2-theories

Lemmas about adjoints Let D be a 2-category which we assume to have strict units and strictly associative composition.

Definition 0.1 A right adjoint to a 1-morphism F : X → Y is a triple F_{r} : Y → X, ϕ_{rl} : Id → F_{r}F, ϕ_{lr} : F F_{r} → Id such that the compositions

F ⟶^{Id∗ϕ_{rl}} F F_{r}F ⟶^{ϕ_{lr}∗Id} F

F_{r} ⟶^{ϕ_{rl}∗Id} F_{r}F F_{r} ⟶^{Id∗ϕ_{lr}} F_{r}

are identity 2-morphisms.

A left adjoint to a 1-morphism F : X → Y is a triple F_{l} : Y → X, ϕ_{rl} : Id → F F_{l}, ϕ_{lr} : F_{l}F → Id such that the compositions

F_{l} ⟶^{Id∗ϕ_{rl}} F_{l}F F_{l} ⟶^{ϕ_{lr}∗Id} F_{l}

F ⟶^{ϕ_{rl}∗Id} F F_{l}F ⟶^{Id∗ϕ_{lr}} F

are identity 2-morphisms.

Lemma 0.2 Consider three 1-morphisms X →^{B} Y →^{A} Z and X →^{C} Z. Then for any right adjoint A_{r} to A there is a canonical bijection

e_{A} : Hom(AB, C) → Hom(B, A_{r}C)

and for any right adjoint B_{r} to B there is a canonical bijection

e_{B} : Hom(C, AB) → Hom(CB_{r}, A)

Proof: Let us prove the second bijection as the less intuitively obvious one. The map e_{B} takes a morphism ϕ : C → AB to the composition CB_{r} →^{ϕ∗Id} ABB_{r} → A. Define e_{B}^{−1} to be the map which takes ψ : CB_{r} → A to the composition C → CB_{r}B →^{ψ∗Id} AB. We have two comutative diagrams

\begin{aligned}C & → & AB \\ ↓ & & ↓ \\ CB_{r}B & → & ABB_{r}B \\ & & ↓ \\ & & AB \end{aligned}  \begin{matrix}CB_{r} \\ ↓ \\ CB_{r}B & → & ABB_{r} \\ ↓ & & ↓ \\ CB_{r} & → & A \end{matrix}

where the compositions of vertical arrows are identities by definition of adjoints. They show that e_{B}^{−1}e_{B} = Id and e_{B}e_{B}^{−1} = Id.

Lemma 0.3 Consider a square of 1-morphisms \begin{matrix}X & →^{B} & Y \\ D ↓ & & ↓A \\ Y′ & →^{C} & Z \end{matrix} and let A_{r} be

a right adjoint to A and D_{r} a right adjoint to D. Then the following square of bijections constructed in Lemma 0.2 commutes

Hom(AB, CD) → Hom(B, A_{r}CD)
↓  ↓
Hom(ABD_{r}, C) → Hom(BD_{r}, A_{r}C)

The obvious left analogs of Lemmas 0.2,0.3 hold. Consider now a more general situation. Let

X = X_{0} →^{A_{1}} X_{1} →^{A_{2}} … →^{A_{n−1}} X_{n−1} →^{A_{n}} X_{n} = Y

X = X′_{0} →^{A_{n+1}} X′_{1} →^{A_{n+2}} … →^{A_{n+m−1}} X′_{n−1} →^{A_{n+m}} X′_{m} = Y

be two composable sequences of 1-morphisms with the same source and target and assume that for each of the A′_{i}s we are given infinite sequences of left and right adjoints which we denote according to the rule

A_{i,0} = A_{i} A_{i,n} = (A_{i,n−1})_{r}

Consider pairs of composable sequences

(A_{i_{1},m_{1}} … A_{i_{k},m_{k}}; A_{j_{1},n_{1}} … A_{j_{l},n_{l}}) m, n ≥ 0

such that the source of A_{i_{1},m_{1}} … A_{i_{k},m_{k}} coincides with the source of A_{j_{1},n_{1}} … A_{j_{l},n_{l}} and the target of A_{i_{1},m_{1}} … A_{i_{k},m_{k}} coincides with the target of A_{j_{1},n_{1}} … A_{j_{l},n_{l}} such that the set of 2-morphisms Hom(A_{i_{1},m_{1}} … A_{i_{k},m_{k}}, A_{j_{1},n_{1}} … A_{j_{l},n_{l}}) is defined. Consider the equivalence realtions on the set of such pairs generated by elementary relations of the form

(A_{i1,m1}...A_{ik,mk};A_{j1,n1}...A_{jl,nl})≅(A_{i2,m2}...A_{ik,mk};A_{i1,m1+1}A_{j1,n1}...A_{jl,nl})
(A_{i1,m1}...A_{ik,mk};A_{j1,n1}...A_{jl,nl})≅(A_{i1,m1}...A_{ik,mk} A_{jl,nl+1};A_{i1,m1+1}A_{j1,n1}...A_{jl−1,nl−1})

By Lemma 0.2 any sequence of elementary equivalences from one pair to another defines a bijection of the sets of 2-morphisms between the corresponding compositions of A_{∗,∗}’s. Lemma 0.3 implies the following “coherence” result.

