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.

Spinorial Algebra


Superspace is to supersymmetry as Minkowski space is to the Lorentz group. Superspace provides the most natural geometrical setting in which to describe supersymmetrical theories. Almost no physicist would utilize the component of Lorentz four-vectors or higher rank tensor to describe relativistic physics.

In a field theory, boson and fermions are to be regarded as diffeomorphisms generating two different vector spaces; the supersymmetry generators are nothing but sets of linear maps between these spaces. We can thus include a supersymmetric theory in a more general geometrical framework defining the collection of diffeomorphisms,

φi : R → RdL, i = 1,…, dL —– (1)

ψαˆ : R → RdR, i = 1,…, dR —– (2)

where the one-dimensional dependence reminds us that we restrict our attention to mechanics. The free vector spaces generated by {φi}i=1dL and {ψαˆ}αˆdR are respectively VL and VR, isomorphic to RdL and RdR. For matrix representations in the following, the two integers are restricted to the case dL = dR = d. Four different linear mappings can act on VL and VR

ML : VL → VR, MR : VR → VL

UL : VL → VL, UR : VR → VR —– (3)

with linear map space dimensions

dimML = dimMR = dRdL = d2,

dimUL = dL2 = d2, dimUR = dR2 = d2 —– (4)

as a consequence of linearity. To relate this construction to a general real (≡ GR) algebraic structure of dimension d and rank N denoted by GR(d,N), two more requirements need to be added.

Defining the generators of GR(d,N) as the family of N + N linear maps

LI ∈ {ML}, I = 1,…, N

RK ∈ {MR}, K = 1,…, N —– (5)

such that ∀ I, K = 1,…, N, we have

LI ◦ RK + LK ◦ RI = −2δIKIVR

RI ◦ LK + RK ◦ LI = −2δIKIVL —– (6)

where IVL and IVR are identity maps on VL and VR. Equations (6) will later be embedded into a Clifford algebra but one point has to be emphasized, we are working with real objects.

After equipping VL and VR with euclidean inner products ⟨·,·⟩VL and ⟨·,·⟩VR, respectively, the generators satisfy the property

⟨φ, RI(ψ)⟩VL = −⟨LI(φ), ψ⟩VR, ∀ (φ, ψ) ∈ VL ⊕ VR —— (7)

This condition relates LI to the hermitian conjugate of RI, namely RI, defined as usual by

⟨φ, RI(ψ)⟩VL = ⟨RI(φ), ψ⟩VR —– (8)

such that

RI = RIt = −LI —– (9)

The role of {UL} and {UR} maps is to connect different representations once a set of generators defined by conditions (6) and (7) has been chosen. Notice that (RILJ)ij ∈ UL and (LIRJ)αˆβˆ ∈ UR. Let us consider A ∈ {UL} and B ∈ {UR} such that

A : φ → φ′ = Aφ

B : ψ → ψ′ = Bψ —– (10)

with Vas an example,

⟨φ, RI(ψ)⟩VL → ⟨Aφ, RI B(ψ)⟩VL

= ⟨φ,A RI B(ψ)⟩VL

= ⟨φ, RI (ψ)⟩VL —– (11)

so a change of representation transforms the generators in the following manner:


RI → RI = ARIB —– (12)

In general (6) and (7) do not identify a unique set of generators. Thus, an equivalence relation has to be defined on the space of possible sets of generators, say {LI, RI} ∼ {LI, RI} iff ∃ A ∈ {UL} and B ∈ {UR} such that L′ = BLIA and R′ = ARIB.

Moving on to how supersymmetry is born, we consider the manner in which algebraic derivations are defined by

δεφi = iεI(RI)iαˆψαˆ

δεψαˆ = −εI(LI)αˆiτφi —– (13)

where the real-valued fields {φi}i=1dL and {ψαˆ}αˆ=1dR can be interpreted as bosonic and fermionic respectively. The fermionic nature attributed to the VR elements implies that ML and MR generators, together with supersymmetry transformation parameters εI, anticommute among themselves. Introducing the dL + dR dimensional space VL ⊕ VR with vectors

Ψ = (ψ φ) —– (14)

(13) reads

δε(Ψ) = (iεRψ εL∂τφ) —– (15)

such that

ε1, δε2]Ψ = iε1Iε2J (RILJτφ LIRJτψ) – iε2Jε1I (RJLIτφ LJRIτψ) = – 2iε1Iε2IτΨ —– (16)

utilizing that we have classical anticommuting parameters and that (6) holds. From (16) it is clear that δε acts as a supersymmetry generator, so that we can set

