Schematic Grothendieck Representation

A spectral Grothendieck representation Rep is said to be schematic if for every triple γ ≤ τ ≤ δ in Top(A), for every A in R^(Ring) we have a commutative diagram in R^:



If Rep is schematic, then, P : Top(A) → R^ is a presheaf with values in R^ over the lattice Top(A)o, for every A in R.

The modality is to restrict attention to Tors(Rep(A)); that is, a lattice in the usual sense; and hence this should be viewed as the commutative shadow of a suitable noncommutative theory.

For obtaining the complete lattice Q(A), a duality is expressed by an order-reversing bijection: (−)−1 : Q(A) → Q((Rep(A))o). (Rep(A))o is not a Grothendieck category. It is additive and has a projective generator; moreover, it is known to be a varietal category (also called triplable) in the sense that it has a projective regular generator P, it is co-complete and has kernel pairs with respect to the functor Hom(P, −), and moreover every equivalence relation in the category is a kernel pair. If a comparison functor is constructed via Hom(P, −) as a functor to the category of sets, it works well for the category of set-valued sheaves over a Grothendieck topology.

Now (−)−1 is defined as an order-reversing bijection between idempotent radicals on Rep(A) and (Rep(A))o, implying we write (Top(A))−1 for the image of Top(A) in Q((Rep( A))o). This is encoded in the exact sequence in Rep(A):

0 → ρ(M) → M → ρ−1(M) → 0

(reversed in (Rep(A))o). By restricting attention to hereditary torsion theories (kernel functors) when defining Tors(−), we introduce an asymmetry that breaks the duality because Top(A)−1 is not in Tors((Rep(A))op). If notationally, TT(G) is the complete lattice of torsion theories (not necessarily hereditary) of the category G; then (TT(G))−1 ≅ TT(Gop). Hence we may view Tors(G)−1 as a complete sublattice of TT(Gop).

Philosophical Identity of Derived Correspondences Between Smooth Varieties.


Let there be a morphism f : X → Y between varieties. Then all the information about f is encoded in the graph Γf ⊂ X × Y of f, which (as a set) is defined as

Γf = {(x, f(x)) : x ∈ X} ⊂ X × Y —– (1)

Now consider the natural projections pX, pY from X × Y to the factors X, Y. Restricted to the subvariety Γf, pX is an isomorphism (since f is a morphism). The fibres of pY restricted to Γf are just the fibres of f; so for example f is proper iff pY | Γf is.

If H(−) is any reasonable covariant homology theory (say singular homology in the complex topology for X, Y compact), then we have a natural push forward map

f : H(X) → H(Y)

This map can be expressed in terms of the graph Γf and the projection maps as

f(α) = pY∗ (pX(α) ∪ [Γf]) —– (2)

where [Γf] ∈ H (X × Y) is the fundamental class of the subvariety [Γf]. Generalizing this construction gives us the notion of a “multi-valued function” or correspondence from X to Y, simply defined to be a general subvariety Γ ⊂ X × Y, replacing the assumption that pX be an isomorphism with some weaker assumption, such as pXf, pY | Γf finite or proper. The right hand side of (2) defines a generalized pushforward map

Γ : H(X) → H(Y)

A subvariety Γ ⊂ X × Y can be represented by its structure sheaf OΓ on X × Y. Associated to the projection maps pX, pY, we also have pullback and pushforward operations on sheaves. The cup product on homology turns out to have an analogue too, namely tensor product. So, appropriately interpreted, (2) makes sense as an operation from the derived category of X to that of Y.

A derived correspondence between a pair of smooth varieties X, Y is an object F ∈ Db(X × Y) with support which is proper over both factors. A derived correspondence defines a functor ΦF by

ΦF : Db(X) → Db(Y)
(−) ↦ RpY∗(LpX(−) ⊗L F)

where (−) could refer to both objects and morphisms in Db(X). F is sometimes called the kernel of the functor ΦF.

The functor ΦF is exact, as it is defined as a composite of exact functors. Since the projection pX is flat, the derived pullback LpX is the same as ordinary pullback pX. Given derived correspondences E ∈ Db(X × Y), F ∈ Db(Y × Z), we obtain functors Φ: Db(X) → Db(Y), Φ: Db(Y) → Db(Z), which can then be composed to get a functor

ΦF ◦ Φ: Db(X) → Db(Z)

which is a two-sided identity with respect to composition of kernels.

Philosophical Equivariance – Sewing Holonomies Towards Equal Trace Endomorphisms.

In d-dimensional topological field theory one begins with a category S whose objects are oriented (d − 1)-manifolds and whose morphisms are oriented cobordisms. Physicists say that a theory admits a group G as a global symmetry group if G acts on the vector space associated to each (d−1)-manifold, and the linear operator associated to each cobordism is a G-equivariant map. When we have such a “global” symmetry group G we can ask whether the symmetry can be “gauged”, i.e., whether elements of G can be applied “independently” – in some sense – at each point of space-time. Mathematically the process of “gauging” has a very elegant description: it amounts to extending the field theory functor from the category S to the category SG whose objects are (d − 1)-manifolds equipped with a principal G-bundle, and whose morphisms are cobordisms with a G-bundle. We regard S as a subcategory of SG by equipping each (d − 1)-manifold S with the trivial G-bundle S × G. In SG the group of automorphisms of the trivial bundle S × G contains G, and so in a gauged theory G acts on the state space H(S): this should be the original “global” action of G. But the gauged theory has a state space H(S,P) for each G-bundle P on S: if P is non-trivial one calls H(S,P) a “twisted sector” of the theory. In the case d = 2, when S = S1 we have the bundle Pg → S1 obtained by attaching the ends of [0,2π] × G via multiplication by g. Any bundle is isomorphic to one of these, and Pg is isomorphic to Pg iff g′ is conjugate to g. But note that the state space depends on the bundle and not just its isomorphism class, so we have a twisted sector state space Cg = H(S,Pg) labelled by a group element g rather than by a conjugacy class.