Proposition 0.4 For two equivalent pairs (A_{i_{1},m_{1}} … A_{i_{k},m_{k}}; A_{j_{1},n_{1}} … A_{j_{l},n_{l}}), (A_{i′_{1},m′_{1}} … A_{i′_{k′},m′_{k′}}; A_{j′_{1},n′_{1}} … A_{j′_{l′},n′_{l′}}) the bijection

Hom(A_{i_{1},m_{1}} … A_{i_{k},m_{k}}; A_{j_{1},n_{1}} … A_{j_{l},n_{l}}) → Hom(A_{i′_{1},m′_{1}} … A_{i′_{k′},m′_{k′}}; A_{j′_{1},n′_{1}} … A_{j′_{l′},n′_{l′}})

does not depend on the choice of the sequence of elementary equivalences connecting the pairs. The same holds when only a finite number of left and right adjoints to each of the morphisms A_{i} is given.

Instead of giving a formal proof of this result we give an illustration. Consider

a square of 1-morphisms \begin{matrix}X & →^{B} & Y \\ D ↓ & & ↓A \\ Y′ & →^{C} & Z \end{matrix} as in Lemma 0.3 but assume now that

for all of them we are given infinite sequences of left and right adjoints. Then the equivalence class of the pair (AB, CD) will look as follows

...→(AllBllDlCl,∗)

↓

...→(BllDlCl,Al)  → (BllDlClA,∗)
↓  ↓
...→(DlCl,BlAl)  →  (DlClA,Bl)  → (DlClAB,∗)
↓  ↓  ↓
...→(Cl,DBlAl)  →  (ClA,DBl)  →  (ClAB,D)  →  (ClABDr,∗)
↓  ↓  ↓  ↓
(∗,CDBlAl)  →  (A,CDBl)  →  (AB,CD)  →  (ABDr,C)  →  (ABDrCr,∗)
↓  ↓  ↓  ↓
(∗,ArCDBl) →  (B,ArCD)  →  (BDr,ArC)  →  (BDrCr,Ar)→...
↓  ↓  ↓
(∗,BrArCD) →  (Dr,BrArC)  →  (DrCr,BrAr)→...
↓  ↓
(∗,DrrBrArC) →  (Cr,DrrBrAr)→...
↓
(∗,CrrDrrBrAr)→...

Each vertice denotes a set of 2-morphisms, each arrow is a bijection of Lemma 0.2 and each square is the square of Lemma 0.3. The right column of this diagram is twice right adjoint to the left one and the whole diagram extends “periodically” to the right and to the left. If we consider the situation of more than four functors (say of a pentagon (ABC, DE) instead of a square (AB, CD)) the digram will become wider in the vertical dimension but other than that will have the same shape. If we are given only finitely many adjoints for each of the morphisms part of the vertices will disappear but always in such a fashion that each of the newly formed connected components is a union of “elementary” squares and the coherence theorem can be again deduced from Lemma 0.3.

2-category of 2-theories Let S be a scheme and Sch^{ft}/S the category of schemes of finite type over S. A 2-theory H on Sch^{ft}/S with values in a 2-category D is defined as follows:

Data

1. 2-functors

\begin{matrix}H^{∗} : (QP/S)^{op} → D & H_{∗} : QP/S → D \\ H_{!} : QP/S → D & H^{!} : (QP/S)^{op} → D \end{matrix}

which coincide on objects

2. 1-morphisms of 2-functors

\begin{matrix}Id → H_{∗}H^{∗} & H^{∗}H_{∗} → Id \\ Id → H^{!}H_{!} & H_{!}H^{!} → Id \end{matrix}

satisfying axioms of adjunctions

3. for any pull back square Q = ( \begin{matrix}X′ & →^{h_{0}} & X \\ v_{0} ↓ & & ↓v_{1} \\ Y′ & →^{h_{1}} & Y \end{matrix} ) a 2-morphism

e_{Q} : H_{!}(v_{1})H_{∗}(h_{0}) → H_{∗}(h_{1})H_{!}(v_{0})

