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

1. Overview Example 2: 2-categories C = 2-GSet , S = free Cat-Gph , T = free Gph-Cat s s A 2 A 1 A 0 vertical comp horizontal comp t t λ A STA TSA 10.

1. Overview Example 2: 2-categories C = 2-GSet , S = free Cat-Gph , T = free Gph-Cat s s A 2 A 1 A 0 vertical comp horizontal comp t t λ A STA TSA A TS -algebra is a 2-category. 10.

1. Overview Example 2: 2-categories C = 2-GSet , S = free Cat-Gph , T = free Gph-Cat s s A 2 A 1 A 0 vertical comp horizontal comp t t λ A STA TSA 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 s t • an S -algebra SA A and a T -algebra TA A 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 s t • an S -algebra SA A and a T -algebra TA A λ A Ts • such that STA TSA TA St t SA A s 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 s t • an S -algebra SA A and a T -algebra TA A λ A Ts • such that STA TSA TA St t SA A s 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 s t • an S -algebra SA A and a T -algebra TA A λ A Ts • such that STA TSA TA St t SA A s 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 s t • an S -algebra SA A and a T -algebra TA A λ A Ts • such that STA TSA TA St t SA A s 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 SA TA and + compatibility via λ ≡ A A 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 SA TA and + compatibility via λ ≡ A A A Eckmann–Hilton structure SA + axiom 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 SA TA and + compatibility via λ ≡ A A A Eckmann–Hilton structure SA + axiom T -algebra part is reconstructed A 12.

3. Eckmann–Hilton Aim: generalise the following structure 13.

3. Eckmann–Hilton Aim: generalise the following structure C = 2-GSet U D = dd-2-GSet D C 13.

3. Eckmann–Hilton Aim: generalise the following structure C = 2-GSet U D = dd-2-GSet D C 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 U D = dd-2-GSet D C S = vertical composition T = horizontal composition Note that S restricts to D but T does not. We have a monad map U D C α S T D C U 13.

3. Eckmann–Hilton Aim: generalise the following structure C = 2-GSet U D = dd-2-GSet D C S = vertical composition T = horizontal composition Note that S restricts to D but T does not. We have a monad map U D C α TUA USA α S T D C U 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 vert units = α interchange = hor units 14.

3. Eckmann–Hilton Key to Eckmann–Hilton: in a doubly degenerate TS -algebra vert units = ε α interchange = hor units We encapsulate this as a natural transformation ε : TU STU “whiskering” 14.

3. Eckmann–Hilton Key to Eckmann–Hilton: in a doubly degenerate TS -algebra vert units = ε α interchange = hor units We encapsulate this as a natural transformation ε : TU STU “whiskering” such that for any T -algebra t ε A ε A TUA STUA St α A SUA 14.

3. Eckmann–Hilton Key to Eckmann–Hilton: in a doubly degenerate TS -algebra vert units = ε α interchange = hor units We encapsulate this as a natural transformation ε : TU STU “whiskering” such that for any T -algebra t and for any S -algebra s ε A ε A ε A λ A TUA STUA TUA STUA TSUA St Ts α A 1 SUA TUA 14.

3. Eckmann–Hilton Definition. Suppose we have λ • monads S and T on C and a distributive law ST TS 15.

3. Eckmann–Hilton Definition. Suppose we have λ • monads S and T on C and a distributive law ST TS U • a functor D C such that SU = US and • S restricts to D along U . 15.

3. Eckmann–Hilton Definition. Suppose we have λ • monads S and T on C and a distributive law ST TS U • a functor D C such that SU = US and • S restricts to D along U . Then an “abstract Eckmann–Hilton structure” consists of α • a monad functor TU US , and ε • a natural transformation TU STU 15.

3. Eckmann–Hilton Definition. Suppose we have λ • monads S and T on C and a distributive law ST TS U • a functor D C such that SU = US and • S restricts to D along U . Then an “abstract Eckmann–Hilton structure” consists of α • a monad functor TU US , and ε • a natural transformation TU STU such that for any T -algebra t and for any S -algebra s ε A ε A ε A λ A TUA STUA TUA STUA TSUA St Ts α A 1 SUA TUA 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.   SUA , TUA Then given any TS -algebra   UA UA 16.

3. Eckmann-Hilton Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure.   SUA , TUA Then given any TS -algebra   UA UA the following triangle commutes α A TUA SUA s t UA 16.

3. Eckmann-Hilton Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure.   SUA , TUA Then given any TS -algebra   UA UA the following triangle commutes α A α A TUA TUA SUA SUA s s t UA UA 16.

3. Eckmann-Hilton Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure.   SUA , TUA Then given any TS -algebra   UA UA the following triangle commutes α A α A TUA TUA SUA SUA ε A hor units St STUA s s t UA UA 16.

3. Eckmann-Hilton Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure.   SUA , TUA Then given any TS -algebra   UA UA the following triangle commutes α A TUA TUA SUA SUA ε A hor units St STUA STUA λ A interchange TSUA TSUA s s t Ts TUA TUA t UA UA 16.

3. Eckmann-Hilton Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure.   SUA , TUA Then given any TS -algebra   UA UA the following triangle commutes α A TUA TUA SUA SUA ε A hor units St STUA STUA λ A 1 interchange TSUA TSUA s t Ts TUA TUA vert units t UA UA 16.

