# 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.

# Hochschild Cohomology Tethers to Closed String Algebra by way of Cyclicity.

When we have an open and closed Topological Field Theory (TFT) each element ξ of the closed algebra C defines an endomorphism ξa = ia(ξ) ∈ Oaa of each object a of B, and η ◦ ξa = ξb ◦ η for each morphism η ∈ Oba from a to b. The family {ξa} thus constitutes a natural transformation from the identity functor 1B : B → B to itself.

For any C-linear category B we can consider the ring E of natural transformations of 1B. It is automatically commutative, for if {ξa}, {ηa} ∈ E then ξa ◦ ηa = ηa ◦ ξa by the definition of naturality. (A natural transformation from 1B to 1B is a collection of elements {ξa ∈ Oaa} such that ξa ◦ f = f ◦ ξb for each morphism f ∈ Oab from b to a. But we can take a = b and f = ηa.) If B is a Frobenius category then there is a map πab : Obb → Oaa for each pair of objects a, b, and we can define jb : Obb → E by jb(η)a = πab(η) for η ∈ Obb. In other words, jb is defined so that the Cardy condition ιa ◦ jb = πab holds. But the question arises whether we can define a trace θ : E → C to make E into a Frobenius algebra, and with the property that

θaa(ξ)η) = θ(ξja(η)) —– (1)

∀ ξ ∈ E and η ∈ Oaa. This is certainly true if B is a semisimple Frobenius category with finitely many simple objects, for then E is just the ring of complex-valued functions on the set of classes of these simple elements, and we can readily define θ : E → C by θ(εa) = θa(1a)2, where a is an irreducible object, and εa ∈ E is the characteristic function of the point a in the spectrum of E. Nevertheless, a Frobenius category need not be semisimple, and we cannot, unfortunately, take E as the closed string algebra in the general case. If, for example, B has just one object a, and Oaa is a commutative local ring of dimension greater than 1, then E = Oaa, and so ιa : E → Oaa is an isomorphism, and its adjoint map ja ought to be an isomorphism too. But that contradicts the Cardy condition, as πaa is multiplication by ∑ψiψi, which must be nilpotent.

The commutative algebra E of natural endomorphisms of the identity functor of a linear category B is called the Hochschild cohomology HH0(B) of B in degree 0. The groups HHp(B) for p > 0, vanish if B is semisimple, but in the general case they appear to be relevant to the construction of a closed string algebra from B. For any Frobenius category B there is a natural homomorphism K(B) → HH0(B) from the Grothendieck group of B, which assigns to an object a the transformation whose value on b is πba(1a) ∈ Obb. In the semisimple case this homomorphism induces an isomorphism K(B) ⊗ C → HH0(B).

For any additive category B the Hochschild cohomology is defined as the cohomology of the cochain complex in which a k-cochain F is a rule that to each composable k-tuple of morphisms

Y0φ1 Y1φ2 ··· →φk Yk —– (2)

assigns F(φ1,…,φk) ∈ Hom(Y0,Yk). The differential in the complex is defined by

(dF)(φ1,…,φk+1) = F(φ2,…,φk+1) ◦ φ1 + ∑i=1k(−1)i F(φ1,…,φi+1 ◦ φi,…,φk+1) + (−1)k+1φk+1 ◦ F(φ1,…,φk) —– (3)

(Notice, in particular, that a 0-cochain assigns an endomorphism FY to each object Y, and is a cocycle if the endomorphisms form a natural transformation. Similarly, a 2-cochain F gives a possible infinitesimal deformation F(φ1, φ2) of the composition law (φ1, φ2) ↦ φ2 ◦ φ1 of the category, and the deformation preserves the associativity of composition iff F is a cocycle.)

