Traced concategories Paul Blain Levy, Sergey Goncharov and Lutz Schr - PowerPoint PPT Presentation

Traced concategories Paul Blain Levy, Sergey Goncharov and Lutz Schr¨ oder December 18, 2018 Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 1 / 21

Outline Varying the notion of category 1 Concategories 2 Symmetric concategories 3 Traced concategories 4 Further work 5 Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 2 / 21

Varying the notion of category Notion Morphism Main example Category f : a → b Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 3 / 21

Varying the notion of category Notion Morphism Main example Category f : a → b f : � − f : − → → Cartesian a → b a → b in a cartesian category multicategory Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 3 / 21

Varying the notion of category Notion Morphism Main example Category f : a → b f : � − f : − → → Cartesian a → b a → b in a cartesian category multicategory f : � − f : − → → Multicategory a → b in a monoidal category a → b Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 3 / 21

Varying the notion of category Notion Morphism Main example Category f : a → b f : � − f : − → → Cartesian a → b a → b in a cartesian category multicategory f : � − f : − → → Multicategory a → b in a monoidal category a → b a → − → f : � − a → � − → f : − → → Concategory b b in a monoidal category Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 3 / 21

Varying the notion of category Notion Morphism Main example Category f : a → b f : � − f : − → → Cartesian a → b a → b in a cartesian category multicategory f : � − f : − → → Multicategory a → b in a monoidal category a → b a → − → f : � − a → � − → f : − → → Concategory b b in a monoidal category f : � − a → ˙ − a → − → → f : − → → Polycategory b b in a linearly distributive category Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 3 / 21

� � � � � � � � � � � � � � � � � � � � � � � Constructions Cartesian Cartesian Cartesian category multicategory operad ⊤ List Monoidal Concategory PRO category ⊤ List � ` = ⊗ Linearly distributive Polycategory Dioperad id category Forget ` Monoidal � Multicategory Operad category Category Monoid Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 4 / 21

Definition of concategory A concategory C consists of the following data. A class ob C of objects. a ; − → a , − → A homset C ( − → b ) for each pair of object lists − → b . Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 5 / 21

Definition of concategory A concategory C consists of the following data. A class ob C of objects. a ; − → a , − → A homset C ( − → b ) for each pair of object lists − → b . a → − → b and g : − → The sequential composite of f : − → b → − → c is f ; g : − → a → − → c . a → − → c → − → The parallel composite of f : − → b and g : − → d is c → − → + − → f ⊠ g : − → + − → a + b + d . Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 5 / 21

Definition of concategory A concategory C consists of the following data. A class ob C of objects. a ; − → a , − → A homset C ( − → b ) for each pair of object lists − → b . a → − → b and g : − → The sequential composite of f : − → b → − → c is f ; g : − → a → − → c . a → − → c → − → The parallel composite of f : − → b and g : − → d is c → − → + − → f ⊠ g : − → + − → a + b + d . a : − → a → − → The sequential identity id − a . → The parallel identity id ⊠ : ε → ε . (Redundant.) Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 5 / 21

The ten commandments Sequential composition is associative and unital. Parallel composition is associative and unital. Interchange between sequential and parallel composition: ( f ; g ) ⊠ ( h ; k ) = ( f ⊠ h ); ( g ⊠ k ) Interchange between sequential identity and parallel composition: id − a ⊠ id − = id − → → + − → → b a + b Interchange between sequential composition and parallel identity: id ⊠ = id ⊠ ; id ⊠ Interchange between sequential and parallel identity: id ε = id ⊠ Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 6 / 21

Why the name? “Category” alludes to sequential composition f ; g : − → a → − → c “Concat” alludes to parallel composition c → − → + − → f ⊠ g : − → + − → a + b + d Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 7 / 21

Why the name? “Category” alludes to sequential composition f ; g : − → a → − → c “Concat” alludes to parallel composition c → − → + − → f ⊠ g : − → + − → a + b + d The overlap alludes to the interchange law. Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 7 / 21

Caviglia’s 2-category of concategories Map of concategories A map F : C → D sends objects to objects and morphisms to morphisms, preserving all structure. Natural transformation A natural transformation sends each object a to α a : [ Fa ] → [ Ga ] . a → − → For f : − → b we require f ; − α b = − → → α a ; f . Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 8 / 21

Examples of concategories Each of the following gives a concategory: 1 A monoidal category. Morphisms go from � − a → � − → → b . Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 9 / 21

Examples of concategories Each of the following gives a concategory: 1 A monoidal category. Morphisms go from � − a → � − → → b . 2 A cartesian multicategory. Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 9 / 21

Examples of concategories Each of the following gives a concategory: 1 A monoidal category. Morphisms go from � − a → � − → → b . 2 A cartesian multicategory. 3 A PRO. It corresponds to a single-object concategory Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 9 / 21

Examples of concategories Each of the following gives a concategory: 1 A monoidal category. Morphisms go from � − a → � − → → b . 2 A cartesian multicategory. 3 A PRO. It corresponds to a single-object concategory 4 A many-sorted list-to-list signature. Acyclic string diagrams modulo isomorphism. Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 9 / 21

Examples of concategories Each of the following gives a concategory: 1 A monoidal category. Morphisms go from � − a → � − → → b . 2 A cartesian multicategory. 3 A PRO. It corresponds to a single-object concategory 4 A many-sorted list-to-list signature. Acyclic string diagrams modulo isomorphism. 5 A dataflow model e.g. Kahn’s or Jonsson’s. Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 9 / 21

PROs A PRO consists of a family of sets ( A m,n ) m,n ∈ N with f ∈ A m,n written f : m → n and sequential and parallel composition and identity satisfying the ten commandments. A PRO A correspond to a single-object concategory ˜ A . Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 10 / 21

Colours Object = “colour”. Concategory = “coloured PRO” Multicategory = “coloured operad” Polycategory = “coloured dioperad” Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 11 / 21

Colours Object = “colour”. Concategory = “coloured PRO” Multicategory = “coloured operad” Polycategory = “coloured dioperad” Category = “coloured monoid” (Tom Leinster satire) Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 11 / 21

Colours Object = “colour”. Concategory = “coloured PRO” Multicategory = “coloured operad” Polycategory = “coloured dioperad” Category = “coloured monoid” (Tom Leinster satire) Monoidal category = “monoidal coloured monoid” Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 11 / 21

Colours Object = “colour”. Concategory = “coloured PRO” Multicategory = “coloured operad” Polycategory = “coloured dioperad” Category = “coloured monoid” (Tom Leinster satire) Monoidal category = “monoidal coloured monoid” In colourful literature, usually: Colours form a set, sometimes a finite set, sometimes fixed in advance. The construction monoidal category → concategory is not prominent. Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 11 / 21

� Concategory vs monoidal category The 2-embedding of MONCAT in CONCAT is reflective. � CONCAT MONCAT ⊤ List List C is a strict monoidal category. Its objects are lists of C -objects. The induced comonad on MONCAT is strictification. So we have resolved strictification into two parts. Levy, Goncharov, Schr¨ oder Traced concategories December 18, 2018 12 / 21