3. Eckmann-Hilton Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure.   SUA , TUA Then given any TS -algebra   UA UA the following triangle commutes α A TUA TUA TUA SUA SUA ε A hor units St STUA STUA λ A 1 interchange TSUA TSUA s t t Ts = TUA TUA vert units t UA UA 16.

3. Eckmann-Hilton Theorem (abstract Eckmann–Hilton argument). Suppose we have an Eckmann–Hilton structure.   SUA , TUA Then given any TS -algebra   UA UA the following triangle commutes α A TUA TUA TUA SUA SUA ε A hor units St STUA STUA λ A 1 interchange TSUA TSUA s t t Ts = TUA TUA vert units So t is redundant. t UA UA 16.

4. Weak Eckmann–Hilton Definition. Suppose we have λ • monads S and T on C and a distributive law ST TS U • a functor D C such that SU = US and • S restricts to D along U . Then a “abstract Eckmann–Hilton structure” consists of α • a monad functor TU US , and ε • a natural transformation TU STU such that for any T -algebra t and for any S -algebra s ε A ε A ε A λ A TUA STUA TUA STUA TSUA St Ts α A 1 SUA TUA 17.

4. Weak Eckmann–Hilton Definition. Suppose we have λ • 2-monads S and T on C and a distributive law ST TS U • a functor D C such that SU = US and • S restricts to D along U . Then a “abstract Eckmann–Hilton structure” consists of α • a monad functor TU US , and ε • a natural transformation TU STU such that for any T -algebra t and for any S -algebra s ε A ε A ε A λ A TUA STUA TUA STUA TSUA St Ts α A 1 SUA TUA 17.

4. Weak Eckmann–Hilton Definition. Suppose we have λ • 2-monads S and T on C and a distributive law ST TS U • a 2-functor D C such that SU = US and • S restricts to D along U . Then a “abstract Eckmann–Hilton structure” consists of α • a monad functor TU US , and ε • a natural transformation TU STU such that for any T -algebra t and for any S -algebra s ε A ε A ε A λ A TUA STUA TUA STUA TSUA St Ts α A 1 SUA TUA 17.

4. Weak Eckmann–Hilton Definition. Suppose we have λ • 2-monads S and T on C and a distributive law ST TS U • a 2-functor D C such that SU = US and • S restricts to D along U . Then a “weak abstract Eckmann–Hilton structure” consists of α • a monad functor TU US , and ε • a natural transformation TU STU such that for any T -algebra t and for any S -algebra s ε A ε A ε A λ A TUA STUA TUA STUA TSUA St Ts α A 1 SUA TUA 17.

4. Weak Eckmann–Hilton Definition. Suppose we have λ • 2-monads S and T on C and a distributive law ST TS U • a 2-functor D C such that SU = US and • S restricts to D along U . Then a “weak abstract Eckmann–Hilton structure” consists of α • a weak monad functor TU US , and ε • a natural transformation TU STU such that for any T -algebra t and for any S -algebra s ε A ε A ε A λ A TUA STUA TUA STUA TSUA St Ts α A 1 SUA TUA 17.

4. Weak Eckmann–Hilton Definition. Suppose we have λ • 2-monads S and T on C and a distributive law ST TS U • a 2-functor D C such that SU = US and • S restricts to D along U . Then a “weak abstract Eckmann–Hilton structure” consists of α • a weak monad functor TU US , and ε • a strictly natural transformation TU STU such that for any T -algebra t and for any S -algebra s ε A ε A ε A λ A TUA STUA TUA STUA TSUA St Ts α A 1 SUA TUA 17.

4. Weak Eckmann–Hilton Definition. Suppose we have λ • 2-monads S and T on C and a distributive law ST TS U • a 2-functor D C such that SU = US and • S restricts to D along U . Then a “weak abstract Eckmann–Hilton structure” consists of α • a weak monad functor TU US , and ε • a strictly natural transformation TU STU and for any T -algebra t and for any S -algebra s ε A ε A ε A λ A TUA STUA TUA STUA TSUA St Ts α A 1 SUA TUA 17.

4. Weak Eckmann–Hilton Definition. Suppose we have λ • 2-monads S and T on C and a distributive law ST TS U • a 2-functor D C such that SU = US and • S restricts to D along U . Then a “weak abstract Eckmann–Hilton structure” consists of α • a weak monad functor TU US , and ε • a strictly natural transformation TU STU and for any T -algebra t and for any S -algebra s ε A ε A ε A λ A TUA STUA TUA STUA TSUA ∼ ∼ St Ts ψ φ α A 1 + axioms SUA TUA 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.   SUA , TUA Then given any TS -algebra   UA UA we have an isomorphism α A TUA SUA ∼ = s t UA 18.

4. Weak Eckmann-Hilton Theorem (weak Eckmann–Hilton argument). Suppose we have a weak Eckmann–Hilton structure.   SUA , TUA Then given any TS -algebra   UA UA we have an isomorphism α A TUA TUA SUA SUA ψ − 1 ∼ ε A hor units St STUA ∼ λ A = 1 φ interchange TSUA s t Ts = TUA vert units t UA UA 18.

5. Weak maps Given a 2-monad T on a 2-category C a weak map of algebras     TA TB             A B 19.

5. Weak maps Given a 2-monad T on a 2-category C a weak map of algebras     TA TB             A B f is a 1-cell A B 19.

5. Weak maps Given a 2-monad T on a 2-category C a weak map of algebras     TA TB             A B f is a 1-cell A B Tf TA TB and a 2-cell A B f + axioms. 19.