Grothendieckian Construction of K-Theory with a Bundle that is Topologically Trivial and Class that is Torsion.

All relativistic quantum theories contain “antiparticles,” and allow the process of particle-antiparticle annihilation. This inspires a physical version of the Grothendieck construction of K-theory. Physics uses topological K-theory of manifolds, whose motivation is to organize vector bundles over a space into an algebraic invariant, that turns out to be useful. Algebraic K-theory started from K_i defined for i, with relations to classical constructions in algebra and number theory, followed by Quillen’s homotopy-theoretic definition ∀ i. The connections to algebra and number theory often persist for larger values of i, but in ways that are subtle and conjectural, such as special values of zeta- and L-functions.

One could also use the conserved charges of a configuration which can be measured at asymptotic infinity. By definition, these are left invariant by any physical process. Furthermore, they satisfy quantization conditions, of which the prototype is the Dirac condition on allowed electric and magnetic charges in Maxwell theory.

There is an elementary construction which, given a physical theory T, produces an abelian group of conserved charges K(T). Rather than considering the microscopic dynamics of the theory, all that is needed to be known is a set S of “particles” described by T, and a set of “bound state formation/decay processes” by which the particles combine or split to form other particles. These are called “binding processes.” Two sets of particles are “physically equivalent” if some sequence of binding processes convert the one to the other. We then define the group K(T) as the abelian group Z^S of formal linear combinations of particles, quotiented by this equivalence relation.

Suppose T contains the particles S = {A,B,C}.

If these are completely stable, we could clearly define three integral conserved charges, their individual numbers, so K(T) ≅ Z³.

Introducing a binding process

A + B ↔ C —– (1)

with the bidirectional arrow to remind us that the process can go in either direction. Clearly K(T) ≅ Z² in this case.

One might criticize this proposal on the grounds that we have assumed that configurations with a negative number of particles can exist. However, in all physical theories which satisfy the constraints of special relativity, charged particles in physical theories come with “antiparticles,” with the same mass but opposite charge. A particle and antiparticle can annihilate (combine) into a set of zero charge particles. While first discovered as a prediction of the Dirac equation, this follows from general axioms of quantum field theory, which also hold in string theory.

Thus, there are binding processes

B + B̄ ↔ Z₁ + Z₂ + · · · .

where B̄ is the antiparticle to a particle B, and Z_i are zero charge particles, which must appear by energy conservation. To define the K-theory, we identify any such set of zero charge particles with the identity, so that

B + B̄ ↔ 0

Thus the antiparticles provide the negative elements of K(T).

Granting the existence of antiparticles, this construction of K-theory can be more simply rephrased as the Grothendieck construction. We can define K(T) as the group of pairs (E, F) ∈ (Z^S, Z^S), subject to the relations (E, F) ≅ (E+B, F +B) ≅ (E+L, F +R) ≅ (E+R, F +L), where (L, R) are the left and right hand side of a binding process (1).

Thinking of these as particles, each brane B must have an antibrane, which we denote by B̄. If B wraps a submanifold L, one expects that B̄ is a brane which wraps a submanifold L of opposite orientation. A potential problem is that it is not a priori obvious that the orientation of L actually matters physically, especially in degenerate cases such as L a point.

Now, let us take X as a Calabi-Yau threefold for definiteness. A physical A-brane, which are branes of the A-model topological string and thereby a TQFT shadow of the D-branes of the superstring, is specified by a pair (L, E) of a special Lagrangian submanifold L with a flat bundle E. The obvious question could be: When are (L₁, E₁) and (L₂, E₂) related by a binding process? A simple heuristic answer to this question is given by the Feynman path integral. Two configurations are connected, if they are connected by a continuous path through the configuration space; any such path (or a small deformation of it) will appear in the functional integral with some non-zero weight. Thus, the question is essentially topological. Ignoring the flat bundles for a moment, this tells us that the K-theory group for A-branes is H₃(Y, Z), and the class of a brane is simply (rank E)·[L] ∈ H₃(Y, Z). This is also clear if the moduli space of flat connections on L is connected.

But suppose it is not, say π₁(L) is torsion. In this case, we need deeper physical arguments to decide whether the K-theory of these D-branes is H₃(Y, Z), or some larger group. But a natural conjecture is that it will be K₁(Y), which classifies bundles on odd-dimensional submanifolds. Two branes which differ only in the choice of flat connection are in fact connected in string theory, consistent with the K-group being H₃(Y, Z). For Y a simply connected Calabi-Yau threefold, K₁(Y) ≅ H₃(Y, Z), so the general conjecture is borne out in this case

There is a natural bilinear form on H₃(Y, Z) given by the oriented intersection number

I(L₁, L₂) = #([L₁] ∩ [L₂]) —– (2)

It has symmetry (−1)ⁿ. In particular, it is symplectic for n = 3. Furthermore, by Poincaré duality, it is unimodular, at least in our topological definition of K-theory.

