Weak functors for degenerate Trimble 3-categories Eugenia Cheng - - PowerPoint PPT Presentation

Weak functors for degenerate Trimble 3-categories

Eugenia Cheng

School of the Art Institute of Chicago

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2.

Trimble n-categories

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Trimble n-categories doubly degenerate 2-categories doubly degenerate 3-categories

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Trimble n-categories doubly degenerate 2-categories doubly degenerate 3-categories Eckmann–Hilton weak Eckmann–Hilton

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Trimble n-categories doubly degenerate 2-categories doubly degenerate 3-categories Eckmann–Hilton weak Eckmann–Hilton distributive laws algebras and maps strict algebras and weak maps

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Plan

• 1. Overview

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Plan

• 1. Overview
• 2. Algebras via distributive laws

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Plan

• 1. Overview
• 2. Algebras via distributive laws
• 3. Eckmann–Hilton

4.

Plan

• 1. Overview
• 2. Algebras via distributive laws
• 3. Eckmann–Hilton
• 4. Weak Eckmann–Hilton

4.

Plan

• 1. Overview
• 2. Algebras via distributive laws
• 3. Eckmann–Hilton
• 4. Weak Eckmann–Hilton
• 5. Weak maps of algebras.

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• 1. Overview

Test principle for weak n-categories Doubly degenerate 3-categories should be somehow equivalent to braided monoidal categories.

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• 1. Overview

Test principle for weak n-categories Doubly degenerate 3-categories should be somehow equivalent to braided monoidal categories. Some known results for classical tricategories

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• 1. Overview

Test principle for weak n-categories Doubly degenerate 3-categories should be somehow equivalent to braided monoidal categories. Some known results for classical tricategories

• DD tricategories ≡ braided monoidal categories (JS, GPS)

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• 1. Overview

Test principle for weak n-categories Doubly degenerate 3-categories should be somehow equivalent to braided monoidal categories. Some known results for classical tricategories

• DD tricategories ≡ braided monoidal categories (JS, GPS)
• Strict interchange/strict units ≡ symmetric

5.

• 1. Overview

Test principle for weak n-categories Doubly degenerate 3-categories should be somehow equivalent to braided monoidal categories. Some known results for classical tricategories

• DD tricategories ≡ braided monoidal categories (JS, GPS)
• Strict interchange/strict units ≡ symmetric
• Weak interchange/strict units ≡ braided

5.

• 1. Overview

Test principle for weak n-categories Doubly degenerate 3-categories should be somehow equivalent to braided monoidal categories. Some known results for classical tricategories

• DD tricategories ≡ braided monoidal categories (JS, GPS)
• Strict interchange/strict units ≡ symmetric
• Weak interchange/strict units ≡ braided
• Strict interchange/weak units ≡ braided (Joyal–Kock)

5.

• 1. Overview

Test principle for weak n-categories Doubly degenerate 3-categories should be somehow equivalent to braided monoidal categories. Some known results for classical tricategories

• DD tricategories ≡ braided monoidal categories (JS, GPS)
• Strict interchange/strict units ≡ symmetric
• Weak interchange/strict units ≡ braided
• Strict interchange/weak units ≡ braided (Joyal–Kock)

Trimble’s definition is most like this

5.

• 1. Overview

Aim: make a 2-category ddTr3Cat with

• 0-cells: doubly degenerate Trimble 3-categories
• 1-cells: weak maps
• 2-cells: icon-like transformations (Lack)

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• 1. Overview

Aim: make a 2-category ddTr3Cat with

• 0-cells: doubly degenerate Trimble 3-categories
• 1-cells: weak maps
• 2-cells: icon-like transformations (Lack)

and prove a biequivalence ddTr3Cat ≃ BrMonCat

6.

