A Sheaf Cohomology Theory for C*-Algebras

∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

A sheaf cohomology theory for C*-algebras

Martin Mathieu

(Queen’s University Belfast)

Banach Algebras 2017 at Oulu on 6 July 2017

Partially supported by UK Engineering and Physical Sciences Research Council Grant No. EP/M02461X/1.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

(Commutative) Topology

∗com category of unital commutative C*-algebras with unital C1 *-homomorphisms as morphisms;

Comp category of compact Hausdorﬀ spaces with continuous mappings as morphisms.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

(Commutative) Topology

∗com category of unital commutative C*-algebras with unital C1 *-homomorphisms as morphisms;

Comp category of compact Hausdorﬀ spaces with continuous mappings as morphisms.

∗com Comp C1

f C C(f ) X ; X −→ Y −→ C(X ); C(Y ) −→ C(X ) C(f )(g) = g ◦ f , g ∈ C(Y )

πˆ Aˆ; Bˆ −→ Aˆ ←−ˆ A; A −→π B πˆ(ϕ) = ϕ ◦ π, ϕ ∈ Bˆ

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Noncommutative Topology

∗ C1 category of unital C*-algebras with unital *-homomorphisms as morphisms;

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Noncommutative Topology

∗ C1 category of unital C*-algebras with unital *-homomorphisms as morphisms; ?

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Noncommutative Topology

A C*-algebra

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Noncommutative Topology

A C*-algebra Prim(A) space of primitive ideals of A with hull-kernel topology T OPrim(A) 3 U A(U) = t∈/U t;

I ¢ A U(I ) = {t ∈ Prim(A) | I * t} ∈ OPrim(A).

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Noncommutative Topology

A C*-algebra Prim(A) space of primitive ideals of A with hull-kernel topology T OPrim(A) 3 U A(U) = t∈/U t;

I ¢ A U(I ) = {t ∈ Prim(A) | I * t} ∈ OPrim(A). hence V ⊆ U ⇒ A(V ) ⊆ A(U);

ringed space( A, X = Spec(A), OX ) “C*-ringed space” (A, Prim(A), A), A is a sheaf of C*-algebras

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Noncommutative Topology

A C*-algebra Prim(A) space of primitive ideals of A with hull-kernel topology T OPrim(A) 3 U A(U) = t∈/U t;

I ¢ A U(I ) = {t ∈ Prim(A) | I * t} ∈ OPrim(A). hence V ⊆ U ⇒ A(V ) ⊆ A(U);

ringed space( A, X = Spec(A), OX ) “C*-ringed space” (A, Prim(A), A), A is a sheaf of C*-algebras Note A separable ⇒ closed prime ideals are primitive

P. Ara, M. Mathieu, Sheaves of C*-algebras, Math. Nachrichten 283 (2010), 21–39.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Sheaves of C*-algebras

Example 1. The multiplier sheaf A C*-algebra with primitive ideal space Prim(A);

∗ MA : OPrim(A) → C1 , MA(U) = M(A(U)),

where M(A(U)) denotes the multiplier algebra of the closed ideal A(U) of A associated to the open subset U ⊆ Prim(A). M(A(U)) → M(A(V )), V ⊆ U, the restriction homomorphisms.

Proposition

The above functor MA deﬁnes a sheaf of C*-algebras.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Sheaves of C*-algebras

Example 2. The injective envelope sheaf let I (B) denote the injective envelope of B;

∗ IA : OPrim(A) → C1 , IA(U) = pU I (A) = I (A(U)),

where pU = pA(U) denotes the unique central open projection in I (A) such that pA(U)I (A) is the injective envelope of A(U).

I (A(U)) → I (A(V )), V ⊆ U, given by multiplication by pV (as pV ≤ pU ).

{pU | U ∈ OPrim(A)} is a complete Boolean algebra isomorphic to the Boolean algebra of regular open subsets of Prim(A), and it is precisely the set of projections of the AW*-algebra Z(I (A)).

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

The Aim: Sheaf Cohomology

Hi (X , F), i ∈ N where X is a topological space, A a sheaf of C*-algebras on X and ∞ F ∈ OModA (X ).

