2-theories

As in your list of main properties of four functors in the theory of derived categories of sheaves one can informally summarize the known functorial properties of the stable homotopy categories over schemes as follows: for any Noetherian scheme S of finite dimension there is a category SH(S) for any morphism f : S_{1} → S_{2} there are functors

f_{∗} : SH(S_{1}) → SH(S_{2}) f^{∗} : SH(S_{2}) → SH(S_{1})

for any quasi-projective morphism f : S_{1} → S_{2} there are functors f_{!} : SH(S_{1}) → SH(S_{2}) f^{!} : SH(S_{2}) → SH(S_{1})

Adjunctions \begin{matrix}Hom(f_{!}A, B) = Hom(A, f^{!}B) \\ Hom(f^{∗}A, B) = Hom(f_{∗}A, B) \end{matrix}

Base change for any pull-back square \begin{matrix}X′ & →^{g} & X \\ f ↓ & & ↓f \\ Y′ & →^{g} & Y \end{matrix} one has

g^{∗}f_{!} = f_{!}g^{∗} if f is quasi-projective
g^{!}f_{∗} = f_{∗}g^{!} if g is quasi-projective

2-functoriality

(fg)_{∗} = f_{∗}g_{∗} (fg)^{∗} = g^{∗}f^{∗}
(fg)_{!} = f_{!}g_{!} (fg)^{!} = g^{!}f^{!}

Identities 1. For a projective morphism f_{∗} = f_{!}

2. For a smooth quasi-projective morphism f : X → S one has

f^{!} = Σ_{Δ}f^{∗} where Σ_{Δ} = Δ^{∗}pr^{!}, X →^{Δ} X ×_{S} X →^{pr} X.

In addition one has: Gluing axioms For a complimentary pair Z →^{i} X ←^{j} U both pairs of func-

tors (j^{∗}, i^{∗}) and (j^{∗}, i^{!}) reflect isomorphisms A^{1}-homotopy invariance For p : A_{S}^{1} → S the adjunctions

Id → p_{∗}p^{∗} p_{!}p^{!} → Id

are isomorphisms

The categories SH(S) are additive (in fact triangulated by this palys no role below), have small direct sums and sets of compact generators. If SH^{c}(S) the subcategory of compact objects in SH(S) then for any morphism f the functor f^{∗} takes SH^{c} to SH^{c} and for any quasi-projective morphism the functor f_{!} takes SH^{c} to SH^{c}. The following is known for varieties over a field with resolution of singularities and I believe should be true always:

Conjecture 1 For any quasi-projective morphism f the functors f_{∗} and f^{!} take SH^{c} to SH^{c}.

The categories SH and functors f_{∗}, f^{∗}, f_{!}.f^{!} give an example of an A^{1}- homotopy invariant “2-theory” with gluing. The rest of this text contains a formal definition of 2-category of 2-theories 2 − T h. Assuming Conjecture 1 the categories SH^{c} also form a 2-theory and I believe the following:

Conjecture 2 For any Noetherian scheme of finite dimension S, SH^{c} is a universal A^{1}-homotopy invariant 2-theory with gluing in the sense that the functor Φ → Hom(SH^{c}, Φ) from 2−T h_{A^{1},gluing}/S to Cat is equivalent to the functor Φ ↦ Φ(S).

E-functors Let C be a category and D a 2-category. Recall that a 2-functor H from C to D is given by

an object H(X) of D for any object X of C

a 1-morphism H(p) of D for any morphism p of C

a 2-morphism h(p, q) : H(pq) → H(p)H(q) of D for any composable pair of morphisms p, q of C

a 2-morphism h_{X} : H(id_{X}) → Id_{H}(X) for any object X of C

A 1-morphism F from a 2-functor H : C → D to a 2-functor H′ : C → D is given by:

a 1-morphism F_{X} : H(X) → H′(X) for any object X of C

a 2-morphism F_{p} : H′(p)F_{X} → F_{X′}H(p) for any morphism p : X → X′ of C

A 2-morphism ϕ : F → F′ between 1-morphisms of 2-functors is given by

a 2-morphism ϕ_{X} : F_{X} → F′_{X} for any object X of C.

These data should satisfy the usual axioms. 2-functors from C to D, their 1-morphisms and 2-morphisms form a 2-category F unct(C, D) which is strict if D is. A contravariant 2-functor from C to D is a 2-functor from C^{op} to D.