• 1. Overview

Aim: make a 2-category ddTr3Cat with

• 0-cells: doubly degenerate Trimble 3-categories
• 1-cells: weak maps
• 2-cells: icon-like transformations (Lack)

and prove a biequivalence ddTr3Cat ≃ BrMonCat Question: what are weak maps?

6.

• 2. Algebras via distributive laws

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• 2. Algebras via distributive laws

Warm-up: doubly degenerate 2-categories

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• 2. Algebras via distributive laws

Warm-up: doubly degenerate 2-categories 0-cells 1-cells 2-cells

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• 2. Algebras via distributive laws

Warm-up: doubly degenerate 2-categories 0-cells 1-cells 2-cells    trivial

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• 2. Algebras via distributive laws

Warm-up: doubly degenerate 2-categories 0-cells 1-cells 2-cells    trivial a set

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• 2. Algebras via distributive laws

Warm-up: doubly degenerate 2-categories 0-cells 1-cells 2-cells    trivial a set horizontal composition vertical composition

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• 2. Algebras via distributive laws

Warm-up: doubly degenerate 2-categories 0-cells 1-cells 2-cells    trivial a set horizontal composition vertical composition two binary operations

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• 2. Algebras via distributive laws

Warm-up: doubly degenerate 2-categories 0-cells 1-cells 2-cells    trivial a set horizontal composition vertical composition two binary operations

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• 2. Algebras via distributive laws

Warm-up: doubly degenerate 2-categories 0-cells 1-cells 2-cells    trivial a set horizontal composition vertical composition two binary operations = =

7.

• 2. Algebras via distributive laws

Warm-up: doubly degenerate 2-categories 0-cells 1-cells 2-cells    trivial a set horizontal composition vertical composition two binary operations = =

7.

• 2. Algebras via distributive laws

Warm-up: doubly degenerate 2-categories 0-cells 1-cells 2-cells    trivial a set horizontal composition vertical composition two binary operations = = =

7.

• 2. Algebras via distributive laws

Warm-up: doubly degenerate 2-categories 0-cells 1-cells 2-cells    trivial a set horizontal composition vertical composition two binary operations = = = = = =

7.

• 2. Algebras via distributive laws

Aim: express this in terms of the monads and algebras

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• 2. Algebras via distributive laws

Let S and T be monads on a category C. A distributive law of S over T is a natural transformation λ : ST TS + axioms. (Beck)

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• 2. Algebras via distributive laws

Let S and T be monads on a category C. A distributive law of S over T is a natural transformation λ : ST TS + axioms. (Beck) Key consequences:

• 1. TS gets the structure of a monad.

9.

• 2. Algebras via distributive laws

Let S and T be monads on a category C. A distributive law of S over T is a natural transformation λ : ST TS + axioms. (Beck) Key consequences:

• 1. TS gets the structure of a monad.
• 2. T lifts to a monad T ′ on S-Alg.

9.

• 2. Algebras via distributive laws

Let S and T be monads on a category C. A distributive law of S over T is a natural transformation λ : ST TS + axioms. (Beck) Key consequences:

• 1. TS gets the structure of a monad.
• 2. T lifts to a monad T ′ on S-Alg.
• 3. TS-algebras coincide with T ′-algebras.

9.

• 2. Algebras via distributive laws

Let S and T be monads on a category C. A distributive law of S over T is a natural transformation λ : ST TS + axioms. (Beck) Key consequences:

• 1. TS gets the structure of a monad.
• 2. T lifts to a monad T ′ on S-Alg.
• 3. TS-algebras coincide with T ′-algebras.

Example 1: Rings

9.

• 2. Algebras via distributive laws

Let S and T be monads on a category C. A distributive law of S over T is a natural transformation λ : ST TS + axioms. (Beck) Key consequences:

• 1. TS gets the structure of a monad.
• 2. T lifts to a monad T ′ on S-Alg.
• 3. TS-algebras coincide with T ′-algebras.

Example 1: Rings C = Set, S = free monoid monad, T = free group monad

9.