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

The Aim: Sheaf Cohomology

Hi (X , F), i ∈ N where X is a topological space, A a sheaf of C*-algebras on X and ∞ F ∈ OModA (X ).

joint work in progress with Pere Ara

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras • every morphism has a kernel and a cokernel; • every morphism can be uniquely factorised as

ϕ E / F > π µ

where π is an epimorphism and µ is a monomorphism.

∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

A key property in abelian categories

in an abelian category,

• the morphism set between any pair of objects is an abelian group;

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras • every morphism can be uniquely factorised as

ϕ E / F > π µ

where π is an epimorphism and µ is a monomorphism.

∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

A key property in abelian categories

in an abelian category,

• the morphism set between any pair of objects is an abelian group; • every morphism has a kernel and a cokernel;

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

A key property in abelian categories

in an abelian category,

• the morphism set between any pair of objects is an abelian group; • every morphism has a kernel and a cokernel; • every morphism can be uniquely factorised as

ϕ E / F > π µ

where π is an epimorphism and µ is a monomorphism.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Operator modules

Deﬁnition A right A-module E which at the same time is an operator space is a right operator A-module if it satisﬁes either: (a) There exist a complete isometry Φ: E −→ B(H, K), for some Hilbert spaces H, K, and a *-homomorphism π : A −→ B(H) such that Φ(x · a) = Φ(x)π(a) for all x ∈ E, a ∈ A. (b) The bilinear mapping E × A −→ E,(x, a) 7→ x · a extends to a complete contraction E ⊗h A −→ E. (c) For each n ∈ N, Mn(E) is a right Banach Mn(A)-module in the canonical way. E is nondegenerate if the linear span of {x · a | x ∈ E, a ∈ A} is dense in E, and it is unital if A is unital and x · 1 = x for all x ∈ E.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Our kind of categories

Categories we need to consider (for a C*-algebra A):

∞ OModA the category with objects the nondegenerate right operator A-modules and morphisms the completely bounded A-module maps;

1 ∞ OModA the subcategory of OModA with morphisms the completely contractive A-module maps.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Our kind of categories

Categories we need to consider (for a C*-algebra A):

∞ OModA the category with objects the nondegenerate right operator A-modules and morphisms the completely bounded A-module maps; additive, ﬁnitely bicomplete

1 ∞ OModA the subcategory of OModA with morphisms the completely contractive A-module maps. non-additive, bicomplete None of these categories is abelian!

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

A kernel

Let f ∈ MorC(A, B) for some A, B ∈ C. A morphism i : K → A is a kernel of f if ﬁ = 0 and for each D ∈ C and g ∈ MorC(D, A) with fg = 0 there is a unique h ∈ MorC(D, K) making the diagram below commutative

D 0 h g

i f K / A 6/ B 0

Any kernel is a monomorphism and is, up to isomorphism, unique.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

A cokernel

Let f ∈ MorC(A, B) for some A, B ∈ C. A morphism p : B → C is a cokernel of f if pf = 0 and for each D ∈ C and g ∈ MorC(B, D) with gf = 0 there is a unique h ∈ MorC(C, D) making the diagram below commutative

0 A / B /( C f p g h 0 & D

Any cokernel is an epimorphism and is, up to isomorphism, unique.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Our kind of categories

the situation in our categories: monomorphisms are injective A-module maps epimorphisms are A-module maps with dense range

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Our kind of categories

the situation in our categories: monomorphisms are injective A-module maps epimorphisms are A-module maps with dense range

1 OModA T : E → F kernel iﬀ is it a complete isometry; T : E → F cokernel iﬀ it is a complete quotient map.

∞ OModA T : E → F kernel iﬀ it is a completely bounded isomorphism onto its image; T : E → F cokernel iﬀ it is surjective and completely open.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Homological Algebra

fundamental concept “short exact sequence”

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Homological Algebra

fundamental concept “short exact sequence”

α β ModA 0 / E / F / G / 0 where E, F , G are modules over a ring A and ker α = 0, im β = G and im α = ker β.

M P OModA E / / F / / G where E, F , G are right operator A-modules over the C*-algebra A and M = Ker P and P = Coker M.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Homological Algebra

fundamental concept “short exact sequence”

α β ModA 0 / E / F / G / 0 where E, F , G are modules over a ring A and ker α = 0, im β = G and im α = ker β.

