Lax monads and generalized multicategory theory Dimitri Chikhladze - - PowerPoint PPT Presentation

Lax monads and generalized multicategory theory Dimitri Chikhladze

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A category (x, a) is a monad in the bicategory Span. It consists of a set of objects x, and a span a x x

p1

• p2
• with a the set of arrows, and p1 and p2 the source and

target maps.

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The composition and unit are given by the span maps: a x x a ×x a a x x a x

• x

x x a 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.

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A multicategory (x, a) consists of a set x, a span a : x

✤ Tx which is a diagram of the form:

a x Tx

p1

• p2
• where a is the set of multimorphisms and p1 and p2 give

sourse and target maps.

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Multicategory composition and units are given by: Ta Tx T 2x Ta ×Tx a a x Tx a x Tx

• mx
• x

x Tx a x Tx

• ex
• 6

There is a functor Set → Span sending a map f : x → y to a span: x x y

1x

• f
• A set monad T extends to Span.

By Kleisli construction we define SpanT, objects of which are sets, and a hom 2-category SpanT(X, Y ) is the category of spans of the form X → TY .

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A multicategory is a ”monad” in SpanT. There is an interaction between Set and SpanT, which allows us to consider morphisms of multicategories.

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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” AT. A generalized multicategory or a T-monoid is a monad in AT 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.

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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.

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A two-sided indexed category A : X

✤ Y is a pseudo-

functor A : Xop × Y

Cat.

There is a tricategory M. With objects categories. And the homcategory M(X, Y ) = [Xop × Y, Cat]. Morphisms of this tricategory are two sided indexed categories. A horizontal composite A.B : X

✤ Z of

A : X

✤ Y and B : Y ✤ Z is defined by a pseudo co-

end: (A.B)(X, Z) =

• A(X, −) × B(−, Z).

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An equipment is a lax monad (X, A) in M. It consists

• f the following data:

X is a category. A : X

X is a two sided indexed fibration.

Pn : A.n

A are 2-cells in M

ξn1,...,nk : Pn1+...+nk → Pk(Pn1 ◦ Pn2 ◦ ... ◦ Pnk) are 3-cells.

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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. Πn : A.n → A is determined by a functor A(x1, x2) × A(x2, x3) × · · · × A(xn, xn+1)

A(x1, xn+1).

We call this the n-ary composition of the equipment, and for its value at a1, a2, ..., an we write a1a2 · · · an. Π0 : X∗

A is determined by functors

X(x, x) → A(x, x.) These give an object ux in A(x, x) for each x.

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There is a pseudofunctor Catop

M, which is identity

• n objects and sends F : X

Y to F ∗ = Y (F−, −).

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 κn : (F.Πn)Φ.n → Φ(Πn.F), These amout to functors: F : A(X, Y ) → B(X, Y ) and F(a1)...F(an) → F(a1...an).

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Equipments, lax functors and lax transformations be- tween them form a 2-category E.

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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.

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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).

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T-induces a monad on the category Mon(X, A), given

• n 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).

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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.

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This Kleisli construction gives: Mnd(E)(((I, I∗), 1I), ((X, A), T)) → E((I, I∗), (X, T ∗.A)) Or: T-Alg → T-Mon On objects it acts as: ((x, b), h) → (x, hrb)

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Let Inc : E → Mnd(E) be a functor which send an equipment (X, A) to ((X, A), 1X). 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. In our situation the counit (Cmp)(Inc)

1 is an isomor-

• phism. The unit 1

(Inc)(Cmp) is given by the family

• f maps ((X, T ∗.A), 1X) → ((X, A), T).

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The functor T-Mon → T-Alg which is the same as E((I, I∗), (X, T ∗.A)) → Mnd(E)(((I, I∗), 1I), ((X, A), T)) is defined by first taking Inc and then precomposing with ((X, T ∗.A), 1X) → ((X, A), T). On objects it acts as: (x, a) → (Tx, mxTa).

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