• 2. Algebras via distributive laws

Let S and T be monads on a category C. A distributive law of S over T is a natural transformation λ : ST TS + axioms. (Beck) Key consequences:

• 1. TS gets the structure of a monad.
• 2. T lifts to a monad T ′ on S-Alg.
• 3. TS-algebras coincide with T ′-algebras.

Example 1: Rings C = Set, S = free monoid monad, T = free group monad STA TSA (a + b)(c + d) ac + bc + ad + bd

9.

• 2. Algebras via distributive laws

Let S and T be monads on a category C. A distributive law of S over T is a natural transformation λ : ST TS + axioms. (Beck) Key consequences:

• 1. TS gets the structure of a monad.
• 2. T lifts to a monad T ′ on S-Alg.
• 3. TS-algebras coincide with T ′-algebras.

Example 1: Rings C = Set, S = free monoid monad, T = free group monad STA TSA (a + b)(c + d) ac + bc + ad + bd A TS-algebra is a ring.

9.

• 2. Algebras via distributive laws

Let S and T be monads on a category C. A distributive law of S over T is a natural transformation λ : ST TS + axioms. (Beck) Key consequences:

• 1. TS gets the structure of a monad.
• 2. T lifts to a monad T ′ on S-Alg.
• 3. TS-algebras coincide with T ′-algebras.

Example 1: Rings C = Set, S = free monoid monad, T = free group monad STA TSA (a + b)(c + d) ac + bc + ad + bd A TS-algebra is a ring. λ ensures distributivity of × over +.

9.

• 1. Overview

Example 2: 2-categories

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• 1. Overview

Example 2: 2-categories C = 2-GSet, A2 A1 A0

s t s t

10.

• 1. Overview

Example 2: 2-categories C = 2-GSet, A2 A1 A0

s t s t

S = free Cat-Gph, vertical comp

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• 1. Overview

Example 2: 2-categories C = 2-GSet, A2 A1 A0

s t s t

S = free Cat-Gph, vertical comp T = free Gph-Cat horizontal comp

10.

• 1. Overview

Example 2: 2-categories C = 2-GSet, A2 A1 A0

s t s t

S = free Cat-Gph, vertical comp T = free Gph-Cat horizontal comp STA TSA

λA

10.

• 1. Overview

Example 2: 2-categories C = 2-GSet, A2 A1 A0

s t s t

S = free Cat-Gph, vertical comp T = free Gph-Cat horizontal comp STA TSA

λA

A TS-algebra is a 2-category.

10.

• 1. Overview

Example 2: 2-categories C = 2-GSet, A2 A1 A0

s t s t

S = free Cat-Gph, vertical comp T = free Gph-Cat horizontal comp STA TSA

λA

A TS-algebra is a 2-category. λ ensures interchange.

10.

• 2. Algebras via distributive laws

Law vs. structure

11.

• 2. Algebras via distributive laws

Law vs. structure More “natural” way to think of a ring: a set with a group structure and a monoid structure compatible via distributivity

11.

• 2. Algebras via distributive laws

Law vs. structure More “natural” way to think of a ring: a set with a group structure and a monoid structure compatible via distributivity

• an S-algebra SA

A

s

and a T-algebra TA A

t

11.

• 2. Algebras via distributive laws

Law vs. structure More “natural” way to think of a ring: a set with a group structure and a monoid structure compatible via distributivity

• an S-algebra SA

A

s

and a T-algebra TA A

t

• such that

STA TSA TA

λA Ts

SA A

s St t

11.

• 2. Algebras via distributive laws

Law vs. structure More “natural” way to think of a ring: a set with a group structure and a monoid structure compatible via distributivity

• an S-algebra SA

A

s

and a T-algebra TA A

t

• such that

STA TSA TA

λA Ts

SA A

s St t

11.