D-branes, which are extended objects defined by mixed Dirichlet-Neumann boundary conditions in string theory, break half of the supersymmetries of the type II superstring and carry a complete set of electric and magnetic Ramond-Ramond charges. The product of the electric and magnetic charges is a single Dirac unit, and that the quantum of charge takes the value required by string duality. Saying that a D-brane has RR-charge means that it is a source for an “RR potential,” a generalized (p + 1)-form gauge potential in ten-dimensional space-time, which can be verified from its world-volume action that contains a minimal coupling term,

∫C^{(p + 1)} —–(3)

where C^{(p + 1)} denotes the gauge potential, and the integral is taken over the (p+1)-dimensional world-volume of the brane. For p = 0, C⁽¹⁾ is a one-form or “vector” potential (as in Maxwell theory), and thus the D0-brane is an electrically charged particle with respect to this 10d Maxwell theory. Upon further compactification, by which, the ten dimensions are R⁴ × X, and a Dp-brane which wraps a p-dimensional cycle L; in other words its world-volume is R × L where R is a time-like world-line in R⁴. Using the Poincaré dual class ω_L ∈ H^2n−p(X, R) to L in X, to rewrite (3) as an integral

∫_{R × X} C^{(p + 1)} ∧ ω_L —– (4)

We can then do the integral over X to turn this into the integral of a one-form over a world-line in R⁴, which is the right form for the minimal electric coupling of a particle in four dimensions. Thus, such a wrapped brane carries a particular electric charge which can be detected at asymptotic infinity. Summarizing the RR-charge more formally,

∫_LC = ∫_XC ∧ ω_L —– (5)

where C ∈ H∗(X, R). In other words, it is a class in H_p(X, R).

In particular, an A-brane (for n = 3) carries a conserved charge in H₃(X, R). Of course, this is weaker than [L] ∈ H₃(X, Z). To see this physically, we would need to see that some of these “electric” charges are actually “magnetic” charges, and study the Dirac-Schwinger-Zwanziger quantization condition between these charges. This amounts to showing that the angular momentum J of the electromagnetic field satisfies the quantization condition J = ħn/2 for n ∈ Z. Using an expression from electromagnetism, J⃗ = E⃗ × B⃗ , this is precisely the condition that (2) must take an integer value. Thus the physical and mathematical consistency conditions agree. Similar considerations apply for coisotropic A-branes. If X is a genuine Calabi-Yau 3-fold (i.e., with strict SU(3) holonomy), then a coisotropic A-brane which is not a special Lagrangian must be five-dimensional, and the corresponding submanifold L is rationally homologically trivial, since H⁵(X, Q) = 0. Thus, if the bundle E is topologically trivial, the homology class of L and thus its K-theory class is torsion.

If X is a torus, or a K3 surface, the situation is more complicated. In that case, even rationally the charge of a coisotropic A-brane need not lie in the middle-dimensional cohomology of X. Instead, it takes its value in a certain subspace of ⊕_p H^p(X, Q), where the summation is over even or odd p depending on whether the complex dimension of X is even or odd. At the semiclassical level, the subspace is determined by the condition

(L − Λ)α = 0, α ∈ ⊕_p H^p(X, Q)

where L and Λ are generators of the Lefschetz SL(2, C) action, i.e., L is the cup product with the cohomology class of the Kähler form, and Λ is its dual.

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The Affinity of Mirror Symmetry to Algebraic Geometry: Going Beyond Formalism

Even though formalism of homological mirror symmetry is an established case, what of other explanations of mirror symmetry which lie closer to classical differential and algebraic geometry? One way to tackle this is the so-called Strominger, Yau and Zaslow mirror symmetry or SYZ in short.

The central physical ingredient in this proposal is T-duality. To explain this, let us consider a superconformal sigma model with target space (M, g), and denote it (defined as a geometric functor, or as a set of correlation functions), as

CFT(M, g)

In physics, a duality is an equivalence

CFT(M, g) ≅ CFT(M′, g′)

which holds despite the fact that the underlying geometries (M,g) and (M′, g′) are not classically diffeomorphic.

T-duality is a duality which relates two CFT’s with toroidal target space, M ≅ M′ ≅ T^d, but different metrics. In rough terms, the duality relates a “small” target space, with noncontractible cycles of length L < l_s, with a “large” target space in which all such cycles have length L > l_s.

This sort of relation is generic to dualities and follows from the following logic. If all length scales (lengths of cycles, curvature lengths, etc.) are greater than l_s, string theory reduces to conventional geometry. Now, in conventional geometry, we know what it means for (M, g) and (M′, g′) to be non-isomorphic. Any modification to this notion must be associated with a breakdown of conventional geometry, which requires some length scale to be “sub-stringy,” with L < l_s. To state T-duality precisely, let us first consider M = M′ = S¹. We parameterise this with a coordinate X ∈ R making the identification X ∼ X + 2π. Consider a Euclidean metric g_R given by ds² = R²dX². The real parameter R is usually called the “radius” from the obvious embedding in R². This manifold is Ricci-flat and thus the sigma model with this target space is a conformal field theory, the “c = 1 boson.” Let us furthermore set the string scale l_s = 1. With this, we attain a complete physical equivalence.