In the case of a category B with a single object whose algebra of endomorphisms is O the cohomology just described is usually called the Hochschild cohomology of the algebra O with coefficients in O regarded as a O-bimodule. This must be carefully distinguished from the Hochschild cohomology with coefficients in the dual O-bimodule O. But if O is a Frobenius algebra it is isomorphic as a bimodule to O, and the two notions of Hochschild cohomology need not be distinguished. The same applies to a Frobenius category B: because Hom(Yk, Y0) is the dual space of Hom(Y0, Yk) we can think of a k-cochain as a rule which associates to each composable k-tuple of morphisms a linear function of an element φ0 ∈ Hom(Yk, Y0). In other words, a k-cochain is a rule which to each “circle” of k + 1 morphisms

···→φ0 Y0φ1 Y1 →φ2···→φk Ykφ0··· —– (4)

assigns a complex number F(φ01,…,φk).

If in this description we restrict ourselves to cochains which are cyclically invariant under rotating the circle of morphisms (φ01,…,φk) then we obtain a sub-cochain complex of the Hochschild complex whose cohomology is called the cyclic cohomology HC(B) of the category B. The cyclic cohomology, which evidently maps to the Hochschild cohomology is a more natural candidate for the closed string algebra associated to B than is the Hochschild cohomology. A very natural Frobenius category on which to test these ideas is the category of holomorphic vector bundles on a compact Calabi-Yau manifold.

# The Natural Theoretic of Electromagnetism. Thought of the Day 147.0

In Maxwell’s theory, the field strength F = 1/2Fμν dxμ ∧ dxν is a real 2-form on spacetime, and thence a natural object at the same time. The homogeneous Maxwell equation dF = 0 is an equation involving forms and it has a well-known local solution F = dA’, i.e. there exists a local spacetime 1-form A’ which is a potential for the field strength F. Of course, if spacetime is contractible, as e.g. for Minkowski space, the solution is also a global one. As is well-known, in the non-commutative Yang-Mills theory case the field strength F = 1/2FAμν TA ⊗ dxμ ∧ dxν is no longer a spacetime form. This is a somewhat trivial remark since the transformation laws of such field strength are obtained as the transformation laws of the curvature of a principal connection with values in the Lie algebra of some (semisimple) non-Abelian Lie group G (e.g. G = SU(n), n 2 ≥ 2). However, the common belief that electromagnetism is to be intended as the particular case (for G =U(1)) of a non-commutative theory is not really physically evident. Even if we subscribe this common belief, which is motivated also by the tremendous success of the quantized theory, let us for a while discuss electromagnetism as a standalone theory.

From a mathematical viewpoint this is a (different) approach to electromagnetism and the choice between the two can be dealt with on a physical ground only. Of course the 1-form A’ is defined modulo a closed form, i.e. locally A” = A’ + dα is another solution.

How can one decide whether the potential of electromagnetism should be considered as a 1-form or rather as a principal connection on a U(1)-bundle? First of all we notice that by a standard hole argument (one can easily define compact supported closed 1-forms, e.g. by choosing the differential of compact supported functions which always exist on a paracompact manifold) the potentials A and A’ represent the same physical situation. On the other hand, from a mathematical viewpoint we would like the dynamical field, i.e. the potential A’, to be a global section of some suitable configuration bundle. This requirement is a mathematical one, motivated on the wish of a well-defined geometrical perspective based on global Variational Calculus.

The first mathematical way out is to restrict attention to contractible spacetimes, where A’ may be always chosen to be global. Then one can require the gauge transformations A” = A’ + dα to be Lagrangian symmetries. In this way, field equations select a whole equivalence class of gauge-equivalent potentials, a procedure which solves the hole argument problem. In this picture the potential A’ is really a 1-form, which can be dragged along spacetime diffeomorphism and which admits the ordinary Lie derivatives of 1-forms. Unfortunately, the restriction to contractible spacetimes is physically unmotivated and probably wrong.

Alternatively, one can restrict electromagnetic fields F, deciding that only exact 2-forms F are allowed. That actually restricts the observable physical situations, by changing the homogeneous Maxwell equations (i.e. Bianchi identities) by requiring that F is not only closed but exact. One should in principle be able to empirically reject this option.