• 2. Algebras via distributive laws

Law vs. structure More “natural” way to think of a ring: a set with a group structure and a monoid structure compatible via distributivity

• an S-algebra SA

A

s

and a T-algebra TA A

t

• such that

STA TSA TA

λA Ts

SA A

s St t

11.

• 2. Algebras via distributive laws

Law vs. structure More “natural” way to think of a ring: a set with a group structure and a monoid structure compatible via distributivity

• an S-algebra SA

A

s

and a T-algebra TA A

t

• such that

STA TSA TA

λA Ts

SA A

s St t

For 2-categories this says a 2-globular set with vertical and horizontal composition compatible via interchange

11.

• 3. Eckmann-Hilton

12.

• 3. Eckmann-Hilton

Warm-up result: Given

• monads S, T on a category C, and
• a distributive law λ : ST

TS,

12.

• 3. Eckmann-Hilton

Warm-up result: Given

• monads S, T on a category C, and
• a distributive law λ : ST

TS, then for algebras TSA A

12.

• 3. Eckmann-Hilton

Warm-up result: Given

• monads S, T on a category C, and
• a distributive law λ : ST

TS, then for algebras TSA A ≡ SA A and TA A + compatibility via λ

12.

• 3. Eckmann-Hilton

Warm-up result: Given

• monads S, T on a category C, and
• a distributive law λ : ST

TS, then for algebras TSA A ≡ SA A and TA A + compatibility via λ Eckmann–Hilton structure SA A + axiom

12.

• 3. Eckmann-Hilton

Warm-up result: Given

• monads S, T on a category C, and
• a distributive law λ : ST

TS, then for algebras TSA A ≡ SA A and TA A + compatibility via λ Eckmann–Hilton structure SA A + axiom T-algebra part is reconstructed

12.

• 3. Eckmann–Hilton

Aim: generalise the following structure

13.

• 3. Eckmann–Hilton

Aim: generalise the following structure C = 2-GSet D = dd-2-GSet D C

U

13.

• 3. Eckmann–Hilton

Aim: generalise the following structure C = 2-GSet D = dd-2-GSet D C

U

S = vertical composition T = horizontal composition Note that S restricts to D but T does not.

13.

• 3. Eckmann–Hilton

Aim: generalise the following structure C = 2-GSet D = dd-2-GSet D C

U

S = vertical composition T = horizontal composition Note that S restricts to D but T does not. We have a monad map D C D C

U U S T α

13.

• 3. Eckmann–Hilton

Aim: generalise the following structure C = 2-GSet D = dd-2-GSet D C

U

S = vertical composition T = horizontal composition Note that S restricts to D but T does not. We have a monad map D C D C

U U S T α

TUA USA

α

13.

• 3. Eckmann–Hilton

Key to Eckmann–Hilton: in a doubly degenerate TS-algebra

α

14.

• 3. Eckmann–Hilton

Key to Eckmann–Hilton: in a doubly degenerate TS-algebra

α vert units = hor units = interchange

14.

• 3. Eckmann–Hilton

Key to Eckmann–Hilton: in a doubly degenerate TS-algebra

α vert units = hor units = interchange

We encapsulate this as a natural transformation ε : TU STU “whiskering” ε

14.

• 3. Eckmann–Hilton

Key to Eckmann–Hilton: in a doubly degenerate TS-algebra

α vert units = hor units = interchange

We encapsulate this as a natural transformation ε : TU STU “whiskering” ε such that for any T-algebra t TUA STUA SUA

εA αA εA St

14.

• 3. Eckmann–Hilton

Key to Eckmann–Hilton: in a doubly degenerate TS-algebra

α vert units = hor units = interchange

We encapsulate this as a natural transformation ε : TU STU “whiskering” ε such that for any T-algebra t TUA STUA SUA

εA αA εA St

and for any S-algebra s TUA STUA TSUA TUA

εA λA Ts 1

14.

• 3. Eckmann–Hilton

Definition. Suppose we have

• monads S and T on C and a distributive law ST

TS

λ

15.

