An introduction to Yoneda structures

An introduction to Yoneda structures Paul-André Melliès CNRS, Université Paris Denis Diderot Groupe de travail Catégories supérieures, polygraphes et homotopie Paris 21 May 2010 1

Bibliography Ross Street and Bob Walters Yoneda structures on 2-categories Journal of Algebra 50:350-379, 1978 Mark Weber Yoneda structures from 2-toposes Applied Categorical Structures 15:259-323, 2007 2

Covariant and contravariant presheaves A few opening words on Isbell conjugacy 3

Ideal completion Every partial order A generates a free complete -lattice P A A P A whose elements are the downward closed subsets of A, with ϕ P A ψ ϕ ψ. P A = A op {0, 1} 4

Free colimit completions of categories Every small category A generates a free cocomplete category P A A P A whose elements are the presheafs over A, with ϕ P A ψ ϕ natural ψ. P A = A op Set 5

Contravariant presheaves ϕ(y ) y ϕ(x) ϕ(f) x Y C f X Replaces downward closed sets 6

Filter completion Every partial order A generates a free complete -lattice Q A A Q A whose elements are the upward closed subsets of A, with ϕ Q A ψ ϕ ψ. Q A = ( A {0, 1} ) op 7

Free limit completions of categories Every small category A generates a free complete category Q A A Q A whose elements are the covariant presheafs over A, with ϕ Q A ψ ϕ natural ψ. Q A = ( P A op ) op = ( A Set ) op 8

Covariant presheaves ϕ(y ) y ϕ(x) ϕ(f) x Y C f X Replaces upward closed sets 9

Related to the Dedekind-MacNeille completion A Galois connection L P A Q A L(ϕ) = { y x ϕ, x A y } R(ψ) = { x y ψ, x A y } R ϕ R(ψ) x ϕ, y ψ, x A y L(ϕ) ψ The completion keeps the pairs (ϕ, ψ) such that ψ = L(ϕ) and ϕ = R(ψ) 10

The Isbell conjugacy An adjunction L P A Q A L(ϕ) : Y P A(ϕ, Y) = ϕ(x) hom(x, Y) x A R(ψ) : X Q A(X, ψ) = ψ(y) hom(x, Y) Y A R P A(ϕ, R(ψ)) X,Y A ϕ(x) ψ(y) hom(x, Y) Q A(L(ϕ), ψ) 11

Yoneda structures An axiomatic approach by Street and Walters 12

General idea Suppose a universe U 1 inside a larger universe U 2. Set is the category of sets in the universe U 1, CAT is the 2-category of categories in the universe U 2. In particular, the category Set is an object of CAT. 13

Admissible functors An object A of CAT is called admissible when its homsets A (A, A ) are all in the category Set. More generally, an arrow in CAT F : A B is called admissible when the homsets B (FA, B) are all in the category Set. 14

Yoneda structures A Yoneda structure in a 2-category K is defined as a class of admissible arrows such that the composite arrow A F B G C is admissible whenever the arrow B G C is admissible. 15

Admissible objects An object A is called admissible when the identity arrow is admissible. id A : A A In the case of the 2-category CAT equipped with its Yoneda structure: admissible objects = locally small categories 16

Yoneda structures every admissible object A induces an admissible arrow y A : A P A 17

Yoneda structures every admissible arrow F : A B from an admissible object A induces a diagram A y P A χ F F B(F,1) B 18

In the traditional case A y P A χ F F B(F,1) B B(F, 1) : b λa. B(Fa, b) χ F a 2 : λa 1. A (a 1, a 2 ) λa 1. B (Fa 1, Fa 2 ) 19

In the traditional case A y P A χ F F B(F,1) B B(F, 1) : b λa. B(Fa, b) χ F a 1 a 2 : A (a 1, a 2 ) B (Fa 1, Fa 2 ) 20

Axiom 1 For A and F accessible, the 2-cell A y P A χ F F B(F,1) exhibits the arrow B(F, 1) as a left extension of y A along the arrow F. B 21

Axiom 1 In other words, every 2-cell factors uniquely as A A y y P A σ F P A χ F F B θ B ϕ ϕ 22

In the traditional case A natural transformation A y P A σ F B ϕ is the same thing as a family of functions σ a1 a 2 : A (a 1, a 2 ) ϕ (Fa 2 ) (a 1 ) natural in a 1 contravariantly and in a 2 covariantly. 23

In the traditional case A natural transformation A y P A σ F B ϕ is the same thing as a family of functions σ a1 a 2 : A (a 1, a 2 ) ϕ (Fa 2 ) (a 1 ) natural in a 1 contravariantly and in a 2 covariantly. 24

In the traditional case A natural transformation θ in A y P A χ F F is the same thing as a family of functions θ B ϕ θ a b : B (Fa, b) ϕ (b) (a) natural in a contravariantly and in b covariantly. 25

In the traditional case This means that every natural transformation σ a1 a 2 : A (a 1, a 2 ) ϕ (Fa 2 ) (a 1 ) factors as χ F a 1 a 2 : A (a 1, a 2 ) B (Fa 1, Fa 2 ) followed by θ a1 Fa 2 : B (Fa 1, Fa 2 ) ϕ (Fa 2 ) (a 1 ) for a unique natural transformation θ which remains to be defined. 26