δQΨ := δεΨ = iεIQIΨ —– (17)

which is equivalent to writing

δQφi = i(εIQIψ)i

δQψαˆ = i(εIQIφ)αˆ —– (18)


Q1 = (0LIH RI0) —– (19)

where H = i∂τ. As a consequence of (16) a familiar anticommutation relation appears

{QI, QJ} = − 2iδIJH —– (20)

confirming that we are about to recognize supersymmetry, and once this is achieved, we can associate to the algebraic derivations (13), the variations defining the scalar supermultiplets. However, the choice (13) is not unique, for this is where we could have a spinorial one,

δQξαˆ = εI(LI)αˆiFi

δQFi = − iεI(RI)iαˆτξαˆ —– (21)

Principal Bundles Preserve Structures…


A bundle P = (P, M ,π; G) is a principal bundle if the standard fiber is a Lie group G and ∃ (at least) one trivialization the transition functions of which act on G by left translations Lg : G → G : h ↦ f  g . h (where . denotes here the group multiplication).

The principal bundles are slightly different from affine bundles and vector bundles. In fact, while in affine bundles the fibers π-1(x) have a canonical structure of affine spaces and in vector bundles the fibers π-1(x) have a canonical structure of vector spaces, in principal bundles the fibers have no canonical Lie group structure. This is due to the fact that, while in affine bundles transition functions act by means of affine transformations and in vector bundles transition functions act by means of linear transformations, in principal bundles transition functions act by means of left translations which are not group automorphisms. Thus the fibers of a principal bundle do not carry a canonical group structure, but rather many non-canonical (trivialization-depending) group structures. In the fibers of a vector bundle there exists a preferred element (the “zero”) the definition of which does not depend on the local trivialization. On the contrary, in the fibers of a principal bundle there is no preferred point which is fixed by transition functions to be selected as an identity. Thus, while in affine bundles affine morphisms are those which preserve the affine structure of the fibers and in vector bundles linear morphisms are the ones which preserve the linear structure of the fibers, in a principal bundle P = (P, M, π; G) principal morphisms preserve instead a structure, the right action of G on P.

Let P = (P, M, π; G) be a principal bundle and {(Uα, t(α)}α∈I a trivialization. We can locally consider the maps

R(α)g : π-1(Uα) → π-1(Uα) : [x, h](α) ↦ [x, h . g](α) —– (1)

∃ a (global) right action Rg of G on P which is free, vertical and transitive on fibers; the local expression in the given trivialization of this action is given by R(α)g .

Using the local trivialization, we set p = [x, h](α) = [x, g(βα)(x) . h]β following diagram commutes:


which clearly shows that the local expressions agree on the overlaps Uαβ, to define a right action. This is obviously a vertical action; it is free because of the following:

Rgp = p => [x, h . g](α) = [x, h](α) => h · g = h => g = e —– (2)

Finally, if p = [x, h1](α) and q = [x, h2](α) are two points in the same fiber of p, one can choose g = h2-1 . h1 (where · denotes the group multiplication) so that p = Rgq. This shows that the right action is also transitive on the fibers.

On the contrary, that a global left action cannot be defined by using the local maps

L(α)g : π-1(Uα) → π-1(Uα) : [x, h](α) ↦ [x, g . h](α) —– (3)

since these local maps do not satisfy a compatibility condition analogous to the condition of the commuting diagram.

let P = (P, M, π; G) and P’ = (P’, M’, π’ ; G’ ) be two principal bundles and θ : G → G’ be a homomorphism of Lie groups. A bundle morphism Φ = (Φ, φ) : P → P’ is a principal morphism with respect to θ if the following diagram is commutative:


When G = G’ and θ = idG we just say that Φ is a principal morphism.

A trivial principal bundle (M x G, M, π; G) naturally admits the global unity section I ∈ Γ(M x G), defined with respect to a global trivialization, I : x ↦ (x, e), e being the unit element of G. Also, principal bundles allow global sections iff they are trivial. In fact, on principal bundles there is a canonical correspondence between local sections and local trivializations, due to the presence of the global right action.

Philosophical Isomorphism of Category Theory. Note Quote.