• 3. Eckmann–Hilton

Definition. Suppose we have

• monads S and T on C and a distributive law ST

TS

λ

• a functor D

C

U

such that SU = US and

• S restricts to D along U.

15.

• 3. Eckmann–Hilton

Definition. Suppose we have

• monads S and T on C and a distributive law ST

TS

λ

• a functor D

C

U

such that SU = US and

• S restricts to D along U.

Then an “abstract Eckmann–Hilton structure” consists of

• a monad functor TU

US

α

, and

• a natural transformation TU

STU

ε

15.

• 3. Eckmann–Hilton

Definition. Suppose we have

• monads S and T on C and a distributive law ST

TS

λ

• a functor D

C

U

such that SU = US and

• S restricts to D along U.

Then an “abstract Eckmann–Hilton structure” consists of

• a monad functor TU

US

α

, and

• a natural transformation TU

STU

ε

such that for any T-algebra t TUA STUA SUA

εA αA εA St

and for any S-algebra s TUA STUA TSUA TUA

εA λA Ts 1

15.

• 3. Eckmann-Hilton

Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure.

16.

• 3. Eckmann-Hilton

Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure. Then given any TS-algebra     SUA UA , TUA UA

16.

• 3. Eckmann-Hilton

Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure. Then given any TS-algebra     SUA UA , TUA UA the following triangle commutes TUA SUA UA

αA t s

16.

• 3. Eckmann-Hilton

Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure. Then given any TS-algebra     SUA UA , TUA UA the following triangle commutes TUA SUA UA

αA t s αA s

TUA SUA UA

16.

• 3. Eckmann-Hilton

Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure. Then given any TS-algebra     SUA UA , TUA UA the following triangle commutes TUA SUA UA

αA t s STUA εA St s αA hor units

TUA SUA UA

16.

• 3. Eckmann-Hilton

Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure. Then given any TS-algebra     SUA UA , TUA UA the following triangle commutes TUA SUA UA

αA t s STUA hor units TSUA TUA εA λA Ts t St s interchange

TUA SUA UA

STUA TSUA TUA

16.

• 3. Eckmann-Hilton

Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure. Then given any TS-algebra     SUA UA , TUA UA the following triangle commutes TUA SUA UA

αA t s STUA hor units TSUA TUA interchange 1 t εA λA Ts vert units

TUA SUA UA

STUA TSUA TUA St

16.

• 3. Eckmann-Hilton

Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure. Then given any TS-algebra     SUA UA , TUA UA the following triangle commutes TUA SUA UA

αA t s STUA hor units TSUA TUA interchange vert units

TUA

t =

TUA SUA UA

STUA TSUA TUA εA St λA Ts t 1

16.

• 3. Eckmann-Hilton

Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure. Then given any TS-algebra     SUA UA , TUA UA the following triangle commutes TUA SUA UA

αA t s STUA hor units TSUA TUA interchange vert units

TUA

t =

TUA SUA UA

STUA TSUA TUA εA St λA Ts t 1

So t is redundant.

16.

• 4. Weak Eckmann–Hilton

Definition. Suppose we have

• monads S and T on C and a distributive law ST

TS

λ

• a functor D

C

U

such that SU = US and

• S restricts to D along U.

Then a “abstract Eckmann–Hilton structure” consists of

• a monad functor TU

US

α

, and

• a natural transformation TU

STU

ε

such that for any T-algebra t TUA STUA SUA

εA αA εA St

and for any S-algebra s TUA STUA TSUA TUA

εA λA Ts 1

17.

• 4. Weak Eckmann–Hilton

Definition. Suppose we have

• 2-monads S and T on C and a distributive law ST

TS

λ

• a functor D

C

U

such that SU = US and

• S restricts to D along U.