CFT(S¹, g_R) ≅ CFT(S¹, g_1/R)

Thus these two target spaces are indistinguishable from the point of view of string theory.

Just to give a physical picture for what this means, suppose for sake of discussion that superstring theory describes our universe, and thus that in some sense there must be six extra spatial dimensions. Suppose further that we had evidence that the extra dimensions factorized topologically and metrically as K₅ × S¹; then it would make sense to ask: What is the radius R of this S¹ in our universe? In principle this could be measured by producing sufficiently energetic particles (so-called “Kaluza-Klein modes”), or perhaps measuring deviations from Newton’s inverse square law of gravity at distances L ∼ R. In string theory, T-duality implies that R ≥ l_s, because any theory with R < l_s is equivalent to another theory with R > l_s. Thus we have a nontrivial relation between two (in principle) observable quantities, R and l_s, which one might imagine testing experimentally. Let us now consider the theory CFT(T^d, g), where T^d is the d-dimensional torus, with coordinates Xⁱ parameterising R^d/2πZ^d, and a constant metric tensor g_ij. Then there is a complete physical equivalence

CFT(T^d, g) ≅ CFT(T^d, g⁻¹)

In fact this is just one element of a discrete group of T-duality symmetries, generated by T-dualities along one-cycles, and large diffeomorphisms (those not continuously connected to the identity). The complete group is isomorphic to SO(d, d; Z).

While very different from conventional geometry, T-duality has a simple intuitive explanation. This starts with the observation that the possible embeddings of a string into X can be classified by the fundamental group π₁(X). Strings representing non-trivial homotopy classes are usually referred to as “winding states.” Furthermore, since strings interact by interconnecting at points, the group structure on π₁ provided by concatenation of based loops is meaningful and is respected by interactions in the string theory. Now π₁(T^d) ≅ Z^d, as an abelian group, referred to as the group of “winding numbers”.

Of course, there is another Z^d we could bring into the discussion, the Pontryagin dual of the U(1)^d of which T^d is an affinization. An element of this group is referred to physically as a “momentum,” as it is the eigenvalue of a translation operator on T^d. Again, this group structure is respected by the interactions. These two group structures, momentum and winding, can be summarized in the statement that the full closed string algebra contains the group algebra C[Z^d] ⊕ C[Z^d].

In essence, the point of T-duality is that if we quantize the string on a sufficiently small target space, the roles of momentum and winding will be interchanged. But the main point can be seen by bringing in some elementary spectral geometry. Besides the algebra structure, another invariant of a conformal field theory is the spectrum of its Hamiltonian H (technically, the Virasoro operator L₀ + L ^̄₀). This Hamiltonian can be thought of as an analog of the standard Laplacian ∆_g on functions on X, and its spectrum on T^d with metric g is

Spec ∆_g = {∑_i,j=1^d g^ijp_ip_j; p_i ∈ Z^d}

On the other hand, the energy of a winding string is (intuitively) a function of its length. On our torus, a geodesic with winding number w ∈ Z^d has length squared

L² = ∑_i,j=1^d g_ijwⁱw^j

Now, the only string theory input we need to bring in is that the total Hamiltonian contains both terms,

H = ∆_g + L² + · · ·

where the extra terms … express the energy of excited (or “oscillator”) modes of the string. Then, the inversion g → g⁻¹, combined with the interchange p ↔ w, leaves the spectrum of H invariant. This is T-duality.

There is a simple generalization of the above to the case with a non-zero B-field on the torus satisfying dB = 0. In this case, since B is a constant antisymmetric tensor, we can label CFT’s by the matrix g + B. Now, the basic T-duality relation becomes

CFT(T^d, g + B) ≅ CFT(T^d, (g + B)⁻¹)

Another generalization, which is considerably more subtle, is to do T-duality in families, or fiberwise T-duality. The same arguments can be made, and would become precise in the limit that the metric on the fibers varies on length scales far greater than l_s, and has curvature lengths far greater than l_s. This is sometimes called the “adiabatic limit” in physics. While this is a very restrictive assumption, there are more heuristic physical arguments that T-duality should hold more generally, with corrections to the relations proportional to curvatures l_s²R and derivatives l_s∂ of the fiber metric, both in perturbation theory and from world-sheet instantons.

Categories of Pointwise Convergence Topology: Theory(ies) of Bundles.

Let H be a fixed, separable Hilbert space of dimension ≥ 1. Lets denote the associated projective space of H by P = P(H). It is compact iff H is finite-dimensional. Let PU = PU(H) = U(H)/U(1) be the projective unitary group of H equipped with the compact-open topology. A projective bundle over X is a locally trivial bundle of projective spaces, i.e., a fibre bundle P → X with fibre P(H) and structure group PU(H). An application of the Banach-Steinhaus theorem shows that we may identify projective bundles with principal PU(H)-bundles and the pointwise convergence topology on PU(H).

