝐵 to 𝐶. Then 𝐵 and 𝐶 are unitary.

Ülo Reimaa Bicategories with lax units and Morita theory

Defjnition Let Γ∶ 𝐵 → 𝐶 be a Morita context. When 𝑄Γ and 𝑅Γ are unitary 1-cells and the 2-cells 𝜄Γ and 𝜚Γ belong to ℰ, we will call Γ an ℰ-Morita context. Proposition The relation of ℰ-equivalence is a transitive and symmetric relation on the objects of a lax-unital bicategory. Proposition Let 𝐵 and 𝐶 be arbitrary objects of a lax-unital bicategory and suppose that there exists an ℰ-Morita context from 𝐵 to 𝐶. Then 𝐵 and 𝐶 are unitary.

Ülo Reimaa Bicategories with lax units and Morita theory

Theorem Suppose that Γ∶ 𝐵 → 𝐶 is a Morita context in a lax-unital bicategory ℬ, where either all left unitors

• r all right unitors are epimorphisms. Then, if

𝜄Γ ∶ 𝑄𝑅 → 𝐽 is in ℰ, it is a monomorphism.

Ülo Reimaa Bicategories with lax units and Morita theory

In practice some lax-unital bicategories have a certain absorption property. Absorption If 𝑁 is right unitary and 𝑂 is left unitary, then the 2-cell 𝑁𝐽𝑂 → 𝑁𝑂 is invertible. Corollary If 𝑁∶ 𝐵 → 𝐶 is right unitary and 𝐶 is unitary, then 𝐵𝐽∶ 𝐵 → 𝐶 is right fjrm. Corollary If 𝐵 is unitary, then 𝐽𝐽∶ 𝐵 → 𝐵 is a fjrm 1-cell.

Ülo Reimaa Bicategories with lax units and Morita theory

In practice some lax-unital bicategories have a certain absorption property. Absorption If 𝑁 is right unitary and 𝑂 is left unitary, then the 2-cell 𝑁𝐽𝑂 → 𝑁𝑂 is invertible. Corollary If 𝑁∶ 𝐵 → 𝐶 is right unitary and 𝐶 is unitary, then 𝐵𝐽∶ 𝐵 → 𝐶 is right fjrm. Corollary If 𝐵 is unitary, then 𝐽𝐽∶ 𝐵 → 𝐵 is a fjrm 1-cell.

Ülo Reimaa Bicategories with lax units and Morita theory

In practice some lax-unital bicategories have a certain absorption property. Absorption If 𝑁 is right unitary and 𝑂 is left unitary, then the 2-cell 𝑁𝐽𝑂 → 𝑁𝑂 is invertible. Corollary If 𝑁∶ 𝐵 → 𝐶 is right unitary and 𝐶 is unitary, then 𝐵𝐽∶ 𝐵 → 𝐶 is right fjrm. Corollary If 𝐵 is unitary, then 𝐽𝐽∶ 𝐵 → 𝐵 is a fjrm 1-cell.

Ülo Reimaa Bicategories with lax units and Morita theory

In practice some lax-unital bicategories have a certain absorption property. Absorption If 𝑁 is right unitary and 𝑂 is left unitary, then the 2-cell 𝑁𝐽𝑂 → 𝑁𝑂 is invertible. Corollary If 𝑁∶ 𝐵 → 𝐶 is right unitary and 𝐶 is unitary, then 𝐵𝐽∶ 𝐵 → 𝐶 is right fjrm. Corollary If 𝐵 is unitary, then 𝐽𝐽∶ 𝐵 → 𝐵 is a fjrm 1-cell.

Ülo Reimaa Bicategories with lax units and Morita theory

Using the (StrongEpis, Monos)-factorizations, we can defjne a lax functor 𝑉∶ ℬ𝑉 → 𝑉ℬ𝑉 . We can turn 1-cells right unitary using the identity

• n objects lax-functor

𝑆∶ ℬ𝑉 → 𝑉 𝑆ℬ𝑉 . 𝑁𝐽 𝑆(𝑁) 𝑁 𝑂𝐽 𝑆(𝑂) 𝑂 .

𝑓𝑁 𝑛𝑁 𝑔1 𝑔 𝑓𝑂 𝑆(𝑔) 𝑛𝑂 𝑠𝑁 𝑠𝑂

Ülo Reimaa Bicategories with lax units and Morita theory

Using the (StrongEpis, Monos)-factorizations, we can defjne a lax functor 𝑉∶ ℬ𝑉 → 𝑉ℬ𝑉 . We can turn 1-cells right unitary using the identity

• n objects lax-functor

𝑆∶ ℬ𝑉 → 𝑉 𝑆ℬ𝑉 . 𝑁𝐽 𝑆(𝑁) 𝑁 𝑂𝐽 𝑆(𝑂) 𝑂 .

𝑓𝑁 𝑛𝑁 𝑔1 𝑔 𝑓𝑂 𝑆(𝑔) 𝑛𝑂 𝑠𝑁 𝑠𝑂

Ülo Reimaa Bicategories with lax units and Morita theory

This lax functor is locally right adjoint to the inclusion 𝑉 𝑆ℬ𝑉 → ℬ𝑉 . If we do the construction of 𝑆 starting from ℬ, we get a locally well-copointed lax functor 𝑆′ ∶ ℬ → ℬ . If the transfjnite sequence of 1-cells ⋯ → 𝑆′2(𝑁) → 𝑆′1(𝑁) → 𝑁 always converges, then we get a lax functor 𝑆∶ ℬ → 𝑉 𝑆ℬ .

Ülo Reimaa Bicategories with lax units and Morita theory

Problem What if we do the same for the transfjnite sequence ⋯ → 𝑁𝐽𝐽 → 𝑁𝐽 → 𝑁 ? Does this converge in the main examples? ⋯ → 𝑁 ⊗ 𝑆 ⊗ 𝑆 → 𝑁 ⊗ 𝑆 → 𝑁 . We can assume that 𝑁 is unitary.

Ülo Reimaa Bicategories with lax units and Morita theory

Problem What if we do the same for the transfjnite sequence ⋯ → 𝑁𝐽𝐽 → 𝑁𝐽 → 𝑁 ? Does this converge in the main examples? ⋯ → 𝑁 ⊗ 𝑆 ⊗ 𝑆 → 𝑁 ⊗ 𝑆 → 𝑁 . We can assume that 𝑁 is unitary.

Ülo Reimaa Bicategories with lax units and Morita theory

We want the inclusions 𝑉ℬ𝑉 → ℬ𝑉 and 𝐺ℬ𝐺 → 𝑉ℬ𝐺 to locally have right adjoints, because that allows us to carry any closed structure on ℬ onto 𝑉ℬ𝑉 and 𝐺ℬ𝐺.

Ülo Reimaa Bicategories with lax units and Morita theory

Theorem Let ℬ be a right closed lax-unital bicategory in which the 2-cell factorizations in ℬ are given by the epimorphic and the monomorphic 2-cells. Then, if two fjrm objects 𝐵 and 𝐶 of ℬ are ℰ-equivalent, the categories 𝐺 𝑆ℬ(𝐷, 𝐵) and 𝐺 𝑆ℬ(𝐷, 𝐶) are also equivalent for any fjrm object 𝐷 of ℬ.

Ülo Reimaa Bicategories with lax units and Morita theory