M P OModA E / / F / / G “kernel–cokernel pair” where E, F , G are right operator A-modules over the C*-algebra A and M = Ker P and P = Coker M.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Exact categories let C be an additive category;

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Exact categories let C be an additive category; if C is abelian every monomorphism is a kernel and every epimorphism is a cokernel;

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Exact categories let C be an additive category; A kernel–cokernel pair (M, P) consists of two composable morphisms in C such that M = Ker P and P = Coker M,

M P E1 / / E2 / / E3

where Ei ∈ C. A monomorphism M arising in such a pair is called admissible

E / /F

and an epimorphism arising in such a pair is called admissible

E / /F

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Exact categories (`ala Quillen)

An exact structure on an additive category C is a class of kernel–cokernel pairs, closed under isomorphisms, satisfying the following axioms.

[E0] ∀ E ∈ C : 1E is an admissible monomorphism; op [E0 ] ∀ E ∈ C : 1E is an admissible epimorphism; [E1] the class of admissible monomorphisms is closed under composition; [E1op ] the class of admissible epimorphisms is closed under composition; [E2] the push-out of an admissible monomorphism along an arbitrary morphism exists and yields an admissible monomorphism; [E2op ] the pull-back of an admissible epimorphism along an arbitrary morphism exists and yields an admissible epimorphism.

Th. Buhler¨ , Exact categories, Expo. Math. 28 (2010), 1–69.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

An exact structure

Deﬁnition

∞ Let A be a C*-algebra. We endow the additive category OModA with the exact structure of all kernel–cokernel pairs

M P E1 / / E2 / / E3

∞ where Ei ∈ obj(OModA ), 1 ≤ i ≤ 3, M is a monomorphism in ∞ OModA with closed range and completely bounded inverse, P is a ∞ completely open mapping in OModA (in particular, surjective) and ker P = im M.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

An exact structure

Theorem Let A be a C*-algebra. The class of all kernel–cokernel pairs in ∞ ∞ OModA is an exact structure on OModA .

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps what is next?

. . . now do the same for sheaves of operator modules

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Sheaves in categories

X topological space; OX category of open subsets (with open subsets U as objects and V → U if and only if V ⊆ U). C bicomplete category with equalisers and coequalisers.

Deﬁnition

A presheaf on X in C is a contravariant functor F: OX → C. A sheaf with values in C is a presheaf F such that F(∅) = 0 and, S for every open subset U of X and every open cover U = i Ui , the maps F(U) → F(Ui ) are the limit of the diagrams F(Ui ) → F(Ui ∩ Uj ) for all i, j.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Limits in categories

Let I −→ C be a small diagram; we write Ei ∈ C for the image of an object i ∈ I and, if ϕij : i → j is a morphism, we denote its

image by Tϕij : Ei → Ej . An object L ∈ C together with morphisms πi : L → Ei , i ∈ I is a limit of the diagram if they make the diagram below commutative and is ﬁnal with this property.

E :5 i ρi πi 0 L / L Tϕij πj ρj $ ) Ej Also called an inverse limits or projective limit.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Sheaves in categories S Let U ∈ OX . Let I be a set and U = i∈I Ui with Ui ∈ OX . Let the index category I (2) consist of unordered pairs of elements in I (we allow the case of singleton sets in I (2)) where {i, j} → {k, `} if {i, j} ⊆ {k, `}. Consider the diagram

(2) (2) I → {Uij = Ui ∩ Uj | {i, j} ∈ I } ⊆ OU , {i, j} 7→ Uij

composed with the functor F: OX → C. Then F(U) is the limit of the diagram

F(Ui ) 7 ) F(U) F(Ui ∩ Uj ) 5 ' F(Uj )

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Sheaves in categories

F(Ui0 ) / F(Ui0 ∩ Ui1 ) : O 7 O µi0i1

ρ Q µ Q F(U) / F(Ui ) / F(Ui ∩ Ui ) ν / 0 1 i∈I (i0,i1)∈I ×I

νi0i1 $ ' F(Ui1 ) / F(Ui0 ∩ Ui1 )

The sheaf property is the requirement that the morphism ρ is the equaliser of the pair (µ, ν).