Then a “abstract Eckmann–Hilton structure” consists of

• a monad functor TU

US

α

, and

• a natural transformation TU

STU

ε

such that for any T-algebra t TUA STUA SUA

εA αA εA St

and for any S-algebra s TUA STUA TSUA TUA

εA λA Ts 1

17.

• 4. Weak Eckmann–Hilton

Definition. Suppose we have

• 2-monads S and T on C and a distributive law ST

TS

λ

• a 2-functor D

C

U

such that SU = US and

• S restricts to D along U.

Then a “abstract Eckmann–Hilton structure” consists of

• a monad functor TU

US

α

, and

• a natural transformation TU

STU

ε

such that for any T-algebra t TUA STUA SUA

εA αA εA St

and for any S-algebra s TUA STUA TSUA TUA

εA λA Ts 1

17.

• 4. Weak Eckmann–Hilton

Definition. Suppose we have

• 2-monads S and T on C and a distributive law ST

TS

λ

• a 2-functor D

C

U

such that SU = US and

• S restricts to D along U.

Then a “weak abstract Eckmann–Hilton structure” consists of

• a monad functor TU

US

α

, and

• a natural transformation TU

STU

ε

such that for any T-algebra t TUA STUA SUA

εA αA εA St

and for any S-algebra s TUA STUA TSUA TUA

εA λA Ts 1

17.

• 4. Weak Eckmann–Hilton

Definition. Suppose we have

• 2-monads S and T on C and a distributive law ST

TS

λ

• a 2-functor D

C

U

such that SU = US and

• S restricts to D along U.

Then a “weak abstract Eckmann–Hilton structure” consists of

• a weak monad functor TU

US

α

, and

• a natural transformation TU

STU

ε

such that for any T-algebra t TUA STUA SUA

εA αA εA St

and for any S-algebra s TUA STUA TSUA TUA

εA λA Ts 1

17.

• 4. Weak Eckmann–Hilton

Definition. Suppose we have

• 2-monads S and T on C and a distributive law ST

TS

λ

• a 2-functor D

C

U

such that SU = US and

• S restricts to D along U.

Then a “weak abstract Eckmann–Hilton structure” consists of

• a weak monad functor TU

US

α

, and

• a strictly natural transformation TU

STU

ε

such that for any T-algebra t TUA STUA SUA

εA αA εA St

and for any S-algebra s TUA STUA TSUA TUA

εA λA Ts 1

17.

• 4. Weak Eckmann–Hilton

Definition. Suppose we have

• 2-monads S and T on C and a distributive law ST

TS

λ

• a 2-functor D

C

U

such that SU = US and

• S restricts to D along U.

Then a “weak abstract Eckmann–Hilton structure” consists of

• a weak monad functor TU

US

α

, and

• a strictly natural transformation TU

STU

ε

and for any T-algebra t TUA STUA SUA

εA αA εA St

and for any S-algebra s TUA STUA TSUA TUA

εA λA Ts 1

17.

• 4. Weak Eckmann–Hilton

Definition. Suppose we have

• 2-monads S and T on C and a distributive law ST

TS

λ

• a 2-functor D

C

U

such that SU = US and

• S restricts to D along U.

Then a “weak abstract Eckmann–Hilton structure” consists of

• a weak monad functor TU

US

α

, and

• a strictly natural transformation TU

STU

ε

and for any T-algebra t TUA STUA SUA

εA αA εA St ∼ ψ

+ axioms and for any S-algebra s TUA STUA TSUA TUA

εA λA Ts 1 ∼ φ

17.

• 4. Weak Eckmann-Hilton

Theorem (weak Eckmann–Hilton argument). Suppose we have a weak Eckmann–Hilton structure.

18.

• 4. Weak Eckmann-Hilton

Theorem (weak Eckmann–Hilton argument). Suppose we have a weak Eckmann–Hilton structure. Then given any TS-algebra     SUA UA , TUA UA we have an isomorphism TUA SUA UA

αA t s

∼ =

18.