We shall call a theory defined on the category SG a G-equivariant Topological Field Theory (TFT). It is important to distinguish the equivariant theory from the corresponding “gauged theory”. In physics, the equivariant theory is obtained by coupling to nondynamical background gauge fields, while the gauged theory is obtained by “summing” over those gauge fields in the path integral.

An alternative and equivalent viewpoint which is especially useful in the two-dimensional case is that SG is the category whose objects are oriented (d − 1)-manifolds S equipped with a map p : S → BG, where BG is the classifying space of G. In this viewpoint we have a bundle over the space Map(S,BG) whose fibre at p is Hp. To say that Hp depends only on the G-bundle pEG on S pulled back from the universal G-bundle EG on BG by p is the same as to say that the bundle on Map(S,BG) is equipped with a flat connection allowing us to identify the fibres at points in the same connected component by parallel transport; for the set of bundle isomorphisms p0EG → p1EG is the same as the set of homotopy classes of paths from p0 to p1. When S = S1 the connected components of the space of maps correspond to the conjugacy classes in G: each bundle Pg corresponds to a specific point pg in the mapping space, and a group element h defines a specific path from pg to phgh−1 .

G-equivariant topological field theories are examples of “homotopy topological field theories”. Using Vladimir Turaev‘s two main results: first, an attractive generalization of the theorem that a two-dimensional TFT “is” a commutative Frobenius algebra, and, secondly, a classification of the ways of gauging a given global G-symmetry of a semisimple TFT.


Definition of the product in the G-equivariant closed theory. The heavy dot is the basepoint on S1. To specify the morphism unambiguously we must indicate consistent holonomies along a set of curves whose complement consists of simply connected pieces. These holonomies are always along paths between points where by definition the fibre is G. This means that the product is not commutative. We need to fix a convention for holonomies of a composition of curves, i.e., whether we are using left or right path-ordering. We will take h(γ1 ◦ γ2) = h(γ1) · h(γ2).

A G-equivariant TFT gives us for each element g ∈ G a vector space Cg, associated to the circle equipped with the bundle pg whose holonomy is g. The usual pair-of-pants cobordism, equipped with the evident G-bundle which restricts to pg1 and pg2 on the two incoming circles, and to pg1g2 on the outgoing circle, induces a product

Cg1 ⊗ Cg2 → Cg1g2 —– (1)


making C := ⊕g∈GCg into a G-graded algebra. Also there is a trace θ: C1  → C defined by the disk diagram with one ingoing circle. The holonomy around the boundary of the disk must be 1. Making the standard assumption that the cylinder corresponds to the unit operator we obtain a non-degenerate pairing

Cg ⊗ Cg−1 → C

A new element in the equivariant theory is that G acts as an automorphism group on C. That is, there is a homomorphism α : G → Aut(C) such that

αh : Cg → Chgh−1 —– (2)

Diagramatically, αh is defined by the surface in the immediately above figure. Now let us note some properties of α. First, if φ ∈ Ch then αh(φ) = φ. The reason for this is diagrammatically in the below figure.


If the holonomy along path P2 is h then the holonomy along path P1 is 1. However, a Dehn twist around the inner circle maps P1 into P2. Therefore, αh(φ) = α1(φ) = φ, if φ ∈ Ch.

Next, while C is not commutative, it is “twisted-commutative” in the following sense. If φ1 ∈ Cg1 and φ2 ∈ Cg2 then

αg212 = φ2φ1 —– (3)

The necessity of this condition is illustrated in the figure below.


The trace of the identity map of Cg is the partition function of the theory on a torus with the bundle with holonomy (g,1). Cutting the torus the other way, we see that this is the trace of αg on C1. Similarly, by considering the torus with a bundle with holonomy (g,h), where g and h are two commuting elements of G, we see that the trace of αg on Ch is the trace of αh on Cg−1. But we need a strengthening of this property. Even when g and h do not commute we can form a bundle with holonomy (g,h) on a torus with one hole, around which the holonomy will be c = hgh−1g−1. We can cut this torus along either of its generating circles to get a cobordism operator from Cc ⊗ Ch to Ch or from Cg−1 ⊗ Cc to Cg−1. If ψ ∈ Chgh−1g−1. Let us introduce two linear transformations Lψ, Rψ associated to left- and right-multiplication by ψ. On the one hand, Lψαg : φ􏰀 ↦ ψαg(φ) is a map Ch → Ch. On the other hand Rψαh : φ ↦ αh(φ)ψ is a map Cg−1 → Cg−1. The last sewing condition states that these two endomorphisms must have equal traces:

TrCh 􏰌Lψαg􏰍 = TrCg−1 􏰌Rψαh􏰍 —– (4)



(4) was taken by Turaev as one of his axioms. It can, however, be reexpressed in a way that we shall find more convenient. Let ∆g ∈ Cg ⊗ Cg−1 be the “duality” element corresponding to the identity cobordism of (S1,Pg) with both ends regarded as outgoing. We have ∆g = ∑ξi ⊗ ξi, where ξi and ξi ru􏰟n through dual bases of Cg and Cg−1. Let us also write

h = ∑ηi ⊗ ηi ∈ Ch ⊗ Ch−1. Then (4) is easily seen to be equivalent to

∑αhii = 􏰟 ∑ηiαgi) —– (5)

in which both sides are elements of Chgh−1g−1.

Superconformal Spin/Field Theories: When Vector Spaces have same Dimensions: Part 1, Note Quote.