If G is a topological group, let GX denote the sheaf of germs of continuous functions G → X, i.e., the sheaf associated to the constant presheaf given by U → F(U) = G. Given a projective bundle P → X and a sufficiently fine good open cover {U_i}_i∈I of X, the transition functions between trivializations P|_Ui can be lifted to bundle isomorphisms g_ij on double intersections U_ij = U_i ∩ U_j which are projectively coherent, i.e., over each of the triple intersections U_ijk = U_i ∩ U_j ∩ U_k the composition g_ki g_jk g_ij is given as multiplication by a U(1)-valued function f_ijk : U_ijk → U(1). The collection {(U_ij, f_ijk)} defines a U(1)-valued two-cocycle called a B-field on X,which represents a class B_P in the sheaf cohomology group H²(X, U(1)_X). On the other hand, the sheaf cohomology H¹(X, PU(H)_X) consists of isomorphism classes of principal PU(H)-bundles, and we can consider the isomorphism class [P] ∈ H¹(X,PU(H)_X).

There is an isomorphism

H¹(X, PU(H)_X) →^≈ H²(X, U(1)_X) provided by the

boundary map [P] ↦ B_P. There is also an isomorphism

H²(X, U(1)_X) →^≈ H³(X, Z_X) ≅ H³(X, Z)

The image δ(P) ∈ H³(X, Z) of B_P is called the Dixmier-Douady invariant of P. When δ(P) = [H] is represented in H³(X, R) by a closed three-form H on X, called the H-flux of the given B-field B_P, we will write P = P_H. One has δ(P) = 0 iff the projective bundle P comes from a vector bundle E → X, i.e., P = P(E). By Serre’s theorem every torsion element of H³(X,Z) arises from a finite-dimensional bundle P. Explicitly, consider the commutative diagram of exact sequences of groups given by

where we identify the cyclic group Z_n with the group of n-th roots of unity. Let P be a projective bundle with structure group PU(n), i.e., with fibres P(Cⁿ). Then the commutative diagram of long exact sequences of sheaf cohomology groups associated to the above commutative diagram of groups implies that the element B_P ∈ H²(X, U(1)_X) comes from H²(X, (Z_n)_X), and therefore its order divides n.

One also has δ(P₁ ⊗ P₂) = δ(P₁) + δ(P₂) and δ(P^∨) = −δ(P). This follows from the commutative diagram

and the fact that P^∨ ⊗ P = P(E) where E is the vector bundle of Hilbert-Schmidt endomorphisms of P . Putting everything together, it follows that the cohomology group H³(X, Z) is isomorphic to the group of stable equivalence classes of principal PU(H)-bundles P → X with the operation of tensor product.

We are now ready to define the twisted K-theory of the manifold X equipped with a projective bundle P → X, such that P_x = P(H) ∀ x ∈ X. We will first give a definition in terms of Fredholm operators, and then provide some equivalent, but more geometric definitions. Let H be a Z₂-graded Hilbert space. We define Fred⁰(H) to be the space of self-adjoint degree 1 Fredholm operators T on H such that T² − 1 ∈ K(H), together with the subspace topology induced by the embedding Fred⁰(H) ֒→ B(H) × K(H) given by T → (T, T² − 1) where the algebra of bounded linear operators B(H) is given the compact-open topology and the Banach algebra of compact operators K = K(H) is given the norm topology.

Let P = P_H → X be a projective Hilbert bundle. Then we can construct an associated bundle Fred⁰(P) whose fibres are Fred⁰(H). We define the twisted K-theory group of the pair (X, P) to be the group of homotopy classes of maps

K⁰(X, H) = [X, Fred⁰(P_H)]

The group K⁰(X, H) depends functorially on the pair (X, P_H), and an isomorphism of projective bundles ρ : P → P′ induces a group isomorphism ρ∗ : K⁰(X, H) → K⁰(X, H′). Addition in K⁰(X, H) is defined by fibre-wise direct sum, so that the sum of two elements lies in K⁰(X, H₂) with [H₂] = δ(P ⊗ P(C²)) = δ(P) = [H]. Under the isomorphism H ⊗ C² ≅ H, there is a projective bundle isomorphism P → P ⊗ P(C²) for any projective bundle P and so K⁰(X, H₂) is canonically isomorphic to K⁰(X, H). When [H] is a non-torsion element of H³(X, Z), so that P = P_H is an infinite-dimensional bundle of projective spaces, then the index map K⁰(X, H) → Z is zero, i.e., any section of Fred⁰(P) takes values in the index zero component of Fred⁰(H).