One philosophical reason for categorification is that it refines our concept of ‘sameness’ by allowing us to distinguish between isomorphism and equality. In a set, two elements are either the same or different. In a category, two objects can be ‘the same in a way’ while still being different. In other words, they can be isomorphic but not equal. Even more importantly, two objects can be the same in more than one way, since there can be different isomorphisms between them. This gives rise to the notion of the ‘symmetry group’ of an object: its group of automorphisms.

Consider, for example, the fundamental groupoid Π1(X) of a topological space X: the category with points of X as objects and homotopy classes of paths with fixed endpoints as morphisms. This category captures all the homotopy-theoretic information about X in dimensions ≤ 1. The group of automorphisms of an object x in this category is just the fundamental group π1(X,x). If we decategorify the fundamental groupoid of X, we forget how points in X are connected by paths, remembering only whether they are, and we obtain the set of components of X. This captures only the homotopy 0-type of X.

This example shows how decategorification eliminates ‘higher-dimensional information’ about a situation. Categorification is an attempt to recover this information. This example also suggests that we can keep track of the homotopy 2-type of X if we categorify further and distinguish between paths that are equal and paths that are merely isomorphic (i.e., homotopic). For this we should work with a ‘2-category’ having points of X as objects, paths as morphisms, and certain equivalence classes of homotopies between paths as 2-morphisms. In a marvelous self-referential twist, the definition of ‘2-category’ is simply the categorification of the definition of ‘category’. Like a category, a 2-category has a class of objects, but now for any pair x,y of objects there is no longer a set hom(x,y); instead, there is a category hom(x,y). Objects of hom(x,y) are called morphisms of C, and morphisms between them are called 2-morphisms of C. Composition is no longer a function, but rather a functor:

◦: hom(x, y) × hom(y, z) → hom(x, z)

For any object x there is an identity 1x ∈ hom(x,x). And now we have a choice. On the one hand, we can impose associativity and the left and right unit laws strictly, as equational laws. If we do this, we obtain the definition of ‘strict 2-category’. On the other hand, we can impose them only up to natural isomorphism, with these natural isomorphisms satisfying the coherence. This is clearly more compatible with the spirit of categorification. If we do this, we obtain the definition of ‘weak 2-category’. (Strict 2-categories are traditionally known as ‘2-categories’, while weak 2-categories are known as ‘bicategories’.)

The classic example of a 2-category is Cat, which has categories as objects, functors as morphisms, and natural transformations as 2-morphisms. The presence of 2-morphisms gives Cat much of its distinctive flavor, which we would miss if we treated it as a mere category. Indeed, Mac Lane has said that categories were originally invented, not to study functors, but to study natural transformations! A good example of two functors that are not equal, but only naturally isomorphic, are the identity functor and the ‘double dual’ functor on the category of finite-dimensional vector spaces. Given a topological space X, we can form a 2-category Π>sub>2(X) called the ‘fundamental 2-groupoid’ of X. The objects of this 2-category are the points of X. Given x, y ∈ X, the morphisms from x to y are the paths f: [0,1] → X starting at x and ending at y. Finally, given f, g ∈ hom(x, y), the 2-morphisms from f to g are the homotopy classes of paths in hom(x, y) starting at f and ending at g. Since the associative law for composition of paths holds only up to homotopy, this 2-category is a weak 2-category. If we decategorify the fundamental 2-groupoid of X, we obtain its fundamental groupoid.

From 2-categories it is a short step to dreaming of n-categories and even ω-categories — but it is not so easy to make these dreams into smoothly functioning mathematical tools. Roughly speaking, an n-category should be some sort of algebraic structure having objects, 1-morphisms between objects, 2-morphisms between 1-morphisms, and so on up to n-morphisms. There should be various ways of composing j-morphisms for 1 ≤ j ≤ n, and these should satisfy various laws. As with 2-categories, we can try to impose these laws either strictly or weakly.


Other approaches to n-categories use j-morphisms with other shapes, such as simplices, or opetopes. We believe that there is basically a single notion of weak n-category lurking behind these different approaches. If this is true, they will eventually be shown to be equivalent, and choosing among them will be merely a matter of convenience. However, the precise meaning of ‘equivalence’ here is itself rather subtle and n-categorical in flavor.