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Sheaves in concrete categories

Notation and Terminology: suppose C is a concrete category

the elements of F(U) are called sections over U ∈ OX ;

by s|V , V ⊆ U open, we mean the “restriction” of s ∈ F(U) to V ; i.e., the image of s in F(V ) under ρVU : F(U) → F(V ); the unique gluing property of a sheaf can be expressed as follows:

for each compatible family of sections si ∈ F(Ui ), i.e., si = sj for all i, j, there is a unique section s ∈ F(U) |Ui ∩Uj |Ui ∩Uj

such that s|Ui = si for all i.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Sheaves in categories

F, G (pre)sheaves on X morphism ϕ: F → G is a natural transformation, i.e.,

ϕU F(U) / G(U)

ρ 0 VU ρVU F(V ) / G(V ) ϕV

is commutative for all U, V ∈ OX , V ⊆ U; hence we have the categories of sheaves on X , Sh(X , C), and of presheaves, PSh(X , C).

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Sheaves in categories

Stalks F presheaf on X , t ∈ X ; stalk of F at t is deﬁned as F = lim F(U), t −→ Ut

where Ut denotes the downward directed family of open neighbourhoods of t and lim denotes the (directed) colimit in C. −→ With stalks we can build bundles and with bundles we can get sheaves again.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Sheaves in categories

An Example

Let F be a presheaf on X . For each U ∈ OX , put p Q F (U) = t∈U Ft and, for V ⊆ U, set ρVU the canonical Q Q morphism from t∈U Ft → t∈V Ft . In this way we obtain the product sheaf associated with F.

p We shall assume that the canonical morphism σU : F(U) → F (U) is a monomorphism whenever F is a sheaf.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras Let ϕ: F → G be a morphism of sheaves. Then

(i) ϕ is a monomorphism if ϕt is a monomorphism for all t ∈ X ; (ii) ϕ is an epimorphism if ϕt is an epimorphism for all t ∈ X ; (iii) ϕ is an isomorphism only if ϕt is an isomorphism for all t ∈ X .

∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Sheaves in categories The stalk functor Suppose F and G are presheaves on X and ϕ: F → G is a morphism of presheaves. For each t ∈ X there is a unique morphism ϕt :Ft → Gt . In this way, we obtain the stalk functor at t : PSh(X , C) → C. Properties: Let ϕ(1), ϕ(2) : F → G be morphisms of sheaves. Then (1) (2) (1) (2) ϕ = ϕ if and only if ϕt = ϕt for all t ∈ X .

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

“C*-ringed spaces”

Sheaves of operator modules over sheaves of C*-algebras X topological space, A sheaf of C*-algebras on X

E: OX −→ Ban1 (sheaf in Ban1) right operator A-module on X

if, for each U ∈ OX , E(U) is a (nondegenerate) right operator A(U)-module and, for U, V ∈ OX with V ⊆ U,

TVU : E(U) → E(V ) is completely contractive and

TVU (x · a) = TVU (x) · πVU (a)(x ∈ E(U), a ∈ A(U)),

where πVU : A(U) → A(V ) are the restriction maps in A.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

The categories we are interested in Categories we need to consider (for a given sheaf A of C*-algebras on X ): ∞ OModA (X ) the category with objects the right operator A-modules and morphisms (at U) the completely bounded A(U)-module maps;

1 ∞ OModA(X ) the subcategory of OModA (X ) with the same objects and morphisms (at U) the completely contractive A(U)-module maps.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

1 ∞ OModA(X ) the subcategory of OModA (X ) with the same objects and morphisms (at U) the completely contractive A(U)-module maps. non-additive, bicomplete

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

1 ∞ OModA(X ) the subcategory of OModA (X ) with the same objects and morphisms (at U) the completely contractive A(U)-module maps. we need this one for constructions

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

1 ∞ OModA(X ) the subcategory of OModA (X ) with the same objects and morphisms (at U) the completely contractive A(U)-module maps.

None of these categories is abelian!

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

∞ An exact structure on OModA (X )

Deﬁnition Let X be a topological space and let A be a sheaf of C*-algebras on X . Let ExA(X ) denote the collection of all kernel–cokernels ∞ pairs in OModA (X )

µ $ E1 / / E2 / / E3 .