Let us now describe some other models for twisted K-theory which will be useful in our physical applications later on. A definition in algebraic K-theory may given as follows. A bundle of projective spaces P yields a bundle End(P) of algebras. However, if H is an infinite-dimensional Hilbert space, then one has natural isomorphisms H ≅ H ⊕ H and

End(H) ≅ Hom(H ⊕ H, H) ≅ End(H) ⊕ End(H)

as left End(H)-modules, and so the algebraic K-theory of the algebra End(H) is trivial. Instead, we will work with the Banach algebra K(H) of compact operators on H with the norm topology. Given that the unitary group U(H) with the compact-open topology acts continuously on K(H) by conjugation, to a given projective bundle P_H we can associate a bundle of compact operators E_H → X given by

E_H = P_H ×_PU K

with δ(E_H) = [H]. The Banach algebra A_H := C₀(X, E_H) of continuous sections of E_H vanishing at infinity is the continuous trace C∗-algebra CT(X, H). Then the twisted K-theory group K^•(X, H) of X is canonically isomorphic to the algebraic K-theory group K_•(A_H).

We will also need a smooth version of this definition. Let A^∞_H be the smooth subalgebra of A_H given by the algebra CT^∞(X, H) = C^∞(X, L¹_PH),

where L¹_PH = P_H ×_PUL¹. Then the inclusion CT^∞(X, H) → CT(X, H) induces an isomorphism K_•CT(X, H) →^≈ K_•CT(X, H) of algebraic K-theory groups. Upon choosing a bundle gerbe connection, one has an isomorphism K_•CT^∞(X, H) ≅ K^•(X, H) with the twisted K-theory defined in terms of projective Hilbert bundles P = P_H over X.

Finally, we propose a general definition based on K-theory with coefficients in a sheaf of rings. It parallels the bundle gerbe approach to twisted K-theory. Let B be a Banach algebra over C. Let E(B, X) be the category of continuous B-bundles over X, and let C(X, B) be the sheaf of continuous maps X → B. The ring structure in B equips C(X, B) with the structure of a sheaf of rings over X. We can therefore consider left (or right) C(X, B)-modules, and in particular the category LF C(X, B) of locally free C(X, B)-modules. Using the functor in the usual way, for X an equivalence of additive categories

E(B, X) ≅ LF (C(X, B))

Since these are both additive categories, we can apply the Grothendieck functor to each of them and obtain the abelian groups K(LF(C(X, B))) and K(E(B, X)). The equivalence of categories ensures that there is a natural isomorphism of groups

K(LF (C(X, B))) ≅ K(E(B, X))

This motivates the following general definition. If A is a sheaf of rings over X, then we define the K-theory of X with coefficients in A to be the abelian group

K(X, A) := K LF(A)

For example, consider the case B = C. Then C(X, C) is just the sheaf of continuous functions X → C, while E(C, X) is the category of complex vector bundles over X. Using the isomorphism of K-theory groups we then have

K(X, C(X,C)) := K(LF (C(X, C))) ≅ K (E(C, X)) = K⁰(X)

The definition of twisted K-theory uses another special instance of this general construction. For this, we define an Azumaya algebra over X of rank m to be a locally trivial algebra bundle over X with fibre isomorphic to the algebra of m × m complex matrices over C, M_m(C). An example is the algebra End(E) of endomorphisms of a complex vector bundle E → X. We can define an equivalence relation on the set A(X) of Azumaya algebras over X in the following way. Two Azumaya algebras A, A′ are called equivalent if there are vector bundles E, E′ over X such that the algebras A ⊗ End(E), A′ ⊗ End(E′) are isomorphic. Then every Azumaya algebra of the form End(E) is equivalent to the algebra of functions C(X) on X. The set of all equivalence classes is a group under the tensor product of algebras, called the Brauer group of X and denoted Br(X). By Serre’s theorem there is an isomorphism

δ : Br(X) →^≈ tor(H³(X, Z))

where tor(H³(X, Z)) is the torsion subgroup of H³(X, Z).

If A is an Azumaya algebra bundle, then the space of continuous sections C(X, A) of X is a ring and we can consider the algebraic K-theory group K(A) := K₀(C(X,A)) of equivalence classes of projective C(X, A)-modules, which depends only on the equivalence class of A in the Brauer group. Under the equivalence, we can represent the Brauer group Br(X) as the set of isomorphism classes of sheaves of Azumaya algebras. Let A be a sheaf of Azumaya algebras, and LF(A) the category of locally free A-modules. Then as above there is an isomorphism

K(X, C(X, A)) ≅ K Proj (C(X, A))

where Proj (C(X, A)) is the category of finitely-generated projective C(X, A)-modules. The group on the right-hand side is the group K(A). For given [H] ∈ tor(H³(X, Z)) and A ∈ Br(X) such that δ(A) = [H], this group can be identified as the twisted K-theory group K⁰(X, H) of X with twisting A. This definition is equivalent to the description in terms of bundle gerbe modules, and from this construction it follows that K⁰(X, H) is a subgroup of the ordinary K-theory of X. If δ(A) = 0, then A is equivalent to C(X) and we have K(A) := K₀(C(X)) = K⁰(X). The projective C(X, A)-modules over a rank m Azumaya algebra A are vector bundles E → X with fibre C^nm ≅ (C^m)^⊕n, which is naturally an M_m(C)-module.