A spin structure on a surface means a double covering of its space of non-zero tangent vectors which is non-trivial on each individual tangent space. On an oriented 1-dimensional manifold S it means a double covering of the space of positively-oriented tangent vectors. For purposes of gluing, this is the same thing as a spin structure on a ribbon neighbourhood of S in an orientable surface. Each spin structure has an automorphism which interchanges its sheets, and this will induce an involution T on any vector space which is naturally associated to a 1-manifold with spin structure, giving the vector space a mod 2 grading by its ±1-eigenspaces. A topological-spin theory is a functor from the cobordism category of manifolds with spin structures to the category of super vector spaces with its graded tensor structure. The functor is required to take disjoint unions to super tensor products, and additionally it is required that the automorphism of the spin structure of a 1-manifold induces the grading automorphism T = (−1)degree of the super vector space. This choice of the supersymmetry of the tensor product rather than the naive symmetry which ignores the grading is forced by the geometry of spin structures if the possibility of a semisimple category of boundary conditions is to be allowed. There are two non-isomorphic circles with spin structure: S1ns, with the Möbius or “Neveu-Schwarz” structure, and S1r, with the trivial or “Ramond” structure. A topological-spin theory gives us state spaces Cns and Cr, corresponding respectively to S1ns and S1r.

There are four cobordisms with spin structures which cover the standard annulus. The double covering can be identified with its incoming end times the interval [0,1], but then one has a binary choice when one identifies the outgoing end of the double covering over the annulus with the chosen structure on the outgoing boundary circle. In other words, alongside the cylinders A+ns,r = S1ns,r × [0,1] which induce the identity maps of Cns,r there are also cylinders Ans,r which connect S1ns,r to itself while interchanging the sheets. These cylinders Ans,r induce the grading automorphism on the state spaces. But because Ans ≅ A+ns by an isomorphism which is the identity on the boundary circles – the Dehn twist which “rotates one end of the cylinder by 2π” – the grading on Cns must be purely even. The space Cr can have both even and odd components. The situation is a little more complicated for “U-shaped” cobordisms, i.e., cylinders with two incoming or two outgoing boundary circles. If the boundaries are S1ns there is only one possibility, but if the boundaries are S1r there are two, corresponding to A±r. The complication is that there seems no special reason to prefer either of the spin structures as “positive”. We shall simply choose one – let us call it P – with incoming boundary S1r ⊔ S1r, and use P to define a pairing Cr ⊗ Cr → C. We then choose a preferred cobordism Q in the other direction so that when we sew its right-hand outgoing S1r to the left-hand incoming one of P the resulting S-bend is the “trivial” cylinder A+r. We shall need to know, however, that the closed torus formed by the composition P ◦ Q has an even spin structure. The Frobenius structure θ on C restricts to 0 on Cr.

There is a unique spin structure on the pair-of-pants cobordism in the figure below, which restricts to S1ns on each boundary circle, and it makes Cns into a commutative Frobenius algebra in the usual way.


If one incoming circle is S1ns and the other is S1r then the outgoing circle is S1r, and there are two possible spin structures, but the one obtained by removing a disc from the cylinder A+r is preferred: it makes Cr into a graded module over Cns. The chosen U-shaped cobordism P, with two incoming circles S1r, can be punctured to give us a pair of pants with an outgoing S1ns, and it induces a graded bilinear map Cr × Cr → Cns which, composing with the trace on Cns, gives a non-degenerate inner product on Cr. At this point the choice of symmetry of the tensor product becomes important. Let us consider the diffeomorphism of the pair of pants which shows us in the usual case that the Frobenius algebra is commutative. When we lift it to the spin structure, this diffeomorphism induces the identity on one incoming circle but reverses the sheets over the other incoming circle, and this proves that the cobordism must have the same output when we change the input from S(φ1 ⊗ φ2) to T(φ1) ⊗ φ2, where T is the grading involution and S : Cr ⊗ Cr → Cr ⊗ Cr is the symmetry of the tensor category. If we take S to be the symmetry of the tensor category of vector spaces which ignores the grading, this shows that the product on the graded vector space Cr is graded-symmetric with the usual sign; but if S is the graded symmetry then we see that the product on Cr is symmetric in the naive sense.

There is an analogue for spin theories of the theorem which tells us that a two-dimensional topological field theory “is” a commutative Frobenius algebra. It asserts that a spin-topological theory “is” a Frobenius algebra C = (Cns ⊕ CrC) with the following property. Let {φk} be a basis for Cns, with dual basis {φk} such that θCkφm) = δmk, and let βk and βk be similar dual bases for Cr. Then the Euler elements χns := ∑ φkφk and χr = ∑ βkβk are independent of the choices of bases, and the condition we need on the algebra C is that χns = χr. In particular, this condition implies that the vector spaces Cns and Cr have the same dimension. In fact, the Euler elements can be obtained from cutting a hole out of the torus. There are actually four spin structures on the torus. The output state is necessarily in Cns. The Euler elements for the three even spin structures are equal to χe = χns = χr. The Euler element χo corresponding to the odd spin structure, on the other hand, is given by χo = ∑(−1)degβkβkβk.

A spin theory is very similar to a Z/2-equivariant theory, which is the structure obtained when the surfaces are equipped with principal Z/2-bundles (i.e., double coverings) rather than spin structures.

It seems reasonable to call a spin theory semisimple if the algebra Cns is semisimple, i.e., is the algebra of functions on a finite set X. Then Cr is the space of sections of a vector bundle E on X, and it follows from the condition χns = χr that the fibre at each point must have dimension 1. Thus the whole structure is determined by the Frobenius algebra Cns together with a binary choice at each point x ∈ X of the grading of the fibre Ex of the line bundle E at x.

We can now see that if we had not used the graded symmetry in defining the tensor category we should have forced the grading of Cr to be purely even. For on the odd part the inner product would have had to be skew, and that is impossible on a 1-dimensional space. And if both Cns and Cr are purely even then the theory is in fact completely independent of the spin structures on the surfaces.

A concrete example of a two-dimensional topological-spin theory is given by C = C ⊕ Cη where η2 = 1 and η is odd. The Euler elements are χe = 1 and χo = −1. It follows that the partition function of a closed surface with spin structure is ±1 according as the spin structure is even or odd.