• 4. Weak Eckmann-Hilton

Theorem (weak Eckmann–Hilton argument). Suppose we have a weak Eckmann–Hilton structure. Then given any TS-algebra     SUA UA , TUA UA we have an isomorphism TUA SUA UA

αA t s STUA hor units TSUA TUA interchange vert units

TUA SUA UA

εA St λA Ts t 1 ∼ ψ−1 ∼ φ = =

18.

• 5. Weak maps

Given a 2-monad T on a 2-category C a weak map of algebras         TA A         TB B

19.

• 5. Weak maps

Given a 2-monad T on a 2-category C a weak map of algebras         TA A         TB B is a 1-cell A B f

19.

• 5. Weak maps

Given a 2-monad T on a 2-category C a weak map of algebras         TA A         TB B is a 1-cell A B f and a 2-cell TA A TB B Tf f + axioms.

19.

• 5. Weak maps

Given 2-monads S, T on a 2-category C and a (strict) distributive law ST TS

α

20.

• 5. Weak maps

Given 2-monads S, T on a 2-category C and a (strict) distributive law ST TS

α

a weak map of TS-algebras       SA A , TA A       SB B ,TB B

20.

• 5. Weak maps

Given 2-monads S, T on a 2-category C and a (strict) distributive law ST TS

α

a weak map of TS-algebras       SA A , TA A       SB B ,TB B is a 1-cell A B f

20.

• 5. Weak maps

Given 2-monads S, T on a 2-category C and a (strict) distributive law ST TS

α

a weak map of TS-algebras       SA A , TA A       SB B ,TB B is a 1-cell A B f and 2-cells TA A TB B Tf f SA A SB B Tf f

20.

• 5. Weak maps

Given 2-monads S, T on a 2-category C and a (strict) distributive law ST TS

α

a weak map of TS-algebras       SA A , TA A       SB B ,TB B is a 1-cell A B f and 2-cells TA A TB B Tf f SA A SB B Tf f + axioms: S-algebra map, T-algebra map, interaction via λ

20.

• 5. Weak maps

Theorem. In the presence of a weak Eckmann–Hilton structure, a weak map of doubly degenerate TS-algebras is of the form

21.

• 5. Weak maps

Theorem. In the presence of a weak Eckmann–Hilton structure, a weak map of doubly degenerate TS-algebras is of the form SA A

as

SB B

bs Sf f σ

TA TB A B

Tf f at bt

21.

• 5. Weak maps

Theorem. In the presence of a weak Eckmann–Hilton structure, a weak map of doubly degenerate TS-algebras is of the form SA A

as

SB B

bs Sf f σ

TA TB A B

Tf f at bt

SA

as

SB

bs Sf αA αB σ ∼ = ∼ =

21.

• 5. Weak maps

Theorem. In the presence of a weak Eckmann–Hilton structure, a weak map of doubly degenerate TS-algebras is of the form SA A

as

SB B

bs Sf f σ

TA TB A B

Tf f at bt

SA

as

SB

bs Sf αA αB σ ∼ = ∼ =

The T-functoriality constraint can be reconstructed from the S-functoriality constraint.

21.

• 5. Weak maps

Trimble doubly degenerate Trimble 3-categories We can construct them via 2-monads on a 2-category and a distributive law.

22.

• 5. Weak maps

Trimble doubly degenerate Trimble 3-categories We can construct them via 2-monads on a 2-category and a distributive law. C = Cat-2-Gph A3 A2 A1 A0

• category

s t s t s t

22.

• 5. Weak maps

Trimble doubly degenerate Trimble 3-categories We can construct them via 2-monads on a 2-category and a distributive law. C = Cat-2-Gph A3 A2 A1 A0

• category

s t s t s t

S = vertical composition T = horizontal composition

• each parametrised by an operad

22.