On non-contractible spacetimes, one is necessarily forced to resort to a more “democratic” attitude. The spacetime is covered by a number of patches Uα. On each patch Uα one defines a potential A(α). In the intersection of two patches the two potentials A(α) and A(β) may not agree. In each patch, in fact, the observer chooses his own conventions and he finds a different representative of the electromagnetic potential, which is related by a gauge transformation to the representatives chosen in the neighbour patch(es). Thence we have a family of gauge transformations, one in each intersection Uαβ, which obey cocycle identities. If one recognizes in them the action of U(1) then one can build a principal bundle P = (P, M, π; U(1)) and interpret the ensuing potential as a connection on P. This leads way to the gauge natural formalism.

Anyway this does not close the matter. One can investigate if and when the principal bundle P, in addition to the obvious principal structure, can be also endowed with a natural structure. If that were possible then the bundle of connections Cp (which is associated to P) would also be natural. The problem of deciding whether a given gauge natural bundle can be endowed with a natural structure is quite difficult in general and no full theory is yet completely developed in mathematical terms. That is to say, there is no complete classification of the topological and differential geometric conditions which a principal bundle P has to satisfy in order to ensure that, among the principal trivializations which determine its gauge natural structure, one can choose a sub-class of trivializations which induce a purely natural bundle structure. Nor it is clear how many inequivalent natural structures a good principal bundle may support. Though, there are important examples of bundles which support at the same time a natural and a gauge natural structure. Actually any natural bundle is associated to some frame bundle L(M), which is principal; thence each natural bundle is also gauge natural in a trivial way. Since on any paracompact manifold one can choose a global Riemannian metric g, the corresponding tangent bundle T(M) can be associated to the orthonormal frame bundle O(M, g) besides being obviously associated to L(M). Thence the natural bundle T(M) may be also endowed with a gauge natural bundle structure with structure group O(m). And if M is orientable the structure can be further reduced to a gauge natural bundle with structure group SO(m).

Roughly speaking, the task is achieved by imposing restrictions to cocycles which generate T(M) according to the prescription by imposing a privileged class of changes of local laboratories and sets of measures. Imposing the cocycle ψ(αβ) to take its values in O(m) rather than in the larger group GL(m). Inequivalent gauge natural structures are in one-to-one correspondence with (non isometric) Riemannian metrics on M. Actually whenever there is a Lie group homomorphism ρ : GU(m) → G for some s onto some given Lie group G we can build a natural G-principal bundle on M. In fact, let (Uα, ψ(α)) be an atlas of the given manifold M, ψ(αβ) be its transition functions and jψ(αβ) be the induced transition functions of L(M). Then we can define a G-valued cocycle on M by setting ρ(jψ(αβ)) and thence a (unique up to fibered isomorphisms) G-principal bundle P(M) = (P(M), M, π; G). The bundle P(M), as well as any gauge natural bundle associated to it, is natural by construction. Now, defining a whole family of natural U(1)-bundles Pq(M) by using the bundle homomorphisms

ρq: GL(m) → U(1): J ↦ exp(iq ln det|J|) —– (1)

where q is any real number and In denotes the natural logarithm. In the case q = 0 the image of ρ0 is the trivial group {I}; and, all the induced bundles are trivial, i.e. P = M x U(1).

The natural lift φ’ of a diffeomorphism φ: M → M is given by

φ'[x, e]α = [φ(x), eiq ln det|J|. e]α —– (2)

where J is the Jacobin of the morphism φ. The bundles Pq(M) are all trivial since they allow a global section. In fact, on any manifold M, one can define a global Riemannian metric g, where the local sections glue together.

Since the bundles Pq(M) are all trivial, they are all isomorphic to M x U(1) as principal U(1)-bundles, though in a non-canonical way unless q = 0. Any two of the bundles Pq1(M) and Pq2(M) for two different values of q are isomorphic as principal bundles but the isomorphism obtained is not the lift of a spacetime diffeomorphism because of the two different values of q. Thence they are not isomorphic as natural bundles. We are thence facing a very interesting situation: a gauge natural bundle C associated to the trivial principal bundle P can be endowed with an infinite family of natural structures, one for each q ∈ R; each of these natural structures can be used to regard principal connections on P as natural objects on M and thence one can regard electromagnetism as a natural theory.