The most common theories defined on surfaces with spin structure are not topological: they are 2-dimensional conformal field theories with N = 1 supersymmetry. It should be noticed that if the theory is not topological then one does not expect the grading on Cns to be purely even: states can change sign on rotation by 2π. If a surface Σ has a conformal structure then a double covering of the non-zero tangent vectors is the complement of the zero-section in a two-dimensional real vector bundle L on Σ which is called the spin bundle. The covering map then extends to a symmetric pairing of vector bundles L ⊗ L → TΣ which, if we regard L and TΣ as complex line bundles in the natural way, induces an isomorphism L ⊗C L ≅ TΣ. An N = 1 superconformal field theory is a conformal-spin theory which assigns a vector space HS,L to the 1-manifold S with the spin bundle L, and is equipped with an additional map

Γ(S,L) ⊗ HS,L → HS,L

(σ,ψ) ↦ Gσψ,

where Γ(S,L) is the space of smooth sections of L, such that Gσ is real-linear in the section σ, and satisfies G2σ = Dσ2, where Dσ2 is the Virasoro action of the vector field σ2 related to σ ⊗ σ by the isomorphism L ⊗C L ≅ TΣ. Furthermore, when we have a cobordism (Σ,L) from (S0,L0) to (S1,L1) and a holomorphic section σ of L which restricts to σi on Si we have the intertwining property

Gσ1 ◦ UΣ,L = UΣ,L ◦ Gσ0


Morphism of Complexes Induces Corresponding Morphisms on Cohomology Objects – Thought of the Day 146.0

Let A = Mod(R) be an abelian category. A complex in A is a sequence of objects and morphisms in A

… → Mi-1 →di-1 Mi →di → Mi+1 → …

such that di ◦ di-1 = 0 ∀ i. We denote such a complex by M.

A morphism of complexes f : M → N is a sequence of morphisms fi : Mi → Ni in A, making the following diagram commute, where diM, diN denote the respective differentials:


We let C(A) denote the category whose objects are complexes in A and whose morphisms are morphisms of complexes.

Given a complex M of objects of A, the ith cohomology object is the quotient

Hi(M) = ker(di)/im(di−1)

This operation of taking cohomology at the ith place defines a functor

Hi(−) : C(A) → A,

since a morphism of complexes induces corresponding morphisms on cohomology objects.

Put another way, an object of C(A) is a Z-graded object

M = ⊕i Mi

of A, equipped with a differential, in other words an endomorphism d: M → M satisfying d2 = 0. The occurrence of differential graded objects in physics is well-known. In mathematics they are also extremely common. In topology one associates to a space X a complex of free abelian groups whose cohomology objects are the cohomology groups of X. In algebra it is often convenient to replace a module over a ring by resolutions of various kinds.

A topological space X may have many triangulations and these lead to different chain complexes. Associating to X a unique equivalence class of complexes, resolutions of a fixed module of a given type will not usually be unique and one would like to consider all these resolutions on an equal footing.

A morphism of complexes f: M → N is a quasi-isomorphism if the induced morphisms on cohomology

Hi(f): Hi(M) → Hi(N) are isomorphisms ∀ i.

Two complexes M and N are said to be quasi-isomorphic if they are related by a chain of quasi-isomorphisms. In fact, it is sufficient to consider chains of length one, so that two complexes M and N are quasi-isomorphic iff there are quasi-isomorphisms

M ← P → N

For example, the chain complex of a topological space is well-defined up to quasi-isomorphism because any two triangulations have a common resolution. Similarly, all possible resolutions of a given module are quasi-isomorphic. Indeed, if

0 → S →f M0 →d0 M1 →d1 M2 → …

is a resolution of a module S, then by definition the morphism of complexes


is a quasi-isomorphism.

The objects of the derived category D(A) of our abelian category A will just be complexes of objects of A, but morphisms will be such that quasi-isomorphic complexes become isomorphic in D(A). In fact we can formally invert the quasi-isomorphisms in C(A) as follows:

There is a category D(A) and a functor Q: C(A) → D(A)

with the following two properties:

(a) Q inverts quasi-isomorphisms: if s: a → b is a quasi-isomorphism, then Q(s): Q(a) → Q(b) is an isomorphism.

(b) Q is universal with this property: if Q′ : C(A) → D′ is another functor which inverts quasi-isomorphisms, then there is a functor F : D(A) → D′ and an isomorphism of functors Q′ ≅ F ◦ Q.

First, consider the category C(A) as an oriented graph Γ, with the objects lying at the vertices and the morphisms being directed edges. Let Γ∗ be the graph obtained from Γ by adding in one extra edge s−1: b → a for each quasi-isomorphism s: a → b. Thus a finite path in Γ∗ is a sequence of the form f1 · f2 ·· · ·· fr−1 · fr where each fi is either a morphism of C(A), or is of the form s−1 for some quasi-isomorphism s of C(A). There is a unique minimal equivalence relation ∼ on the set of finite paths in Γ∗ generated by the following relations:

(a) s · s−1 ∼ idb and s−1 · s ∼ ida for each quasi-isomorphism s: a → b in C(A).

(b) g · f ∼ g ◦ f for composable morphisms f: a → b and g: b → c of C(A).

Define D(A) to be the category whose objects are the vertices of Γ∗ (these are the same as the objects of C(A)) and whose morphisms are given by equivalence classes of finite paths in Γ∗. Define a functor Q: C(A) → D(A) by using the identity morphism on objects, and by sending a morphism f of C(A) to the length one path in Γ∗ defined by f. The resulting functor Q satisfies the conditions of the above lemma.

The second property ensures that the category D(A) of the Lemma is unique up to equivalence of categories. We define the derived category of A to be any of these equivalent categories. The functor Q: C(A) → D(A) is called the localisation functor. Observe that there is a fully faithful functor

J: A → C(A)

which sends an object M to the trivial complex with M in the zeroth position, and a morphism F: M → N to the morphism of complexes


Composing with Q we obtain a functor A → D(A) which we denote by J. This functor J is fully faithful, and so defines an embedding A → D(A). By definition the functor Hi(−): C(A) → A inverts quasi-isomorphisms and so descends to a functor

Hi(−): D(A) → A

establishing that composite functor H0(−) ◦ J is isomorphic to the identity functor on A.