 

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 HoCW_n 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, emb_1∗ : H₀(t_<1K) → H₀(t_<∞K) is an isomorphism and H_r(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 emb_1∗ is an isomorphism on H⁰ as K is path connected. Clearly H_r(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(id_K) = [id_k⁰] 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, emb_2∗ : H_r(t_<2K) → H_r(t_<∞K) is an isomorphism for r = 0, 1, and H_r(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 emb_2∗ is an isomorphism both on H₀ as K is path connected and on H₁ as H₁(k⁰) = 0 = H₁(K), while trivially H_r(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.

Metaphysics of the Semantics of HoTT. Thought of the Day 73.0

Types and tokens are interpreted as concepts (rather than spaces, as in the homotopy interpretation). In particular, a type is interpreted as a general mathematical concept, while a token of a given type is interpreted as a more specific mathematical concept qua instance of the general concept. This accords with the fact that each token belongs to exactly one type. Since ‘concept’ is a pre-mathematical notion, this interpretation is admissible as part of an autonomous foundation for mathematics.

Expressions in the language are the names of types and tokens. Those naming types correspond to propositions. A proposition is ‘true’ just if the corresponding type is inhabited (i.e. there is a token of that type, which we call a ‘certificate’ to the proposition). There is no way in the language of HoTT to express the absence or non-existence of a token. The negation of a proposition P is represented by the type P → 0, where P is the type corresponding to proposition P and 0 is a type that by definition has no token constructors (corresponding to a contradiction). The logic of HoTT is not bivalent, since the inability to construct a token of P does not guarantee that a token of P → 0 can be constructed, and vice versa.

The rules governing the formation of types are understood as ways of composing concepts to form more complex concepts, or as ways of combining propositions to form more complex propositions. They follow from the Curry-Howard correspondence between logical operations and operations on types. However, we depart slightly from the standard presentation of the Curry-Howard correspondence, in that the tokens of types are not to be thought of as ‘proofs’ of the corresponding propositions but rather as certificates to their truth. A proof of a proposition is the construction of a certificate to that proposition by a sequence of applications of the token construction rules. Two different such processes can result in construction of the same token, and so proofs and tokens are not in one-to-one correspondence.

When we work formally in HoTT we construct expressions in the language according to the formal rules. These expressions are taken to be the names of tokens and types of the theory. The rules are chosen such that if a construction process begins with non-contradictory expressions that all name tokens (i.e. none of the expressions are ‘empty names’) then the result will also name a token (i.e. the rules preserve non-emptiness of names).

Since we interpret tokens and types as concepts, the only metaphysical commitment required is to the existence of concepts. That human thought involves concepts is an uncontroversial position, and our interpretation does not require that concepts have any greater metaphysical status than is commonly attributed to them. Just as the existence of a concept such as ‘unicorn’ does not require the existence of actual unicorns, likewise our interpretation of tokens and types as mathematical concepts does not require the existence of mathematical objects. However, it is compatible with such beliefs. Thus a Platonist can take the concept, say, ‘equilateral triangle’ to be the concept corresponding to the abstract equilateral triangle (after filling in some account of how we come to know about these abstract objects in a way that lets us form the corresponding concepts). Even without invoking mathematical objects to be the ‘targets’ of mathematical concepts, one could still maintain that concepts have a mind-independent status, i.e. that the concept ‘triangle’ continues to exist even while no-one is thinking about triangles, and that the concept ‘elliptic curve’ did not come into existence at the moment someone first gave the definition. However, this is not a necessary part of the interpretation, and we could instead take concepts to be mind-dependent, with corresponding implications for the status of mathematics itself.

Grothendieck’s Abstract Homotopy Theory

Let E be a Grothendieck topos (think of E as the category, Sh(X), of set valued sheaves on a space X). Within E, we can pick out a subcategory, C, of locally finite, locally constant objects in E. (If X is a space with E = Sh(X), C corresponds to those sheaves whose espace étale is a finite covering space of X.) Picking a base point in X generalises to picking a ‘fibre functor’ F : C → Sets_fin, a functor satisfying various conditions implying that it is pro-representable. (If x₀ ∈ X is a base point {x₀} → X induces a ‘fibre functor’ Sh(X) → Sh{x₀} ≅ Sets, by pullback.)