The first challenge to any theory of n-categories is to give an adequate treatment of coherence laws. Composition in an n-category should satisfy equational laws only at the top level, between n-morphisms. Any law concerning j-morphisms for j < n should hold only ‘up to equivalence’. Here a n-morphism is defined to be an ‘equivalence’ if it is invertible, while for j < n a j-morphism is recursively defined to be an equivalence if it is invertible up to equivalence. Equivalence is generally the correct substitute for the notion of equality in n-categorical mathematics. When laws are formulated as equivalences, these equivalences should in turn satisfy coherence laws of their own, but again only up to equivalence, and so on. This becomes ever more complicated and unmanageable with increasing n unless one takes a systematic approach to coherence laws.

The second challenge to any theory of n-categories is to handle certain key examples. First, for any n, there should be an (n + 1)-category nCat, whose objects are (small) n-categories, whose morphisms are suitably weakened functors between these, whose 2-morphisms are suitably weakened natural transformations, and so on. Here by ‘suitably weakened’ we refer to the fact that all laws should hold only up to equivalence. Second, for any topological space X, there should be an n-category Πn(X) whose objects are points of X, whose morphisms are paths, whose 2-morphisms are paths of paths, and so on, where we take homotopy classes only at the top level. Πn(X) should be an ‘n-groupoid’, meaning that all its j-morphisms are equivalences for 0 ≤ j ≤ n. We call Πn(X) the ‘fundamental n-groupoid of X’. Conversely, any n-groupoid should determine a topological space, its ‘geometric realization’.

In fact, these constructions should render the study of n-groupoids equivalent to that of homotopy n-types. A bit of the richness inherent in the concept of n-category becomes apparent when we make the following observation: an (n + 1)-category with only one object can be regarded as special sort of n-category. Suppose that C is an (n+1)-category with one object x. Then we can form the n-category C ̃ by re-indexing: the objects of C ̃ are the morphisms of C, the morphisms of C ̃ are the 2-morphisms of C, and so on. The n-categories we obtain this way have extra structure. In particular, since the objects of C ̃ are really morphisms in C from x to itself, we can ‘multiply’ (that is, compose) them.

The simplest example is this: if C is a category with a single object x, C ̃ is the set of endomorphisms of x. This set is actually a monoid. Conversely, any monoid can be regarded as the monoid of endomorphisms of x for some category with one object x. We summarize this situation by saying that ‘a one-object category is a monoid’. Similarly, a one-object 2-category is a monoidal category. It is natural to expect this pattern to continue in all higher dimensions; in fact, it is probably easiest to cheat and define a monoidal n-category to be an (n + 1)-category with one object.

Things get even more interesting when we iterate this process. Given an (n + k)-category C with only one object, one morphism, and so on up to one (k − 1)-morphism, we can form an n-category whose j-morphisms are the (j + k)-morphisms of C. In doing so we obtain a particular sort of n-category with extra structure and properties, which we call a ‘k-tuply monoidal’ n-category. Table below shows what we expect these to be like for low values of n and k. For example, the Eckmann-Hilton argument shows that a 2-category with one object and one morphism is a commutative monoid. Categorifying this argument, one can show that a 3-category with one object and one morphism is a braided monoidal category. Similarly, we expect that a 4-category with one object, one morphism and one 2-morphism is a symmetric monoidal category, though this has not been worked out in full detail, because of our poor understanding of 4-categories. The fact that both braided and symmetric monoidal categories appear in this table seems to explain why both are natural concepts.


In any reasonable approach to n-categories there should be an n-category nCatk whose objects are k-tuply monoidal weak n-categories. One should also be able to treat nCatk as a full sub-(n + k)-category of (n + k)Cat, though even for low n, k this is perhaps not as well known as it should be. Consider for example n = 0, k = 1. The objects of 0Cat1 are one-object categories, or monoids. The morphisms of 0Cat1 are functors between one-object categories, or monoid homomorphisms. But 0Cat1 also has 2-morphisms corresponding to natural transformations.

• Decategorification: (n, k) → (n − 1, k). Let C be a k-tuply monoidal n-category C. Then there should be a k-tuply monoidal (n − 1)-category DecatC whose j-morphisms are the same as those of C for j < n − 1, but whose (n − 1)-morphisms are isomorphism classes of (n − 1)-morphisms of C.

• Discrete categorification: (n, k) → (n + 1, k). There should be a ‘discrete’ k-tuply monoidal (n + 1)-category DiscC having the j-morphisms of C as its j-morphisms for j ≤ n, and only identity (n + 1)-morphisms. The decategorification of DiscC should be C.