Ringed Spaces (2)


Let |M| be a topological space. A presheaf of commutative algebras F on X is an assignment

U ↦ F(U), U open in |M|, F(U) is a commutative algebra, such that the following holds,

(1) If U ⊂ V are two open sets in |M|, ∃ a morphism rV, U: F(V) → F(U), called the restriction morphism and often denoted by rV, U(ƒ) = ƒ|U, such that

(i) rU, U = id,

(ii) rW, U = rV, U ○ rW, V

A presheaf ƒ is called a sheaf if the following holds:

(2) Given an open covering {Ui}i∈I of U and a family {ƒi}i∈I, ƒi ∈ F(Ui) such that ƒi|Ui ∩ Uj = ƒj|Ui ∩ Uj ∀ i, j ∈ I, ∃ a unique ƒ ∈ F(U) with ƒ|Ui = ƒi

The elements in F(U) are called sections over U, and with U = |M|, these are termed global sections.

The assignments U ↦ C(U), U open in the differentiable manifold M and U ↦ OX(U), U open in algebraic variety X are examples of sheaves of functions on the topological spaces |M| and |X| underlying the differentiable manifold M and the algebraic variety X respectively.

In the language of categories, the above definition says that we have defined a functor, F, from top(M) to (alg), where top(M) is the category of the open sets in the topological space |M|, the arrows given by the inclusions of open sets while (alg) is the category of commutative algebras. In fact, the assignment U ↦ F(U) defines F on the objects while the assignment

U ⊂ V ↦ rV, U: F(V) → F(U)

defines F on the arrows.

Let |M| be a topological space. We define a presheaf of algebras on |M| to be a functor

F: top(M)op → (alg)

The suffix “op” denotes as usual the opposite category; in other words, F is a contravariant functor from top(M) to (alg). A presheaf is a sheaf if it satisfies the property (2) of the above definition.

If F is a (pre)sheaf on |M| and U is open in |M|, we define F|U, the (pre)sheaf F restricted to U, as the functor F restricted to the category of open sets in U (viewed as a topological space itself).

Let F be a presheaf on the topological space |M| and let x be a point in |M|. We define the stalk Fx of F, at the point x, as the direct limit

lim F(U)

where the direct limit is taken ∀ the U open neighbourhoods of x in |M|. Fx consists of the disjoint union of all pairs (U, s) with U open in |M|, x ∈ U, and s ∈ F(U), modulo the equivalence relation: (U, s) ≅ (V, t) iff ∃ a neighbourhood W of x, W ⊂ U ∩ V, such that s|W = t|W.

The elements in Fx are called germs of sections.

Let F and G be presheaves on |M|. A morphism of presheaves φ: F → G, for each open set U in |M|, such that ∀ V ⊂ U, the following diagram commutes


Equivalently and more elegantly, one can also say that a morphism of presheaves is a natural transformation between the two presheaves F and G viewed as functors.

A morphism of sheaves is just a morphism of the underlying presheaves.

Clearly any morphism of presheaves induces a morphism on the stalks: φx: Fx → Gx. The sheaf property, i.e., property (2) in the above definition, ensures that if we have two morphisms of sheaves φ and ψ, such that φx = ψx ∀ x, then φ = ψ.

We say that the morphism of sheaves is injective (resp. surjective) if x is injective (resp. surjective).

On the notion of surjectivity, however, one should exert some care, since we can have a surjective sheaf morphism φ: F → G such that φU: F(U) → G(U) is not surjective for some open sets U. This strange phenomenon is a consequence of the following fact. While the assignment U ↦ ker(φ(U)) always defines a sheaf, the assignment

U ↦ im( φ(U)) = F(U)/G(U)

defines in general only a presheaf and not all the presheaves are sheaves. A simple example is given by the assignment associating to an open set U in R, the algebra of constant real functions on U. Clearly this is a presheaf, but not a sheaf.

We can always associate, in a natural way, to any presheaf a sheaf called its sheafification. Intuitively, one may think of the sheafification as the sheaf that best “approximates” the given presheaf. For example, the sheafification of the presheaf of constant functions on open sets in R is the sheaf of locally constant functions on open sets in R. We construct the sheafification of a presheaf using the étalé space, which we also need in the sequel, since it gives an equivalent approach to sheaf theory.

Let F be a presheaf on |M|. We define the étalé space of F to be the disjoint union ⊔x∈|M| Fx. Let each open U ∈ |M| and each s ∈ F(U) define the map šU: U ⊔x∈|U| Fx, šU(x) = sx. We give to the étalé space the finest topology that makes the maps š continuous, ∀ open U ⊂ |M| and all sections s ∈ F(U). We define Fet to be the presheaf on |M|:

U ↦ Fet(U) = {šU: U → ⊔x∈|U| Fx, šU(x) = sx ∈ Fx}

Let F be a presheaf on |M|. A sheafification of F is a sheaf F~, together with a presheaf morphism α: F → Fsuch that

(1) any presheaf morphism ψ: F → G, G a sheaf factors via α, i.e. ψ: F →α F~ → G,

(2) F and Fare locally isomorphic, i.e., ∃ an open cover {Ui}i∈I of |M| such that F(Ui) ≅ F~(Ui) via α.

Let F and G be sheaves of rings on some topological space |M|. Assume that we have an injective morphism of sheaves G → F such that G(U) ⊂ F(U) ∀ U open in |M|. We define the quotient F/G to be the sheafification of the image presheaf: U ↦ F(U)/G(U). In general F/G (U) ≠ F(U)/G(U), however they are locally isomorphic.

Ringed space is a pair M = (|M|, F) consisting of a topological space |M| and a sheaf of commutative rings F on |M|. This is a locally ringed space, if the stalk Fx is a local ring ∀ x ∈ |M|. A morphism of ringed spaces φ: M = (|M|, F) → N = (|N|, G) consists of a morphism |φ|: |M| → |N| of the topological spaces and a sheaf morphism φ*: ON → φ*OM, where φ*OM is a sheaf on |N| and defined as follows:

*OM)(U) = OM-1(U)) ∀ U open in |N|

Morphism of ringed spaces induces a morphism on the stalks for each

x ∈ |M|: φx: ON,|φ|(x) → OM,x

If M and N are locally ringed spaces, we say that the morphism of ringed spaces φ is a morphism of locally ringed spaces if φx is local, i.e. φ-1x(mM,x) = mN,|φ|(x), where mN,|φ|(x) and mM,x are the maximal ideals in the local rings ON,|φ|(x) and OM,x respectively.

