# 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.

# Super Lie Algebra

A super Lie algebra L is an object in the category of super vector spaces together with a morphism [ , ] : L ⊗ L → L, often called the super bracket, or simply, the bracket, which satisfies the following conditions

Anti-symmetry,

[ , ] + [ , ] ○ cL,L = 0

which is the same as

[x, y] + (-1)|x||y|[y, x] = 0 for x, y ∈ L homogenous.

Jacobi identity,

[, [ , ]] + [, [ , ]] ○ σ + [, [ , ]] ○ σ2 = 0,

where σ ∈ S3 is a three-cycle, i.e. taking the first entity of [, [ , ]] to the second, and the second to the third, and then the third to the first. So, for x, y, z ∈ L homogenous, this reads

[x + [y, z]] + (-1)|x||y| + |x||z|[y, [z, x]] + (-1)|y||z| + |x||z|[z, [x, y]] = 0

It is important to note that in the super category, these conditions are modifications of the properties of the bracket in a Lie algebra, designed to accommodate the odd variables. We can immediately extend this definition to the case where L is an A-module for A a commutative superalgebra, thus defining a Lie superalgebra in the category of A-modules. In fact, we can make any associative superalgebra A into a Lie superalgebra by taking the bracket to be

[a, b] = ab – (-1)|a||b|ba,

i.e., we take the bracket to be the difference τ – τ ○ cA,A, where τ is the multiplication morphism on A.

A left A-module is a super vector space M with a morphism A ⊗ M → M, a ⊗ m ↦ am, of super vector spaces obeying the usual identities; that is, ∀ a, b ∈ A and x, y ∈ M, we have

a (x + y) = ax + ay

(a + b)x = ax + bx

(ab)x  = a(bx)

1x = x

A right A-module is defined similarly. Note that if A is commutative, a left A-module is also a right A-module if we define (the sign rule)

m . a = (-1)|m||a|a . m

for m ∈ M and a ∈ A. Morphisms of A-modules are defined in the obvious manner: super vector space morphisms φ: M → N such that φ(am) = aφ(m) ∀ a ∈ A and m ∈ M. So, we have the category of A-modules. For A commutative, the category of A-modules admits tensor products: for M1, M2 A-modules, M1 ⊗ M2 is taken as the tensor product of M1 as a right module with M2 as a left module.

Turning our attention to free A-modules, we have the notion of super vector kp|q over k, and so we define Ap|q := A ⊗ kp|q where

(Ap|q)0 = A0 ⊗ (kp|q)0 ⊕ A1 ⊗ (kp|q)1

(Ap|q)1 = A1 ⊗ (kp|q)0 ⊕ A0 ⊗ (kp|q)1

We say that an A-module M is free if it is isomorphic (in the category of A-modules) to Ap|q for some (p, q). This is equivalent to saying that M contains p even elements {e1, …, ep} and q odd elements {ε1, …, εq} such that

M0 = spanA0{e1, …, ep} ⊕ spanA11, …, εq}

M1 = spanA1{e1, …, ep} ⊕ spanA01, …, εq}

We shall also say M as the free module generated over A by the even elements e1, …, eand the odd elements ε1, …, εq.

Let T: Ap|q → Ar|s be a morphism of free A-modules and then write ep+1, …., ep+q for the odd basis elements ε1, …, εq. Then T is defined on the basis elements {e1, …, ep+q} by

T(ej) = ∑i=1p+q eitij

Hence T can be represented as a matrix of size (r + s) x (p + q)

T = (T1 T2 T3 T4)

where T1 is an r x p matrix consisting of even elements of A, T2 is an r x q matrix of odd elements, T3 is an s x p matrix of odd elements, and T4 is an s x q matrix of even elements. When we say that T is a morphism of super A-modules, it means that it must preserve parity, and therefore the parity of the blocks, T1 & T4, which are even and T2 & T3, which are odd, is determined. When we define T on the basis elements, the basis elements precedes the coordinates tij. This is important to keep the signs in order and comes naturally from composing morphisms. In other words if the module is written as a right module with T acting from the left, composition becomes matrix product in the usual manner:

(S . T)(ej) = S(∑i eitij) = ∑i,keksiktij

hence for any x ∈ Ap|q , we can express x as the column vector x = ∑eixi and so T(x) is given by the matrix product T x.

# 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

where

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.

# Superalgebras

Let k be an algebraically closed field. Given a superalgebra A we will denote with A0 the even part, with A1 the odd part and with IAodd the ideal generated by the odd part.

A superalgebra is said to be commutative (or supercommutative) if

xy = (−1)p(x)p(y)yx, ∀ homogeneous x, y

where p denotes the parity of an homogeneous element (p(x) = 0 if x ∈ A0, p(x) = 1 if x ∈ A1).

Let’s denote with A the category of affine superalgebras that is commutative superalgebras such that, modulo the ideal generated by their odd part, they are affine algebras (an affine algebra is a finitely generated reduced commutative algebra).

Define affine algebraic supervariety over k a representable functor V from the category A of affine superalgebras to the category S of sets. Let’s call k[V] the commutative k-superalgebra representing the functor V,

V (A) = Homk−superalg(k[V], A), A ∈ A

We will call V (A) the A-points of the variety V. A morphism of affine supervarieties is identified with a morphism between the representing objects, that is a morphism of affine superalgebras.

We also define the functor Vred associated to V from the category Ac of affine k-algebras to the category of sets:

Vred(Ac)= Homk−alg(k[V]/Ik[V]odd, Ac), Ac ∈ Ac

Vred is an affine algebraic variety and it is called the reduced variety associated to V. If the algebra k[V] representing the functor V has the additional structure of a commutative Hopf superalgebra, we say that V is an affine algebraic supergroup.

Let G be an affine algebraic supergroup. As in the classical setting, the condition k[G] being a commutative Hopf superalgebra makes the functor group valued, that is the product of two morphisms is still a morphism. In fact let A be a commutative superalgebra and let x, y ∈ Homk−superalg(k[G], A) be two points of G(A). The product of x and y is defined as:

x · y = defmA · x ⊗ y · ∆

where mA is the multiplication in A and ∆ the comultiplication in k[G]. One can find that x · y ∈ Homk−superalg(k[G], A), that is:

(x · y)(ab) = (x · y)(a)(x · y)(b)

The non commutativity of the Hopf algebra in the quantum setting does not allow to multiply morphisms(=points). In fact in the quantum (super)group setting the product of two morphisms is not in general a morphism.

Let V be an affine algebraic supervariety. Let k0 ⊂ k be a subfield of k. We say that V is a k0-variety if there exists a k0-superalgebra k0[V] such that k[V] ≅ k0[V] ⊗k0 k and

V(A) = Homk0 − superalg(k0[V], A) = Homk−superalg(k[V], A), A ∈ A.

We obtain a functor that we still denote by V from the category Ak0 of affine k0-superalgebras to the category of sets:

V(Ak0) = Homk0−superalg(k0[V], Ak0), A ∈ Ak0

thus opening up for consideration of rationality on supervariety.

# 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).