Existence Given the natural transformation σ, the natural transformation transports every morphism θ a b : B (Fa, b) ϕ (Fa) (b) f : F a b to the result of the action of f on the element σ a a ( a id a ) ϕ (Fa) (a) 27

Uniqueness... 28

Axiom 2 For A and F accessible, the 2-cell A y P A χ F F B(F,1) exhibits the arrow F as an absolute left lifting of y A through B(F, 1). B 29

Axiom 3(i) For A accessible, the identity 2-cell A y P A id y id P A exhibits the identity arrow as a left extension of y A along y A. 30

Axiom 3(ii) For A, B, F, G accessible, the 2-cell A y P A F χ B(1,F) F B y P B χ G G C(G,1) exhibits the arrow F C(G, 1) as a left extension of y A along G F. C 31

The inverse arrow Given an arrow between admissible objects A and B F : A B the arrow F : P B P A is defined as follows: y A F P A χ B (1,F) F = P B ( B (1,F), 1 ) B y P B where B (1, F) denotes the composite arrow y F. 32

The 2-dimensional cell A F The inverse arrow y P A χ B(1,F) factors as a pair of Kan extensions: F = P B ( B (1,F), 1 ) B y P B y A F P A χ F B B(F,1) B(F,1) P A B F y P B 33

Existential image Given an arrow between admissible objects A and B F : A B the arrow F : P A P B is defined as follows: P A F P B A F y B y 34

Universal image Given an arrow between admissible objects A and B the arrow is defined as follows: F : A B F : P A P B A y P A χ F F F χ B(F,1) y B P B 35

Monads with arity An idea by Mark Weber 36

Category with arity A fully faithful and dense functor where Θ 0 is a small category. i 0 : Θ 0 A 37

Category with arity This induces a fully faithful functor A(i 0, 1) : A P Θ 0 which transports every object A of the category A into the presheaf A(i 0, A) : Θ 0 Set p A(i 0 p, A) 38

Monads with arity A monad T on a category with arity (A, i 0 ) such that: (1) the natural transformation exhibits the functor T T i 0 A id Θ 0 i 0 as a left kan extension of the functor T i 0 along the functor i 0, T A 39

Monads with arity (2) this left kan extension is preserved by the inclusion functor to P Θ 0. P Θ 0 A(i 0,1) A T i 0 T id Θ 0 i 0 A 40

Equivalently... The natural transformation P Θ 0 A(i 0,1) A T i 0 T id Θ 0 i 0 exhibits A(i 0, 1) T as a left kan extension of A(i 0, 1) T i 0 along i 0. A 41

Equivalently... For every object A, the canonical morphism p Θ0 A(i0 n, Ti 0 p) A(i 0 p, A) A(i 0 n, TA) is an isomorphism. 42

Unique factorization up to zig-zag Every morphism in the category A decomposes as i 0 n TA i 0 n e Ti 0 p T f TA for a pair of morphisms e : i 0 n Ti 0 p and f : i 0 p A. 43

Unique factorization up to zig-zag The factorization should be unique up to zig-zag of i 0 n e 1 e 2 Ti 0 p Ti 0 q T f 1 Ti 0 u T f 2 TA 44

An abstract Segal condition A general theorem by Mark Weber axiomatizing a theorem by Clemens Berger on higher dimensional categories 45

Motivating example: the free category monad Cat 0 Graph 46

Morphism between categories with arity A morphism between categories with arity (F, l) : (A, i 0 ) (B, i 1 ) is defined as a pair of functors (F, l) making the diagram Θ 1 i 1 B l F commute. Θ 0 i 0 A 47

Morphism between categories with arity This induces a commutative diagram P Θ 1 l P Θ 0 i 1 id i 0 P B F P A which is required to be an exact square in the sense of Guitart. 48

Morphism between categories with arity This means that the Beck-Chevalley condition holds, which states that the canonical natural transformation P Θ 1 i1 P B l F P Θ 0 i0 P A is reversible. 49

Proposition A. For every morphism (F, l) between categories with arity (F, l) : (A, i 0 ) (B, i 1 ) the adjunction i 1 i 1 induces an adjunction between the full subcategory of presheaves of B whose restriction along F is representable in A, the full subcategory of presheaves of Θ 1 whose restriction along l is representable along i 0. Moreover, this adjunction defines an equivalence when the functor F is essentially surjective. 50

Proposition B. Every monad T with arity i 0 induces a commutative diagram Θ T i T A T l F Θ 0 i 0 where the pair (F, l) defines a morphism (F, l) : (A, i 0 ) (A T, i T ) of categories with arity. A 51

Algebraic theories with arity A 2-dimensional approach to Lawvere theories 52

Algebraic theory with arity An algebraic theory L with arity i 0 : Θ 0 A is an identity-on-object functor L : Θ 0 Θ L such that the endofunctor P Θ 0 L P Θ L L P Θ 0 maps a presheaf representable along i 0 to a presheaf representable along i 0. 53

Algebraic theories with arity A(i 0,X) Set Θ op 0 L Kan extension Θ op L 54

Model of an algebraic theory A model A of a Lawvere theory L is a presheaf A : Θ op L such that the induced presheaf Set Θ op 0 i 0 Θ op L A Set is representable along i 0. 55

Main theorem the category of algebraic theories with arity (A, i 0 ) is equivalent to the category of monads with arity (A, i 0 ) 56