Lax monads and generalized multicategory theory Dimitri Chikhladze 1
� � A category ( x, a ) is a monad in the bicategory Span. It consists of a set of objects x , and a span a p 1 p 2 x x with a the set of arrows, and p 1 and p 2 the source and target maps. 2
� � � � � � � � � � � � � � The composition and unit are given by the span maps: a × x a a a a x x x x x x a x x x x 3
Let ( T, m, e ) be the free monoid monad on Set. The monad T extends to Cat by T ( x, a ) = ( Tx, Ta ). A stirct monoidal category (( x, a ) , h ) is an algebra for this monad. So ( x, a ) is a category, and ( h, σ ) : ( Tx, Ta ) → ( x, a ) gives the monoidal structure. 4
� � A multicategory ( x, a ) consists of a set x , a span a : ✤ � Tx which is a diagram of the form: x a p 1 p 2 x Tx where a is the set of multimorphisms and p 1 and p 2 give sourse and target maps. 5
� � � � � � � � � � � � � � � Multicategory composition and units are given by: Ta × Tx a a a Ta x T 2 x x Tx Tx m x Tx x a e x x x Tx Tx 6
� � There is a functor Set → Span sending a map f : x → y to a span: x f 1 x y x A set monad T extends to Span. By Kleisli construction we define Span T , objects of which are sets, and a hom 2-category Span T ( X, Y ) is the category of spans of the form X → TY . 7
A multicategory is a ”monad” in Span T . There is an interaction between Set and Span T , which allows us to consider morphisms of multicategories. 8
The idea is to replace Set by an abstract category X , to replace Span by a ”bicategory like structure” A and replace the free monoid monad by a general monad. The Kleisli construction gives a ”bicategory like struc- ture” A T . A generalized multicategory or a T -monoid is a monad in A T We get diverse examples: multicategories, topological spaces, metric spaces, lawere theories, globular oper- ads, and more. Generalized version of a strict monoidal category is called a T -algebra. 9
For Ordinary multicategories we have an adjunction: Multicat ← � MonCat . The motivation of this work is to generlize this adjunc- tion to generalized multicategories so that the monad (monoid) nature of them is emphasized. T -Mon ← � T -Alg . 10
✤ � Y is a pseudo- A two-sided indexed category A : X functor A : X op × Y � Cat. There is a tricategory M . With objects categories. And the homcategory M ( X, Y ) = [ X op × Y, Cat]. Morphisms of this tricategory are two sided indexed ✤ � Z of categories. A horizontal composite A.B : X ✤ � Y and B : Y ✤ � Z is defined by a pseudo co- A : X end: � ( A.B )( X, Z ) = A ( X, − ) × B ( − , Z ) . 11
An equipment is a lax monad ( X, A ) in M . It consists of the following data: X is a category. � X is a two sided indexed fibration. A : X � A are 2-cells in M P n : A .n ξ n 1 ,...,n k : P n 1 + ... + n k → P k ( P n 1 ◦ P n 2 ◦ ... ◦ P n k ) are 3-cells. 12
We write an object a of A ( X, Y ) as a : X → Y . Correspondingly, morphisms of A ( X, Y ) will be written as α : a ⇒ b . Π n : A .n → A is determined by a functor � A ( x 1 , x n +1 ) . A ( x 1 , x 2 ) × A ( x 2 , x 3 ) × · · · × A ( x n , x n +1 ) We call this the n -ary composition of the equipment, and for its value at a 1 , a 2 , ..., a n we write a 1 a 2 · · · a n . Π 0 : X ∗ � A is determined by functors X ( x, x ) → A ( x, x. ) These give an object u x in A ( x, x ) for each x . 13
There is a pseudofunctor Cat op � M , which is identity � Y to F ∗ = Y ( F − , − ). on objects and sends F : X A lax functor between equipments ( X, A ) → ( Y, B ) con- sists of a functor X → Y , a 2-cell Φ : AF ∗ → F ∗ B , and for every n ≥ 0 a 3-cell κ n : ( F.Π n ) Φ .n → Φ( Π n .F ) , These amout to functors: F : A ( X, Y ) → B ( X, Y ) and F ( a 1 ) ...F ( a n ) → F ( a 1 ...a n ) . 14
Equipments, lax functors and lax transformations be- tween them form a 2-category E . 15
Equipments allow the Kleisli construction: Let T be a monad on the equipment ( X, A ) in E . By monad composition ( X, T ∗ .A ) becomes an equip- ment. The Kleisli construction can be extended to a 2-functor: Cmp : Mnd( E ) → E . 16
Let I denote the terminal category. ( I, I ∗ ) is a terminal equipment. A monoid in ( X, A ) is a lax functor ( I, I ∗ ) → ( X, A ). This amount to an object x of X and an element a : x → x of A , a multiplication and a unit satisfying three axioms. The category of monoids is denoted by Mon( X, A ). 17
T -induces a monad on the category Mon( X, A ), given on objects by T ( x, a ) = ( Tx, Ta ). An algebra for this category is called a T -algebra. This amount to a monoid ( x, a ) and an action ( h, σ ) : ( Tx, Ta ) → ( x, a ). 18
A T -monoid is a monad in ( X, T ∗ .A ). This amount to an object x of X and an element a : x → Tx of A , a multiplication and a unit satisfying three axioms. 19
This Kleisli construction gives: Mnd( E )((( I, I ∗ ) , 1 I ) , (( X, A ) , T )) → E (( I, I ∗ ) , ( X, T ∗ .A )) Or: T -Alg → T -Mon On objects it acts as: (( x, b ) , h ) �→ ( x, h r b ) 20
Let Inc : E → Mnd( E ) be a functor which send an equipment ( X, A ) to (( X, A ) , 1 X ). Inc is a left lax 2-adjoint to Cmp. A lax adjunction between 2-categories has lax natural transformations for its unit and counit, and the triangle identities are replaced by appropriately directed 2-cells. � 1 is an isomor- In our situation the counit (Cmp)(Inc) � (Inc)(Cmp) is given by the family phism. The unit 1 of maps (( X, T ∗ .A ) , 1 X ) → (( X, A ) , T ). 21
The functor T -Mon → T -Alg which is the same as E (( I, I ∗ ) , ( X, T ∗ .A )) → Mnd( E )((( I, I ∗ ) , 1 I ) , (( X, A ) , T )) is defined by first taking Inc and then precomposing with (( X, T ∗ .A ) , 1 X ) → (( X, A ) , T ). On objects it acts as: ( x, a ) �→ ( Tx, m x Ta ) . 22
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