Definition 0.1 An exchange structure on a category C is given by a pair of subcategories E_{v}, E_{h} each of them containing all objects and all isomorphisms of C and a class of commutative squares E with vertical sides in E_{v} and horizontal sides in E_{h} which contains all squares with either vertical or horizontal sides being isomorphisms and is closed under the horizontal and vertical compositions.

An E-functor H : (C, E) → D from C to D with respect to a given exchange data (E_{v}, E_{h}, E) is given by

a covariant 2-functor H_{c} : E_{v} → D and a contravariant 2-functor H : E_{h} → D which coincide on objects

a 2-morphism e_{Q} : H(h_{1})H_{c}(v_{1}) → H_{c}(v_{0})H(h_{0}) for any square Q = ( \begin{matrix}X′ & →^{h_{0}} & X \\ v_{0} ↓ & & ↓v_{1} \\ Y′ & →^{h_{1}} & Y \end{matrix} )

from E These data should satisfy the following four axioms Vertical composition for a vertically composable pair of exchange 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. If D is strict this means that the pentagon

\begin{aligned}H(h_{2})H_{c}(w_{1}v_{1}) & → & H(h_{2})H_{c}(w_{1})H_{c}(v_{1}) \\ & & ↓ \\ ↓ & & H_{c}(w_{0})H(h_{1})H_{c}(v_{1}) \\ & & ↓ \\ H_{c}(w_{0}v_{0})H(h_{0}) & → & H_{c}(w_{0})H_{c}(v_{0})H(h_{0}) \end{aligned} of 2-morphisms commutes

Vertical unit for any morphism h : X′ → X in E_{h} the diagram

commutes in D

i.e. the pentagon of 2-morphisms \begin{aligned}H(h)H_{c}(id_{X′}) & → & H(h)Id_{H(X′)} \\ & & ↓ \\ ↓ & & H(h) \\ & & ↑ \\ H_{c}(id_{X})H(h) & → & Id_{H(X)}H(h) \end{aligned}

commutes. Horizontal composition for a horizontally composable pair of exchange

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. If D is strict this means that the pentagon of 2-

morphisms \begin{aligned}H(h_{1}g_{1})H_{c}(v_{2}) & → & H(g_{1})H(h_{1})H_{c}(v_{2}) \\ & & ↓ \\ ↓ & & H(g_{1})H_{c}(v_{1})H(h_{0}) \\ & & ↓ \\ H_{c}(v_{0})H(h_{0}g_{0}) & → & H_{c}(v_{0})H(g_{0})H(h_{0}) \end{aligned} commutes.

Horizontal unit for any morphism v : X → Y in E_{v} the diagram commutes in D

i.e. the pentagon of 2-morphisms \begin{aligned}H(id_{Y})H_{c}(v) & → & Id_{H(Y)}H_{c}(v) \\ & & ↓ \\ ↓ & & H_{c}(v) \\ & & ↑ \\ H_{c}(v)H(id_{X}) & → & H_{c}(v)Id_{H(X)} \end{aligned}

commutes. A 1-morphism F : (H, H_{c}) → (H′, H′_{c}) is a pair of 1-morphisms of 2-functors F : H → H′, F_{c} : H_{c} → H′ which coincide on objects and such that for any

square Q = ( \begin{matrix}X′ & →^{h_{0}} & X \\ v_{0} ↓ & & ↓v_{1} \\ Y′ & →^{h_{1}} & Y \end{matrix} ) from E the cube

commutes in D. If D is strict this means that the following hexagon of 2-

morphisms \begin{aligned}H′(h_{1})H_{c}(v_{0})F_{X} & → & H′(h_{1})F_{Y} H_{c}(v_{0}) & → & F_{Y′}H(h_{1})H_{c}(v_{0}) \\ e′_{Q}↓ & & & & ↓e_{Q} \\ H′_{c}(v_{1})H′(h_{0})F_{X} & → & H′_{c}(v_{1})F_{X′}H(h_{0}) & → & F_{Y′}H_{c}(v_{1})H(h_{0}) \end{aligned} com-

mutes.

A 2-morphism ϕ : F → F′ is just a pair of 2-morphisms F → F′, F_{c} → F′_{c} coinciding on objects.

Proposition 0.2 E-functors, their 1-morphisms and 2-morphisms form a 2-category F unct_{E}(C, D).

E-functors and the 2-categories of adjunctions Let D be a 2-category. Define the associated 2-categories of adjunctions A(D) as follows. Objects of A(D) are the same as object of D.