Category of Super Vector Spaces Becomes a Tensor Category


The theory of manifolds and algebraic geometry are ultimately based on linear algebra. Similarly the theory of supermanifolds needs super linear algebra, which is linear algebra in which vector spaces are replaced by vector spaces with a Z/2Z-grading, namely, super vector spaces.

A super vector space is a Z/2Z-graded vector space

V = V0 ⊕ V1

where the elements of Vare called even and that of Vodd.

The parity of v ∈ V , denoted by p(v) or |v|, is defined only on non-zero homogeneous elements, that is elements of either V0 or V1:

p(v) = |v| = 0 if v ∈ V0

= 1 if v ∈ V1

The superdimension of a super vector space V is the pair (p, q) where dim(V0) = p and dim(V1) = q as ordinary vector spaces. We simply write dim(V) = p|q.

If dim(V) = p|q, then we can find a basis {e1,…., ep} of V0 and a basis {ε1,….., εq} of V1 so that V is canonically isomorphic to the free k-module generated by {e1,…., ep, ε1,….., εq}. We denote this k-module by kp|q and we will call {e1,…., ep, ε1,….., εq} the canonical basis of kp|q. The (ei) form a basis of kp = k0p|q and the (εj) form a basis for kq = k1p|q.

A morphism from a super vector space V to a super vector space W is a linear map from V to W preserving the Z/2Z-grading. Let Hom(V, W) denote the vector space of morphisms V → W. Thus we have formed the category of super vector spaces that we denote by (smod). It is important to note that the category of super vector spaces also admits an “inner Hom”, which we denote by Hom(V, W); for super vector spaces V, W, Hom(V, W) consists of all linear maps from V to W ; it is made into a super vector space itself by:

Hom(V, W)0 = {T : V → W|T preserves parity}  (= Hom(V, W))

Hom(V, W)1 = {T : V → W|T reverses parity}

If V = km|n, W = kp|q we have in the canonical basis (ei, εj):

Hom(V, W)0 = (A 0 0 D) and Hom(V, W)1 = (0 B C 0)

where A, B, C , D are respectively (p x m), (p x n), (q x m), (q x n) – matrices with entries in k.

In the category of super vector spaces we have the parity reversing functor ∏(V → ∏V) defined by

(∏V)0 = V1, (∏V)1 = V0

The category of super vector spaces admits tensor products: for super vector spaces V, W, V ⊗ W is given the Z/2Z-grading as

(V ⊗ W)0 = (V0 ⊗ W0) ⊕ (V1 ⊗ W1),

(V ⊗ W)1 = (V0 ⊗ W1) ⊕ (V1 ⊗ W0)

The assignment V, W ↦ V ⊗ W is additive and exact in each variable as in the ordinary vector space category. The object k functions as a unit element with respect to tensor multiplication ⊗ and tensor multiplication is associative, i.e., the two products U ⊗ (V ⊗ W) and (U ⊗ V) ⊗ W are naturally isomorphic. Moreover, V ⊗ W ≅ W ⊗ V by the commutative map,

cV,W : V ⊗ W → W ⊗ V


v ⊗ w ↦ (-1)|v||w|w ⊗ v

If we are working with the category of vector spaces, the commutativity isomorphism takes v ⊗ w to w ⊗ v. In super linear algebra we have to add the sign factor in front. This is a special case of the general principle called the “sign rule”. The principle says that in making definitions and proving theorems, the transition from the usual theory to the super theory is often made by just simply following this principle, which introduces a sign factor whenever one reverses the order of two odd elements. The functoriality underlying the constructions makes sure that the definitions are all consistent.

The commutativity isomorphism satisfies the so-called hexagon diagram:


where, if we had not suppressed the arrows of the associativity morphisms, the diagram would have the shape of a hexagon.

The definition of the commutativity isomorphism, also informally referred to as the sign rule, has the following very important consequence. If V1, …, Vn are the super vector spaces and σ and τ are two permutations of n-elements, no matter how we compose associativity and commutativity morphisms, we always obtain the same isomorphism from Vσ(1) ⊗ … ⊗ Vσ(n) to Vτ(1) ⊗ … ⊗ Vτ(n) namely:

Vσ(1) ⊗ … ⊗ Vσ(n) → Vτ(1) ⊗ … ⊗ Vτ(n)

vσ(1) ⊗ … ⊗ vσ(n) ↦ (-1)N vτ(1) ⊗ … ⊗ vτ(n)

where N is the number of pair of indices i, j such that vi and vj are odd and σ-1(i) < σ-1(j) with τ-1(i) > τ-1(j).

The dual V* of V is defined as

V* := Hom (V, k)

If V is even, V = V0, V* is the ordinary dual of V consisting of all even morphisms V → k. If V is odd, V = V1, then V* is also an odd vector space and consists of all odd morphisms V1 → k. This is because any morphism from V1 to k = k1|0 is necessarily odd and sends odd vectors into even ones. The category of super vector spaces thus becomes what is known as a tensor category with inner Hom and dual.



Many important spaces in topology and algebraic geometry have no odd-dimensional homology. For such spaces, functorial spatial homology truncation simplifies considerably. On the theory side, the simplification arises as follows: To define general spatial homology truncation, we used intermediate auxiliary structures, the n-truncation structures. For spaces that lack odd-dimensional homology, these structures can be replaced by a much simpler structure. Again every such space can be embedded in such a structure, which is the analogon of the general theory. On the application side, the crucial simplification is that the truncation functor t<n will not require that in truncating a given continuous map, the map preserve additional structure on the domain and codomain of the map. In general, t<n is defined on the category CWn⊃∂, meaning that a map must preserve chosen subgroups “Y ”. Such a condition is generally necessary on maps, for otherwise no truncation exists. So arbitrary continuous maps between spaces with trivial odd-dimensional homology can be functorially truncated. In particular the compression rigidity obstructions arising in the general theory will not arise for maps between such spaces.