If F is pro-representable by P, then π₁(E, F) is defined to be Aut(P), which is a profinite group. Grothendieck proves there is an equivalence of categories C ≃ π₁(E) − Sets_fin, the category of finite π₁(E)-sets. If X is a locally nicely behaved space such as a CW-complex and E = Sh(X), then π₁(E) is the profinite completion of π₁(X). This profinite completion occurs only because Grothendieck considers locally finite objects. Without this restriction, a covering space Y of X would correspond to a π₁(X) – set, Y′, but if Y is a finite covering of X then the homomorphism from π₁(X) to the finite group of transformations of Y factors through the profinite completion of π₁(X). This is defined by : if G is a group, G^ˆ = lim(G/H : H ◅ G, H of finite index) is its profinite completion. This idea of using covering spaces or their analogue in E raises several important points:

a) These are homotopy theoretic results, but no paths are used. The argument involving sheaf theory, the theory of (pro)representable functors, etc., is of a purely categorical nature. This means it is applicable to spaces where the use of paths, and other homotopies is impossible because of bad (or unknown) local properties. Such spaces have been studied within Shape Theory and Strong Shape Theory, although not by using Grothendieck’s fundamental group, nor using sheaf theory.

b) As no paths are used, these methods can also be applied to non-spaces, e.g. locales and possibly to their non-commutative analogues, quantales. For instance, classically one could consider a field k and an algebraic closure K of k and then choose C to be a category of étale algebras over k, in such a way that π₁(E) ≅ Gal(K/k), the Galois group of k. It, in fact, leads to a classification theorem for Grothendieck toposes. From this viewpoint, low dimensional homotopy theory is ssen as being part of Galois theory, or vice versa.

c) This underlines the fact that π₁(X) classifies covering spaces – but for i > 1, π_i(X) does not seem to classify anything other than maps from Sⁱ into X!

This is abstract homotopy theory par excellence.

Obstruction Theory

Obstruction is a concept in homotopy theory where an invariant equals zero if a corresponding problem is solvable and is non-zero otherwise. Let Y be a space, and assume for convenience that Y is n-simple for every n, that is, the action of π₁(y) on π_n(y) is trivial for every n. Under this hypothesis we can forget about base points for homotopy groups, and any map ƒ: S → Y determines an element of π_n(Y).

Let B be a complex and A a subcomplex. Write Xⁿ for A U Bⁿ, where Bⁿ denotes the n-skeleton of B. Let σ be an (n + I)-cell of B which is not in A. Let g = g_σ be the attaching map σ^. = Sⁿ → Xⁿ ⊂ B.

Given a map ƒ: Xⁿ → Y, denote by c(ƒ) the cochain in Cⁿ⁺¹ (B, A; π_n(Y)) given by c(ƒ): σ → [f º g_σ]. Then it is clear that ƒ may be extended over Xⁿ ∪ _gσ σ iff f º g_σ is null-homotopic, that is, iff c(ƒ)(σ) = 0, and therefore that ƒ can be extended over Xⁿ⁺¹ = A ∪ Bⁿ⁺¹ if the cochain c(ƒ) is the zero cochain. It is a theorem of obstruction theory that c(ƒ) is a cocycle. It is called the obstruction cocycle or “the obstruction to extending ƒ over Bⁿ⁺¹“

There are two immediate applications. First, any map of an n-dimensional complex K into an n-connected space X is null-homotopic.

Take (B, A) = (K x I, K x i) and define ƒ:A → X by the given map K → X on one piece and a constant map on the other piece; then ƒ can be extended over B because the obstructions lie in the trivial groups π_i(X).

Second, as a particular case, a finite-dimensional complex K is contractible iff π_i(K) is trivial for all i < dim K.

Suppose ƒ, g are two maps Xⁿ → Y which agree on X^n-1. Then for each n-cell of B which is not in A, we get a map Sⁿ → Y by taking ƒ and g on the two hemispheres. The resulting cochain of Cⁿ(B, A; π_n(Y)) is called the difference cochain of ƒ and g, denoted d(ƒ, g).

Truncation Functors

Let A be an abelian category, and let D = D(A) be the derived category. For any complex A• in A, and n ∈ Z, we let τ_≤nA• be the truncated complex

··· → Aⁿ⁻² → Aⁿ⁻¹ → ker(Aⁿ → Aⁿ⁺¹)→ 0 → 0 → ··· , and dually we let τ_≥nA be the complex

··· → 0 → 0 → coker(Aⁿ⁻¹ → Aⁿ) → Aⁿ⁺¹ → Aⁿ⁺² → ···

Note that

H^m(τ_≤nA•) = H^m(A•) if m ≤ n

= 0 if m > n

and that

H^m(τ_≥nA•) = H^m(A•)  if m ≥ n

= 0 if m < n

One checks that τ_≥n (respectively τ_≤n) extends naturally to an additive functor of complexes which preserves homotopy and takes quasi-isomorphisms to quasi-isomorphisms, and hence induces an additive functor D → D. In fact if D_≤n (respectively D_≥n) is the full subcategory of D whose objects are the complexes A• such that H^m(A•) = 0 for m > n (respectively m < n) then we have additive functors

τ_≤n : D → D_≤n ⊂ D

τ_≥n : D → D_≥n ⊂ D

together with obvious functorial maps

iⁿ_A : τ_≤n A• → A•

jⁿ_A : A• → τ_≥n A•

The preceding iⁿ_A , jⁿ_A induce functorial isomorphisms

Hom_D≤n (B•,τ_≤nA•) →^~ Hom_D(B•, A•) (B• ∈ D_≤n) —– (1)

Hom_D≥n (τ_≥nA•,C•) →^~ Hom_D(A•,C• ) (C• ∈ D_≥n) —– (2)

Bijectivity of (1) means that any map φ : B• → A• (in D) with B• ∈ D_≤n factors uniquely via i_A := iⁿ_A

Given φ, we have a commutative diagram

and since B• ∈ D_≤n, therefore i_B is an isomorphism in D, so we can write

φ = i ◦ (τ_≤nφ ◦ i⁻¹_B),

and thus (1) is surjective.