Now that the mathematical situation has been a little bit clarified, it is again a matter of physical interpretation. One can in fact restrict to electromagnetic potentials which are a priori connections on a trivial structure bundle P ≅ M x U(1) or to accept that more complicated situations may occur in Nature. But, non-trivial situations are still empirically unsupported, at least at a fundamental level.

# Homotopically Truncated Spaces.

The Eckmann–Hilton dual of the Postnikov decomposition of a space is the homology decomposition (or Moore space decomposition) of a space.

A Postnikov decomposition for a simply connected CW-complex X is a commutative diagram

such that pn∗ : πr(X) → πr(Pn(X)) is an isomorphism for r ≤ n and πr(Pn(X)) = 0 for r > n. Let Fn be the homotopy fiber of qn. Then the exact sequence

πr+1(PnX) →qn∗ πr+1(Pn−1X) → πr(Fn) → πr(PnX) →qn∗ πr(Pn−1X)

shows that Fn is an Eilenberg–MacLane space K(πnX, n). Constructing Pn+1(X) inductively from Pn(X) requires knowing the nth k-invariant, which is a map of the form kn : Pn(X) → Yn. The space Pn+1(X) is then the homotopy fiber of kn. Thus there is a homotopy fibration sequence

K(πn+1X, n+1) → Pn+1(X) → Pn(X) → Yn

This means that K(πn+1X, n+1) is homotopy equivalent to the loop space ΩYn. Consequently,

πr(Yn) ≅ πr−1(ΩYn) ≅ πr−1(K(πn+1X, n+1) = πn+1X, r = n+2,

= 0, otherwise.

and we see that Yn is a K(πn+1X, n+2). Thus the nth k-invariant is a map kn : Pn(X) → K(πn+1X, n+2)

Note that it induces the zero map on all homotopy groups, but is not necessarily homotopic to the constant map. The original space X is weakly homotopy equivalent to the inverse limit of the Pn(X).

Applying the paradigm of Eckmann–Hilton duality, we arrive at the homology decomposition principle from the Postnikov decomposition principle by changing:

• the direction of all arrows
• π to H
• loops Ω to suspensions S
• fibrations to cofibrations and fibers to cofibers
• Eilenberg–MacLane spaces K(G, n) to Moore spaces M(G, n)
• inverse limits to direct limits

A homology decomposition (or Moore space decomposition) for a simply connected CW-complex X is a commutative diagram

such that jn∗ : Hr(X≤n) → Hr(X) is an isomorphism for r ≤ n and Hr(X≤n) = 0 for

r > n. Let Cn be the homotopy cofiber of in. Then the exact sequence

Hr(X≤n−1) →in∗ Hr(X≤n) → Hr(Cn) →in∗ Hr−1(X≤n−1) → Hr−1(X≤n)

shows that Cn is a Moore space M(HnX, n). Constructing X≤n+1 inductively from X≤n requires knowing the nth k-invariant, which is a map of the form kn : Yn → X≤n.

The space X≤n+1 is then the homotopy cofiber of kn. Thus there is a homotopy cofibration sequence

Ynkn X≤nin+1 X≤n+1 → M(Hn+1X, n+1)

This means that M(Hn+1X, n+1) is homotopy equivalent to the suspension SYn. Consequently,

H˜r(Yn) ≅ Hr+1(SYn) ≅ Hr+1(M(Hn+1X, n+1)) = Hn+1X, r = n,

= 0, otherwise

and we see that Yn is an M(Hn+1X, n). Thus the nth k-invariant is a map kn : M(Hn+1X, n) → X≤n

It induces the zero map on all reduced homology groups, which is a nontrivial statement to make in degree n:

kn∗ : Hn(M(Hn+1X, n)) ∼= Hn+1(X) → Hn(X) ∼= Hn(X≤n)