Let ICW be the full subcategory of CW whose objects are simply connected CW-complexes K with finitely generated even-dimensional homology and vanishing odd-dimensional homology for any coefficient group. We call ICW the interleaf category.

For example, the space K = S22 e3 is simply connected and has vanishing integral homology in odd dimensions. However, H3(K;Z/2) = Z/2 ≠ 0.

Let X be a space whose odd-dimensional homology vanishes for any coefficient group. Then the even-dimensional integral homology of X is torsion-free.

Taking the coefficient group Q/Z, we have

Tor(H2k(X),Q/Z) = H2k+1(X) ⊗ Q/Z ⊕ Tor(H2k(X),Q/Z) = H2k+1(X;Q/Z) = 0.

Thus H2k(X) is torsion-free, since the group Tor(H2k(X),Q/Z) is isomorphic to the torsion subgroup of H2k(X).

Any simply connected closed 4-manifold is in ICW. Indeed, such a manifold is homotopy equivalent to a CW-complex of the form


where the homotopy class of the attaching map ƒ : S3 → Vi=1k Si2 may be viewed as a symmetric k × k matrix with integer entries, as π3(Vi=1kSi2) ≅ M(k), with M(k) the additive group of such matrices.

Any simply connected closed 6-manifold with vanishing integral middle homology group is in ICW. If G is any coefficient group, then H1(M;G) ≅ H1(M) ⊗ G ⊕ Tor(H0M,G) = 0, since H0(M) = Z. By Poincaré duality,

0 = H3(M) ≅ H3(M) ≅ Hom(H3M,Z) ⊕ Ext(H2M,Z),

so that H2(M) is free. This implies that Tor(H2M,G) = 0 and hence H3(M;G) ≅ H3(M) ⊗ G ⊕ Tor(H2M,G) = 0. Finally, by G-coefficient Poincaré duality,

H5(M;G) ≅ H1(M;G) ≅ Hom(H1M,G) ⊕ Ext(H0M,G) = Ext(Z,G) = 0

Any smooth, compact toric variety X is in ICW: Danilov’s Theorem implies that H(X;Z) is torsion-free and the map A(X) → H(X;Z) given by composing the canonical map from Chow groups to homology, Ak(X) = An−k(X) → H2n−2k(X;Z), where n is the complex dimension of X, with Poincaré duality H2n−2k(X;Z) ≅ H2k(X;Z), is an isomorphism. Since the odd-dimensional cohomology of X is not in the image of this map, this asserts in particular that Hodd(X;Z) = 0. By Poincaré duality, Heven(X;Z) is free and Hodd(X;Z) = 0. These two statements allow us to deduce from the universal coefficient theorem that Hodd(X;G) = 0 for any coefficient group G. If we only wanted to establish Hodd(X;Z) = 0, then it would of course have been enough to know that the canonical, degree-doubling map A(X) → H(X;Z) is onto. One may then immediately reduce to the case of projective toric varieties because every complete fan Δ has a projective subdivision Δ, the corresponding proper birational morphism X(Δ) → X(Δ) induces a surjection H(X(Δ);Z) → H(X(Δ);Z) and the diagram



Let G be a complex, simply connected, semisimple Lie group and P ⊂ G a connected parabolic subgroup. Then the homogeneous space G/P is in ICW. It is simply connected, since the fibration P → G → G/P induces an exact sequence

1 = π1(G) → π1(G/P) → π0(P) → π0(G) = 0,

which shows that π1(G/P) → π0(P) is a bijection. Accordingly, ∃ elements sw(P) ∈ H2l(w)(G/P;Z) (“Schubert classes,” given geometrically by Schubert cells), indexed by w ranging over a certain subset of the Weyl group of G, that form a basis for H(G/P;Z). (For w in the Weyl group, l(w) denotes the length of w when written as a reduced word in certain specified generators of the Weyl group.) In particular Heven(G/P;Z) is free and Hodd(G/P;Z) = 0. Thus Hodd(G/P;G) = 0 for any coefficient group G.

The linear groups SL(n, C), n ≥ 2, and the subgroups S p(2n, C) ⊂ SL(2n, C) of transformations preserving the alternating bilinear form

x1yn+1 +···+ xny2n −xn+1y1 −···−x2nyn

on C2n × C2n are examples of complex, simply connected, semisimple Lie groups. A parabolic subgroup is a closed subgroup that contains a Borel group B. For G = SL(n,C), B is the group of all upper-triangular matrices in SL(n,C). In this case, G/B is the complete flag manifold

G/B = {0 ⊂ V1 ⊂···⊂ Vn−1 ⊂ Cn}

of flags of subspaces Vi with dimVi = i. For G = Sp(2n,C), the Borel subgroups B are the subgroups preserving a half-flag of isotropic subspaces and the quotient G/B is the variety of all such flags. Any parabolic subgroup P may be described as the subgroup that preserves some partial flag. Thus (partial) flag manifolds are in ICW. A special case is that of a maximal parabolic subgroup, preserving a single subspace V. The corresponding quotient SL(n, C)/P is a Grassmannian G(k, n) of k-dimensional subspaces of Cn. For G = Sp(2n,C), one obtains Lagrangian Grassmannians of isotropic k-dimensional subspaces, 1 ≤ k ≤ n. So Grassmannians are objects in ICW. The interleaf category is closed under forming fibrations.

Functoriality in Low Dimensions. Note Quote.