To prove that (1) is also injective, we assume that i_A ◦ τ_≤n φ = 0 and deduce that τ_≤n φ = 0. The assumption means that there is a commutative diagram in K(A)

where s and s′′ are quasi-isomorphisms, and f/s = τ_≤nφ

Applying the (idempotent) functor τ_≥n, we get a commutative diagram

Since τ_≤ns and τ_≤ns′′ are quasi-isomorphisms, we have

τ_≤nφ = τ_≤n f/τ_≤ns = 0/τ_≤ns′′ = 0

as desired.

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 1_x ∈ 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 Π_n(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. Π_n(X) should be an ‘n-groupoid’, meaning that all its j-morphisms are equivalences for 0 ≤ j ≤ n. We call Π_n(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 nCat_k whose objects are k-tuply monoidal weak n-categories. One should also be able to treat nCat_k 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: nCat_k → (n − 1)Cat_k+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: nCat_k → nCat_k−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: nCat_k → nCat_k+1.

• Forming the generalized center: (n,k) → (n,k+1). Thinking of C as an object of the (n+k)-category nCat_k, there should be a (k+1)-tuply monoidal n-category ZC, the ‘generalized center’ of C, given by Ω^k(end(C)). In other words, ZC is the largest sub-(n + k + 1)-category of (n + k)Cat having C as its only object, 1_C as its only morphism, 1_1C 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).

Conjuncted: Of Topos and Torsors

The condition that each stalk F_x be equivalent to a classifying space BG can be summarized by saying that F is a gerbe on X: more precisely, it is a gerbe banded by the constant sheaf G associated to G.

For larger values of n, even the language of stacks is not sufficient to describe the nature of the sheaf F associated to the fibration X^~ → X. To address the situation, Grothendieck proposed (in his infamous letter to Quillen; see [35]) that there should be a theory of n-stacks on X, for every integer n ≥ 0. Moreover, for every sheaf of abelian groups G on X, the cohomology group Hⁿ⁺¹_sheaf(X;G) should have an interpreation as sheaf classifying a special type of n-stack: namely, the class of n-gerbes banded by G. In the special case where the space X is a point (and where we restrict our attention to n-stacks in groupoids), the theory of n-stacks on X should recover the classical homotopy theory of n-types: that is, CW complexes Z such that the homotopy groups π_i(Z, z) vanish for i > n (and every base point z ∈ Z). More generally, we should think of an n-stack (in groupoids) on a general space X as a “sheaf of n-types” on X.

When n = 0, an n-stack on a topological space X simply means a sheaf of sets on X. The collection of all such sheaves can be organized into a category Shv_Set(X), and this category is a prototypical example of a Grothendieck topos. The main goal of this book is to obtain an analogous understanding of the situation for n > 0. More precisely, we would like answers to the following questions:

(Q1) Given a topological space X, what should we mean by a “sheaf of n-types” on X?

(Q2)  Let Shv_≤n(X) denote the collection of all sheaves of n-types on X. What sort of a mathematical object is Shv_≤n(X)?

(Q3)  What special features (if any) does Shv_≤n(X) possess?

Our answers to questions (Q2) and (Q3) may be summarized as follows:

(A2)  The collection Shv_≤n(X) has the structure of an ∞-category.

(A3)  The ∞-category Shv_≤n(X) is an example of an (n+1)-topos: that is, an ∞-category which satisfies higher categorical analogues of Giraud’s axioms for Grothendieck topoi.

Grothendieck’s vision has been realized in various ways, thanks to the work of a number of mathematicians (most notably Jardine), and their work can also be used to provide answers to questions (Q1) and (Q2). Question (Q3) has also been addressed (at least in limiting case n = ∞) by Toën and Vezzosi. 

To provide more complete versions of the answers (A2) and (A3), we will need to develop the language of higher category theory. This is generally regarded as a technical and forbidding subject. More precisely, we will need a theory of (∞, 1)-categories: higher categories C for which the k-morphisms of C are required to be invertible for k > 1.

Classically, category theory is a useful tool not so much because of the light it sheds on any particular mathematical discipline, but instead because categories are so ubiquitous: mathematical objects in many different settings (sets, groups, smooth manifolds, etc.) can be organized into categories. Moreover, many elementary mathematical concepts can be described in purely categorical terms, and therefore make sense in each of these settings. For example, we can form products of sets, groups, and smooth manifolds: each of these notions can simply be described as a Cartesian product in the relevant category. Cartesian products are a special case of the more general notion of limit, which plays a central role in classical category theory.