∞ We call this the canonical exact structure on OModA (X ).

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

∞ An exact structure on OModA (X )

Deﬁnition Let X be a topological space and let A be a sheaf of C*-algebras on X . Let ExA(X ) denote the collection of all kernel–cokernels ∞ pairs in OModA (X )

µ $ E1 / / E2 / / E3 .

∞ We call this the canonical exact structure on OModA (X ).

Theorem

The class ExA(X ) of all kernel–cokernel pairs deﬁnes an exact ∞ structure on OModA (X ).

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Exact sequences

in ExA(X ), every kernel–cokernel pair

µ $ E1 / / E2 / / E3 ∞ where Ei ∈ OModA (X ) is called a short exact sequence. Deﬁnition ∞ The morphism ϕ ∈ CBA(E, F), E, F ∈ OModA (X ) is called admissible if it can be factorised as ϕ E / F ? $ µ ? G for some admissible monomorphism µ and some admissible ∞ epimorphism $ in OModA (X ).

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Exact sequences

Deﬁnition ∞ A sequence of admissible morphisms in OModA (X )

ϕ1 ϕ2 E / E / E 1 : 2 : 3 $1 $ $ : µ1 $2 $ $ : µ2 G1 G2

µ1 $2 is said to be exact if the short sequence G1 / / E2 / / G2 is exact. An arbitrary sequence of admissible morphisms in ∞ OModA (X ) is exact if the sequences given by any two consecutive morphisms are exact.

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ♦ to introduce injective sheaves; ♦ to construct injective resolutions; • ∞ ♦ to deﬁne the homology of a complex F in OModA (X )

δi−1 δi ... / Fi−1 / Fi / Fi+1 / ...

♦ to use homological algebra (such as the Horseshoe Lemma) in ∞ OModA (X ) to ensure that homotopic injective resolutions yield the same homology ♦ to introduce the cohomology groups as the right derived functor of the global section functor applied to an injective resolution of ∞ F ∈ OModA (X )

∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps what are the next steps in our programme?

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ♦ to construct injective resolutions; • ∞ ♦ to deﬁne the homology of a complex F in OModA (X )

δi−1 δi ... / Fi−1 / Fi / Fi+1 / ...

∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps what are the next steps in our programme?

♦ to introduce injective sheaves;

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras • ∞ ♦ to deﬁne the homology of a complex F in OModA (X )

δi−1 δi ... / Fi−1 / Fi / Fi+1 / ...

∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps what are the next steps in our programme?

♦ to introduce injective sheaves; ♦ to construct injective resolutions;

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ♦ to use homological algebra (such as the Horseshoe Lemma) in ∞ OModA (X ) to ensure that homotopic injective resolutions yield the same homology ♦ to introduce the cohomology groups as the right derived functor of the global section functor applied to an injective resolution of ∞ F ∈ OModA (X )

∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps what are the next steps in our programme?

♦ to introduce injective sheaves; ♦ to construct injective resolutions; • ∞ ♦ to deﬁne the homology of a complex F in OModA (X )

δi−1 δi ... / Fi−1 / Fi / Fi+1 / ...

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ♦ to introduce the cohomology groups as the right derived functor of the global section functor applied to an injective resolution of ∞ F ∈ OModA (X )

∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps what are the next steps in our programme?

♦ to introduce injective sheaves; ♦ to construct injective resolutions; • ∞ ♦ to deﬁne the homology of a complex F in OModA (X )

δi−1 δi ... / Fi−1 / Fi / Fi+1 / ...

♦ to use homological algebra (such as the Horseshoe Lemma) in ∞ OModA (X ) to ensure that homotopic injective resolutions yield the same homology

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps what are the next steps in our programme?

♦ to introduce injective sheaves; ♦ to construct injective resolutions; • ∞ ♦ to deﬁne the homology of a complex F in OModA (X )

δi−1 δi ... / Fi−1 / Fi / Fi+1 / ...

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Progress!

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras ∞ Topology Exact categories Sheaves Our kind of categories OModA (X ) is exact The next steps

Martin Mathieu (Queen’s University Belfast) A sheaf cohomology theory for C*-algebras