Let CW be the category of CW-complexes and cellular maps, let CW0 be the full subcategory of path connected CW-complexes and let CW1 be the full subcategory of simply connected CW-complexes. Let HoCW denote the category of CW-complexes and homotopy classes of cellular maps. Let HoCWn denote the category of CW-complexes and rel n-skeleton homotopy classes of cellular maps. Dimension n = 1: It is straightforward to define a covariant truncation functor

t<n = t<1 : CW0 → HoCW together with a natural transformation

emb1 : t<1 → t<∞,

where t<∞ : CW0 → HoCW is the natural “inclusion-followed-by-quotient” functor given by t<∞(K) = K for objects K and t<∞(f) = [f] for morphisms f, such that for all objects K, emb1∗ : H0(t<1K) → H0(t<∞K) is an isomorphism and Hr(t<1K) = 0 for r ≥ 1. The details are as follows: For a path connected CW-complex K, set t<1(K) = k0, where k0 is a 0-cell of K. Let emb1(K) : t<1(K) = k0 → t<∞(K) = K be the inclusion of k0 in K. Then emb1∗ is an isomorphism on H0 as K is path connected. Clearly Hr(t<1K) = 0 for r ≥ 1. Let f : K → L be a cellular map between objects of CW0. The morphism t<1(f) : t<1(K) = k0 → l0 = t<1(L) is the homotopy class of the unique map from a point to a point. In particular, t<1(idK) = [idk0] and for a cellular map g : L → P we have t<1(gf) = t<1(g) ◦ t<1(f), so that t<1 is indeed a functor. To show that emb1 is a natural transformation, we need to see that


that is


commutes in HoCW. This is where we need the functor t<1 to have values only in HoCW, not in CW, because the square need certainly not commute in CW. (The points k0 and l0 do not know anything about f, so l0 need not be the image of k0 under f.) Since L is path connected, there is a path ω : I → L from l0 = ω(0) to f (k0) = ω(1). Then H : {k0} × I → L, H(k0, t) = ω(t), defines a homotopy from

k0 → l0 → L to k0 → K →f L.

Dimension n = 2: We will define a covariant truncation functor t<n = t<2 : CW1 → HoCW

together with a natural transformation
emb2 : t<2 → t<∞,

where t<∞ : CW1 → HoCW is as above (only restricted to simply connected spaces), such that for all objects K, emb2∗ : Hr(t<2K) → Hr(t<∞K) is an isomorphism for r = 0, 1, and Hr(t<2K) = 0 for r ≥ 2. For a simply connected CW-complex K, set t<2(K) = k0, where k0 is a 0-cell of K. Let emb2(K) : t<2(K) = k0 → t<∞(K) = K be the inclusion as in the case n = 1. It follows that emb2∗ is an isomorphism both on H0 as K is path connected and on H1 as H1(k0) = 0 = H1(K), while trivially Hr(t<2K) = 0 for r ≥ 2. On a cellular map f, t<2(f) is defined as in the case n = 1. As in the case n = 1, this yields a functor and emb2 is a natural transformation.

Hypercoverings, or Fibrant Homotopies




Given that a Grothendieck topology is essentially about abstracting a notion of ‘covering’, it is not surprising that modified Čech methods can be applied. Artin and Mazur used Verdier’s idea of a hypercovering to get, for each Grothendieck topos, E, a pro-object in Ho(S) (i.e. an inverse system of simplicial sets), which they call the étale homotopy type of the topos E (which for them is ‘sheaves for the étale topology on a variety’). Applying homotopy group functors gives pro-groups πi(E) such that π1(E) is essentially the same as Grothendieck’s π1(E).

Grothendieck’s nice π1 has thus an interpretation as a limit of a Čech type, or shape theoretic, system of π1s of ‘hypercoverings’. Can shape theory be useful for studying ́etale homotopy type? Not without extra work, since the Artin-Mazur-Verdier approach leads one to look at inverse systems in proHo(S), i.e. inverse systems in a homotopy category not a homotopy category of inverse systems as in Strong Shape Theory.

One of the difficulties with this hypercovering approach is that ‘hypercovering’ is a difficult concept and to the ‘non-expert’ seem non-geometric and lacking in intuition. As the Grothendieck topos E ‘pretends to be’ the category of Sets, but with a strange logic, we can ‘do’ simplicial set theory in Simp(E) as long as we take care of the arguments we use. To see a bit of this in action we can note that the object [0] in Simp(E) will be the constant simplicial sheaf with value the ordinary [0], “constant” here taking on two meanings at the same time, (a) constant sheaf, i.e. not varying ‘over X’ if E is thought of as Sh(X), and (b) constant simplicial object, i.e. each Kn is the same and all face and degeneracy maps are identities. Thus [0] interpreted as an étale space is the identity map X → X as a space over X. Of course not all simplicial objects are constant and so Simp(E) can store a lot of information about the space (or site) X. One can look at the homotopy structure of Simp(E). Ken Brown showed it had a fibration category structure (i.e. more or less dual to the axioms) and if we look at those fibrant objects K in which the natural map

p : K → [0]

is a weak equivalence, we find that these K are exactly the hypercoverings. Global sections of p give a simplicial set, Γ(K) and varying K amongst the hypercoverings gives a pro-simplicial set (still in proHo(S) not in Hopro(S) unfortunately) which determines the Artin-Mazur pro-homotopy type of E.

This makes the link between shape theoretic methods and derived category theory more explicit. In the first, the ‘space’ is resolved using ‘coverings’ and these, in a sheaf theoretic setting, lead to simplicial objects in Sh(X) that are weakly equivalent to [0]; in the second, to evaluate the derived functor of some functor F : C → A, say, on an object C, one takes the ‘average’ of the values of F on objects weakly equivalent to G, i.e. one works with the functor

F′ : W(C) → A

(where W(C) has objects, α : C → C′, α a weak equivalence, and maps, the commuting ‘triangles’, and this has a ‘domain’ functor δ : W(C) → C, δ(α) = C′ and F′ is the composite Fδ). This is in many cases a pro-object in A – unfortunately standard derived functor theory interprets ‘commuting triangles’ in too weak a sense and thus corresponds to shape rather than strong shape theory – one thus, in some sense, arrives in proHo(A) instead of in Ho(proA).