• 5. Weak maps

Trimble doubly degenerate Trimble 3-categories We can construct them via 2-monads on a 2-category and a distributive law. C = Cat-2-Gph A3 A2 A1 A0

• category

s t s t s t

S = vertical composition T = horizontal composition

• each parametrised by an operad

Distributive law ST TS is parametrised interchange

22.

• 5. Weak maps

Trimble doubly degenerate Trimble 3-categories We can construct them via 2-monads on a 2-category and a distributive law. C = Cat-2-Gph A3 A2 A1 A0

• category

s t s t s t

S = vertical composition T = horizontal composition

• each parametrised by an operad

Distributive law ST TS is parametrised interchange We look at the category of strict TS-algebras and strict maps: TS-Alg ∼ = Tr3Cat

Weak 3-categories strict functors.

22.

• 5. Weak maps

Weak functors of DD Trimble 3-categories

23.

• 5. Weak maps

Weak functors of DD Trimble 3-categories A weak map of TS-algebras is a priori

• a weak map of S-algebras: vertical functoriality constraint
• a weak map of T-algebras: horizontal functoriality constraint

+ compatibility axiom.

23.

• 5. Weak maps

Weak functors of DD Trimble 3-categories A weak map of TS-algebras is a priori

• a weak map of S-algebras: vertical functoriality constraint
• a weak map of T-algebras: horizontal functoriality constraint

+ compatibility axiom. NB Weak enough if underlying object is doubly degenerate.

23.

• 5. Weak maps

Weak functors of DD Trimble 3-categories A weak map of TS-algebras is a priori

• a weak map of S-algebras: vertical functoriality constraint
• a weak map of T-algebras: horizontal functoriality constraint

+ compatibility axiom. NB Weak enough if underlying object is doubly degenerate. In that case:

• Eckmann–Hilton says the T-map constraint is redundant.

23.

• 5. Weak maps

Weak functors of DD Trimble 3-categories A weak map of TS-algebras is a priori

• a weak map of S-algebras: vertical functoriality constraint
• a weak map of T-algebras: horizontal functoriality constraint

+ compatibility axiom. NB Weak enough if underlying object is doubly degenerate. In that case:

• Eckmann–Hilton says the T-map constraint is redundant.
• Compatibility becomes an extra condition on the S-map

constraint.

23.

• 5. Weak maps

Weak functors of DD Trimble 3-categories A weak map of TS-algebras is a priori

• a weak map of S-algebras: vertical functoriality constraint
• a weak map of T-algebras: horizontal functoriality constraint

+ compatibility axiom. NB Weak enough if underlying object is doubly degenerate. In that case:

• Eckmann–Hilton says the T-map constraint is redundant.
• Compatibility becomes an extra condition on the S-map

constraint. This corresponds to the condition on a monoidal functor making it braided.

23.

• 5. Weak maps

This helps us construct the 2-category ddTr3Cat:

• 0-cells: doubly degenerate Trimble 3-categories
• 1-cells: weak maps
• 2-cells: icon-like transformations (Lack)

24.

• 5. Weak maps

This helps us construct the 2-category ddTr3Cat:

• 0-cells: doubly degenerate Trimble 3-categories
• 1-cells: weak maps
• 2-cells: icon-like transformations (Lack)

and we get a biequivalence ddTr3Cat ≃ BrMonCat The proof follows the methodology of Joyal–Kock. Abstract E–H: avoid fiddling around with reparametrisations.

24.

• 5. Weak maps

Future work (with Nick Gurski)

• Express this at the level of operads and relate it to the

little n-cubes operad.

• Examine dependence on weakness of horizontal units,

vertical units and distributive law separately.

• Explore using lax duoidal structures.

(Batanin–Cisinki–Weber, Garner–L´

• pez Franco)
• Investigate what type of monads work. (Kelly)
• Better abstract description.
• Relationship between different Eckmann–Hilton structures
• n the same data.
• Braiding vs. symmetry

25.