Skew monoidal structures on categories of algebras Marcelo Fiore and - PowerPoint PPT Presentation

Skew monoidal structures on categories of algebras Marcelo Fiore and Philip Saville University of Cambridge Department of Computer Science and Technology 10th July 2018 1 / 29

Skew monoidal categories A version of monoidal categories (Szlachányi (2012)) Structural transformations need not be invertible: α : p A b B q b C Ñ A b p B b C q λ : I b A Ñ A ρ : A Ñ A b I 2 / 29

Skew monoidal categories A version of monoidal categories (Szlachányi (2012)) Structural transformations need not be invertible: α : p A b B q b C Ñ A b p B b C q λ : I b A Ñ A ρ : A Ñ A b I Example § For C with coproducts, p X { C q with a b inl a ` b p X Ý Ñ A q ‘ p X Ñ B q : “ X Ý Ý Ñ X ` X Ý Ý Ñ A ` B § For C cocomplete, r J , C s with unit J and tensor ` ˘ F ‹ G : “ p lan J F q ˝ G Altenkirch et al. (2010) . 2 / 29

Skew monoidal categories A version of monoidal categories (Szlachányi (2012)) Structural transformations need not be invertible: α : p A b B q b C Ñ A b p B b C q λ : I b A Ñ A ρ : A Ñ A b I Recently studied very actively ( list not exhaustive! ): Coherence properties: Lack & Street (2014), Andrianopoulos (2017), Bourke (2017), Uustalu (2017, 2018), . . . Extensions, theory and examples: Street (2013), Campbell (2018), . . . 2 / 29

Past work Linton (’69), Kock (’71a, ’71b), Guitart (’80), Jacobs (’94), Seal (’13), . . . C monoidal T a monoidal monad C T monoidal ñ reflexive coequalizers in C + preservation conditions 3 / 29

Past work Linton (’69), Kock (’71a, ’71b), Guitart (’80), Jacobs (’94), Seal (’13), . . . C monoidal T a monoidal monad C T monoidal ñ reflexive coequalizers in C + preservation conditions This work C T skew C skew monoidal monoidal T a strong monad ñ reflexive coequalizers in C + preservation conditions 3 / 29

Past work Linton (’69), Kock (’71a, ’71b), Guitart (’80), Jacobs (’94), Seal (’13), . . . C monoidal T a monoidal monad C T monoidal ñ reflexive coequalizers in C + preservation conditions This work C T skew C skew monoidal monoidal T a strong monad ñ monoids are reflexive coequalizers in C + T -monoids preservation conditions 3 / 29

Monoidal case ( C , T monoidal) Definition (Kock (1971)) For p A , a q , p B , b q , p C , c q P C T a map h : A b B Ñ C in C is bilinear if it is linear in each argument: T p A qb η Th κ T p A q b B T p A q b T p B q T p A b B q TC c a b B A b B C h η b T p B q Th κ A b T p B q T p A q b T p B q T p A b B q TC c A b b A b B C h 4 / 29

Monoidal case ( C , T monoidal) Aim Construct p´q ‹ p“q : C T ˆ C T Ñ C T satisfying 1. C T p A ‹ B , C q – Bilin C p A , B ; C q 2. A suitable preservation property to guarantee coherence 5 / 29

Monoidal case ( C , T monoidal) Aim Construct p´q ‹ p“q : C T ˆ C T Ñ C T satisfying 1. C T p A ‹ B , C q – Bilin C p A , B ; C q 2. A suitable preservation property to guarantee coherence Construction (Linton 1969) Reflexive coequalizer in C T : µ coeq. T κ ` ˘ T 2 p A b B q T T p A q b T p B q T p A b B q A ‹ B T p a b b q NB: U : C T Ñ C creates reflexive coequalizers if T preserves them 5 / 29

Monoidal case ( C , T monoidal) Aim Construct p´q ‹ p“q : C T ˆ C T Ñ C T satisfying 1. C T p A ‹ B , C q – Bilin C p A , B ; C q 2. if every p´q b X and X b p´q preserve reflexive coequalizers, so do p´q ‹ p A , a q and p A , a q ‹ p´q Construction (Linton 1969) Reflexive coequalizer in C T : µ coeq. T κ ` ˘ T 2 p A b B q T T p A q b T p B q T p A b B q A ‹ B T p a b b q NB: U : C T Ñ C creates reflexive coequalizers if T preserves them 5 / 29

Monoidal case ( C , T monoidal) Proposition (Guitart (’80), Seal (’13)) Suppose that § C has all reflexive coequalizers, § T preserves reflexive coequalizers, § Every p´q b X and X b p´q preserves reflexive coequalizers Then p C T , ‹ , TI q is a monoidal category. Other versions are available: e.g. closed, symmetric, cartesian. . . 6 / 29

Skew monoidal case ( C skew monoidal, T strong) Classify left-linear maps Construct an action C T ˆ C Ñ C T Extend to a skew monoidal structure on C T 7 / 29

Skew monoidal case ( C skew monoidal, T strong) Classify left-linear maps Construct an action C T ˆ C Ñ C T Extend to a skew monoidal structure on C T Background assumption : ` ˘ C skew monoidal, T strong st : T p A q b B Ñ T p A b B q 7 / 29

Factoring the proof C closed or α invertible C has reflexive C T has reflexive C has a p 1 , 2 , 3 q -left coequalizers, coequalizers linear classifier which T preserves p´q b X preserves reflexive coeqs. C closed C acts on C T or α invertible or p´q b X preserves reflexive coeqs. C T skew monoidal 8 / 29