• Delooping: (n, k) → (n + 1, k − 1). There should be a (k − 1)-tuply monoidal (n + 1)-category BC with one object obtained by reindexing, the j-morphisms of BC being the (j + 1)-morphisms of C. We use the notation ‘B’ and call BC the ‘delooping’ of C because of its relation to the classifying space construction in topology.

• Looping: (n, k) → (n − 1, k + 1). Given objects x, y in an n-category, there should be an (n − 1)-category hom(x, y). If x = y this should be a monoidal (n−1)-category, and we denote it as end(x). For k > 0, if 1 denotes the unit object of the k-tuply monoidal n-category C, end(1) should be a (k + 1)-tuply monoidal (n − 1)-category. We call this process ‘looping’, and denote the result as ΩC, because of its relation to loop space construction in topology. For k > 0, looping should extend to an (n + k)-functor Ω: nCatk → (n − 1)Catk+1. The case k = 0 is a bit different: we should be able to loop a ‘pointed’ n-category, one having a distinguished object x, by letting ΩC = end(x). In either case, the j-morphisms of ΩC correspond to certain (j − 1)-morphisms of C.

• Forgetting monoidal structure: (n, k) → (n, k−1). By forgetting the kth level of monoidal structure, we should be able to think of C as a (k−1)-tuply monoidal n-category FC. This should extend to an n-functor F: nCatk → nCatk−1.

• Stabilization: (n, k) → (n, k + 1). Though adjoint n-functors are still poorly understood, there should be a left adjoint to forgetting monoidal structure, which is called ‘stabilization’ and denoted by S: nCatk → nCatk+1.

• Forming the generalized center: (n,k) → (n,k+1). Thinking of C as an object of the (n+k)-category nCatk, there should be a (k+1)-tuply monoidal n-category ZC, the ‘generalized center’ of C, given by Ωk(end(C)). In other words, ZC is the largest sub-(n + k + 1)-category of (n + k)Cat having C as its only object, 1C as its only morphism, 11C as its only 2-morphism, and so on up to dimension k. This construction gets its name from the case n = 0, k = 1, where ZC is the usual center of the monoid C. Categorifying leads to the case n = 1, k = 1, which gives a very important construction of braided monoidal categories from monoidal categories. In particular, when C is the monoidal category of representations of a Hopf algebra H, ZC is the braided monoidal category of representations of the quantum double D(H).

Emancipating Microlinearity from within a Well-adapted Model of Synthetic Differential Geometry towards an Adequately Restricted Cartesian Closed Category of Frölicher Spaces. Thought of the Day 15.0


Differential geometry of finite-dimensional smooth manifolds has been generalized by many authors to the infinite-dimensional case by replacing finite-dimensional vector spaces by Hilbert spaces, Banach spaces, Fréchet spaces or, more generally, convenient vector spaces as the local prototype. We know well that the category of smooth manifolds of any kind, whether finite-dimensional or infinite-dimensional, is not cartesian closed, while Frölicher spaces, introduced by Frölicher, do form a cartesian closed category. It seems that Frölicher and his followers do not know what a kind of Frölicher space, besides convenient vector spaces, should become the basic object of research for infinite-dimensional differential geometry. The category of Frölicher spaces and smooth mappings should be restricted adequately to a cartesian closed subcategory.


Synthetic differential geometry is differential geometry with a cornucopia of nilpotent infinitesimals. Roughly speaking, a space of nilpotent infinitesimals of some kind, which exists only within an imaginary world, corresponds to a Weil algebra, which is an entity of the real world. The central object of study in synthetic differential geometry is microlinear spaces. Although the notion of a manifold (=a pasting of copies of a certain linear space) is defined on the local level, the notion of microlinearity is defined absolutely on the genuinely infinitesimal level. What we should do so as to get an adequately restricted cartesian closed category of Frölicher spaces is to emancipate microlinearity from within a well-adapted model of synthetic differential geometry.