Axioms

Level 0

1. for a composable pair of morphisms f, g

H^{∗}(f)H^{∗}(g) → H^{∗}(gf) is adjoint to H_{∗}(fg) → H_{∗}(g)H_{∗}(f)

H_{!}(g)H_{!}(f) → H_{!}(gf) is adjoint to H^{!}(gf) → H^{!}(f)H^{!}(g)

2. for an object X H^{∗}(id_{X}) → Id_{H(X)} is adjoint to Id_{H(X)} → H_{∗}(id_{X})

H_{!}(id_{X}) → Id_{H(X)} is adjoint to Id_{H(X)} → H^{!}(id_{X})

3. for a horizontally composable pair of of pull back squares \begin{matrix}X′′ & →^{g_{0}} & X′ & →^{h_{0}} & X \\ v_{0} ↓ & & v_{1} ↓ & & ↓v_{2} \\ Y′′ & →^{g_{1}} & Y′ & →^{h_{1}} & Y \end{matrix} the diagram commutes in D.

4. for a vertically composable pair of pull-back squares \begin{matrix}X′ & →^{h_{0}} & X \\ v_{0} ↓ & & ↓v_{1} \\ Y′ & →^{h_{1}} & Y \\ w_{0} ↓ & & ↓w_{1} \\ Z′ & →^{h_{2}} & Z \end{matrix}

the diagram commutes in D.

5. for a morphism v : X → Y the diagram commutes in D 6. for a morphism h : X′ → X in E_{h} the diagram

commutes in D Level 1

1. for any pull-back square Q as above the left base change morphism

e_{l,Q} : H^{∗}(h_{1})H_{!}(v_{1}) → H_{!}(v_{0})H^{∗}(h_{0})

and the right base change morphism

e_{r,Q} : H_{∗}(h_{0})H^{!}(v_{0}) → H^{!}(v_{1})H_{∗}(h_{1})

adjoint to e_{Q} are isomorphisms

2. for any pull back square Q such that v_{0}, v_{1} are projective the exchange morphism e_{Q} is an isomorphism

3. for any pull-back square Q such that v_{0}, v_{1} are smooth the inverse

exchange morphism

e_{Q}^{inv} : H^{∗}(h_{0})H^{!}(v_{1}) → H^{!}(v_{0})H^{∗}(h_{1})

adjoint to e_{l,Q}^{−1} or equivalently to e_{r,Q}^{−1} is an isomorphism. A 1-morphism F : H → G is defined as follows:

Data

Four 1-morphisms of 2-functors

\begin{matrix}F^{∗} : H^{∗} → G^{∗} & F_{∗} : H_{∗} → G_{∗} \\ F_{!} : H_{!} → G_{!} & F^{!} : H^{!} → G^{!} \end{matrix}

which coincide on objects.

Axioms 1. for any morphism f : X → Y

G^{∗}(f)F_{Y} → F_{X}H^{∗}(f) is adjoint to F_{Y} H_{∗}(f) → G_{∗}(f)F_{X}

G_{!}(f)F_{Y} → F_{X}H_{!}(f) is adjoint to F_{Y} H^{!}(f) → G^{!}(f)F_{X}

2. for any pull back square Q = ( \begin{matrix}X′ & →^{h_{0}} & X \\ v_{0} ↓ & & ↓v_{1} \\ Y′ & →^{h_{1}} & Y \end{matrix} ) the cube

with vertical sides of the form H_{!}, G_{!} and horizontal of the form H_{∗}, G_{∗} commutes in D.

Lemma 0.5 Let F : H_{1} → H_{2}, G : H_{2} → H_{3} be a pair of 1-morphisms of 2-theories. Then the 1-morphisms of 2-functors G^{∗}F^{∗}, G_{∗}F_{∗}, G^{!}F^{!}, G_{!}F_{!} satisfy the axioms of a 1-morphism of 2-theories.

The 1-morphism GF : H_{1} → H_{3} defined according to Lemma 0.5 by four 1-morphisms of 2-functors G^{∗}F^{∗}, G_{∗}F_{∗}, G^{!}F^{!}, G_{!}F_{!} is called the composition of F and G. The identity 1-morphism of a 2-theory is given by the identity 1-morphisms of the corresponding four 2-functors.

A 2-morphism ϕ : F → F′ is just four 2-morphisms of 1-morphisms between 2-functors which coincide on objects. Compositions 2-morphisms are defined as compositions of the corresponding 2-morphisms between 2functors. The identity 2-morphism of a 1-morphism is the identity 2-morphism of the corresponding four 1-morphisms of 2-functors.

Proposition 0.6 2-theories on Sch^{ft}/S with values in a given strict 2category D, their 1-morphisms, compositions of 1-morphisms, 2-morphisms compositions of 2-morphisms and identities form a 2-category T T (S, D).