Factoring the proof C closed or α invertible C has reflexive C T has reflexive C has a p 1 , 2 , 3 q -left coequalizers, coequalizers linear classifier which T preserves p´q b X preserves reflexive coeqs. C closed C acts on C T or α invertible or p´q b X preserves reflexive coeqs. C T skew monoidal 8 / 29

Left-linear maps Definition ( c.f. Kock (1971)) For p A , a q , p B , b q P C T and P P C , a map h : A b P Ñ C is left linear if st A , B Th T p A q b P T p A b P q TB a b P b A b P B h 9 / 29

Left-linear classifiers Definition ( c.f. Guitart (’80), Jacobs (’94), Seal (’13)) A left-linear classifier is a family of maps σ A , P : A b P Ñ A ‹ P such that 1. p A ‹ P , τ A , P q P C T 2. σ A , B is left-linear, 3. Every left-linear map A b P Ñ B factors uniquely: σ A b P A ‹ P D ! algebra map @ left-linear maps B Determines an isomorphism C T p A ‹ P , B q – LeftLin C p A , P ; B q . 10 / 29

Left-linear classifiers Definition ( c.f. Guitart (’80), Jacobs (’94), Seal (’13)) A left-linear classifier is a family of maps σ A , P : A b P Ñ A ‹ P such that 1. p A ‹ P , τ A , P q P C T 2. σ A , B is left-linear, 3. Every left-linear map A b P Ñ B factors uniquely: σ A b P A ‹ P D ! algebra map @ left-linear maps B Determines an isomorphism C T p A ‹ P , B q – LeftLin C p A , P ; B q . ù Need to build in a preservation property to guarantee coherence 10 / 29

n -left linear maps Definition For p A , a q , p B , b q P C T and P 1 , . . . , P n P C , a map ` ` ˘ ˘ h : ¨ ¨ ¨ p A b P 1 q b P 2 ¨ ¨ ¨ b P n ´ 1 b P n Ñ B is n -left linear if st b n Th T p A q b P 1 b ¨ ¨ ¨ b P n T p A b P 1 b ¨ ¨ ¨ b P n q TB a b P 1 b¨¨¨b P n b A b P 1 b ¨ ¨ ¨ b P n B h where st b 1 : “ st and st bp n ` 1 q : “ st ˝ st b n . 11 / 29

n -left linear maps Definition For p A , a q , p B , b q P C T and P 1 , . . . , P n P C , a map ` ` ˘ ˘ h : ¨ ¨ ¨ p A b P 1 q b P 2 ¨ ¨ ¨ b P n ´ 1 b P n Ñ B is n -left linear if st b n Th T p A q b P 1 b ¨ ¨ ¨ b P n T p A b P 1 b ¨ ¨ ¨ b P n q TB a b P 1 b¨¨¨b P n b A b P 1 b ¨ ¨ ¨ b P n B h where st b 1 : “ st and st bp n ` 1 q : “ st ˝ st b n . ù An n -parameter version of left-linearity. 11 / 29

n -left linear classifiers Definition A n -left linear classifier is a family of maps σ A , P 1 : A b P 1 Ñ A ‹ P 1 such that 1. p A ‹ P 1 , τ A , P 1 q P C T 2. σ A , B is left-linear, ` ` ˘ ˘ 3. Every n -left linear map ¨ ¨ ¨ p A b P 1 q b P 2 ¨ ¨ ¨ b P n Ñ B factors uniquely: σ b P 2 b¨¨¨b P n A b P 1 b ¨ ¨ ¨ b P n p A ‹ P 1 q b P 2 b ¨ ¨ ¨ b P n D ! p n ´ 1 q -left linear map @ n -left linear B A p 1 , . . . , n q -left linear classifier is a 1-left linear classifier that is also an i -left linear classifier (1 ď i ď n ). 12 / 29

n -left linear classifiers Lemma ` ` ˘ ˘ If h : ¨ ¨ ¨ p A b P 1 q b P 2 ¨ ¨ ¨ b P n ` 1 Ñ B is p n ` 1 q -left linear, then (if they exist) 1. The transpose ˜ h : A b P 1 b ¨ ¨ ¨ b P n Ñ r P n ` 1 , B s is n -left linear, 2. h ˝ α ´ 1 : p A b P 1 ¨ ¨ ¨ b P n ´ 1 q b p P n b P n ` 1 q Ñ B is n -left linear Lemma If C has an n -left linear classifier and satisfies either § C is closed, or § α is invertible Then C has an p n ` 1 q -left linear classifier. 12 / 29

Factoring the proof C closed or α invertible C has reflexive C T has reflexive C has a p 1 , 2 , 3 q -left coequalizers, coequalizers linear classifier which T preserves p´q b X preserves ? reflexive coeqs. C closed C acts on C T or α invertible or p´q b X preserves reflexive coeqs. C T skew monoidal 13 / 29

From classifier to action Proposition If C has a p 1 , 2 , 3 q -left linear classifier σ A , B : A b B Ñ A ‹ B , then 1. ‹ : C T ˆ C Ñ C T is a skew action, and 2. The free-forgetful adjunction F : C Ô C T : U is strong. 14 / 29

From classifier to action Proposition If C has a p 1 , 2 , 3 q -left linear classifier σ A , B : A b B Ñ A ‹ B , then 1. ‹ : C T ˆ C Ñ C T is a skew action, and 2. The free-forgetful adjunction F : C Ô C T : U is strong. Holds in particular if C has a 1-left linear classifier and § C is closed, or § α is invertible 14 / 29

Factoring the proof C closed or α invertible C has reflexive C T has reflexive C has a p 1 , 2 , 3 q -left coequalizers, coequalizers linear classifier which T preserves p´q b X preserves reflexive coeqs. C closed C acts on C T or α invertible or ? p´q b X preserves reflexive coeqs. C T skew monoidal 15 / 29