Although nilpotent infinitesimals exist only within a well-adapted model of synthetic differential geometry, the notion of Weil functor was formulated for finite-dimensional manifolds and for infinite-dimensional manifolds. This is the first step towards microlinearity for Frölicher spaces. Therein all Frölicher spaces which believe in fantasy that all Weil functors are really exponentiations by some adequate infinitesimal objects in imagination form a cartesian closed category. This is the second step towards microlinearity for Frölicher spaces. Introducing the notion of “transversal limit diagram of Frölicher spaces” after the manner of that of “transversal pullback” is the third and final step towards microlinearity for Frölicher spaces. Just as microlinearity is closed under arbitrary limits within a well-adapted model of synthetic differential geometry, microlinearity for Frölicher spaces is closed under arbitrary transversal limits.

From Vector Spaces to Categories. Part 6.


We began by thinking of categories as “posets with extra arrows”. This analogy gives excellent intuition for the general facts about adjoint functors. However, our intuition from posets is insufficient to actually prove anything about adjoint functors.

To complete the proofs we will switch to a new analogy between categories and vector spaces. Let V be a vector space over a field K and let V ∗ be the dual space consisting of K-linear functions V → K. Now consider any K-bilinear function ⟨−,−⟩ ∶ V × V → K. We say that the function ⟨−,−⟩ is non-degenerate in both coordinates if we have

⟨u1,v⟩ = ⟨u2,v⟩ ∀ v ∈ V ⇒ u1 = u2, ⟨u,v1⟩ = ⟨u,v2⟩ ∀ u ∈ V ⇒ v1 = v2

We say that two K-linear operators L ∶ V ⇄ V ∶ R define an adjunction with respect to ⟨−, −⟩ if, ∀ vectors u,v ∈ V, we have

⟨u, R(v)⟩ = ⟨L(u), v⟩

Uniqueness of Adjoint Operators. Let L ⊣ R be an adjoint pair of operators with respect to a non-degenerate bilinear function ⟨−, −⟩ ∶ V × V → K. Then each of L and R determines

the other uniquely.

Proof: To show that R determines L, suppose that L′ ⊣ R is another adjoint pair. Thus, ∀ vectors u,v ∈ V we have

⟨L(u), v⟩ = ⟨u, R(v)⟩ = ⟨L′(u), v⟩

Now consider any vector u ∈ V. The non-degeneracy of ⟨−, −⟩ tells us that

⟨L(u), v⟩ = ⟨L′(u), v⟩ ∀ v ∈ V ⇒ L(u) = L′(u)

and since this is true ∀ u ∈ V we conclude that L = L′

RAPL for Operators:

Suppose that the function ⟨−, −⟩ ∶ V × V → K is non-degenerate and continuous. Now let T ∶ V → V be any linear operator. If T has a left or a right adjoint, then T is continuous.


Suppose that T ∶ V → V has a left adjoint L ⊣ T, and suppose that the sequence of vectors vi ∈ V has a limit limivi ∈ V. Furthermore, suppose that the limit limiT(vi) ∈ V exists. Then for each u ∈ V, the continuity of ⟨−, −⟩ in the second coordinate tells us that

⟨u, T (limivi)⟩ = ⟨L(u), limivi

= limi⟨L(u), vi

= limi⟨u,T(vi)⟩

= ⟨u, limiT (vi)⟩

Since this is true for all u ∈ V, non-degeneracy gives

T (limivi) = limiT (vi)

The theorem can be made rigorous if we work with topological vector spaces. If (V, ∥ − ∥) is a normed (real or complex) vector space, then an operator T ∶ V → V is bounded if and only if it is continuous. Furthermore, if (V,⟨−,−⟩) is a Hilbert space then an operator T ∶ V → V having an adjoint is necessarily bounded, hence continuous. Many theorems of category have direct analogues in functional analysis. After all, Grothendieck began as a functional analyst.

We can summarize these two results as follows. Let ⟨−,−⟩ ∶ V ×V → K be a K-bilinear function. Then for each vector v ∈ V we have two elements of the dual space Hv, Hv ∈ V defined by

Hv ∶= ⟨v,−⟩ ∶ V → K,

Hv ∶= ⟨−,v⟩ ∶ V → K

The mappings v ↦ Hv and v ↦ Hv thus define two K-linear functions from V to V : H(−) ∶V → V and H(−) ∶ V → V

Furthermore, if the function is ⟨−,−⟩ is non-degenerate and continuous then the functions H(−), H(−) ∶ V → V are both injective and continuous.

the hom bifunctor

HomC(−,−) ∶ Cop × C → Set behaves like a “non-degenerate and continuous bilinear function”……