The original space X is homotopy equivalent to the direct limit of the X≤n. The Eckmann–Hilton duality paradigm, while being a very valuable organizational principle, does have its natural limitations. Postnikov approximations possess rather good functorial properties: Let pn(X) : X → Pn(X) be a stage-n Postnikov approximation for X, that is, pn(X) : πr(X) → πr(Pn(X)) is an isomorphism for r ≤ n and πr(Pn(X)) = 0 for r > n. If Z is a space with πr(Z) = 0 for r > n, then any map g : X → Z factors up to homotopy uniquely through Pn(X). In particular, if f : X → Y is any map and pn(Y) : Y → Pn(Y) is a stage-n Postnikov approximation for Y, then, taking Z = Pn(Y) and g = pn(Y) ◦ f, there exists, uniquely up to homotopy, a map pn(f) : Pn(X) → Pn(Y) such that

homotopy commutes. Let X = S22 e3 be a Moore space M(Z/2,2) and let Y = X ∨ S3. If X≤2 and Y≤2 denote stage-2 Moore approximations for X and Y, respectively, then X≤2 = X and Y≤2 = X. We claim that whatever maps i : X≤2 → X and j : Y≤2 → Y such that i : Hr(X≤2) → Hr(X) and j : Hr(Y≤2) → Hr(Y) are isomorphisms for r ≤ 2 one takes, there is always a map f : X → Y that cannot be compressed into the stage-2 Moore approximations, i.e. there is no map f≤2 : X≤2 → Y≤2 such that

commutes up to homotopy. We shall employ the universal coefficient exact sequence for homotopy groups with coefficients. If G is an abelian group and M(G, n) a Moore space, then there is a short exact sequence

0 → Ext(G, πn+1Y) →ι [M(G, n), Y] →η Hom(G, πnY) → 0,

where Y is any space and [−,−] denotes pointed homotopy classes of maps. The map η is given by taking the induced homomorphism on πn and using the Hurewicz isomorphism. This universal coefficient sequence is natural in both variables. Hence, the following diagram commutes:

Here we will briefly write E2(−) = Ext(Z/2,−) so that E2(G) = G/2G, and EY (−) = Ext(−, π3Y). By the Hurewicz theorem, π2(X) ∼= H2(X) ∼= Z/2, π2(Y) ∼= H2(Y) ∼= Z/2, and π2(i) : π2(X≤2) → π2(X), as well as π2(j) : π2(Y≤2) → π2(Y), are isomorphisms, hence the identity. If a homomorphism φ : A → B of abelian groups is onto, then E2(φ) : E2(A) = A/2A → B/2B = E2(B) remains onto. By the Hurewicz theorem, Hur : π3(Y) → H3(Y) = Z is onto. Consequently, the induced map E2(Hur) : E23Y) → E2(H3Y) = E2(Z) = Z/2 is onto. Let ξ ∈ E2(H3Y) be the generator. Choose a preimage x ∈ E23Y), E2(Hur)(x) = ξ and set [f] = ι(x) ∈ [X,Y]. Suppose there existed a homotopy class [f≤2] ∈ [X≤2, Y≤2] such that

j[f≤2] = i[f].

Then

η≤2[f≤2] = π2(j)η≤2[f≤2] = ηj[f≤2] = ηi[f] = π2(i)η[f] = π2(i)ηι(x) = 0.

Thus there is an element ε ∈ E23Y≤2) such that ι≤2(ε) = [f≤2]. From ιE2π3(j)(ε) = jι≤2(ε) = j[f≤2] = i[f] = iι(x) = ιEY π2(i)(x)

we conclude that E2π3(j)(ε) = x since ι is injective. By naturality of the Hurewicz map, the square

commutes and induces a commutative diagram upon application of E2(−):

It follows that

ξ = E2(Hur)(x) = E2(Hur)E2π3(j)(ε) = E2H3(j)E2(Hur)(ε) = 0,

a contradiction. Therefore, no compression [f≤2] of [f] exists.

Given a cellular map, it is not always possible to adjust the extra structure on the source and on the target of the map so that the map preserves the structures. Thus the category theoretic setup automatically, and in a natural way, singles out those continuous maps that can be compressed into homologically truncated spaces.

# Category of a Quantum Groupoid

For a quantum groupoid H let Rep(H) be the category of representations of H, whose objects are finite-dimensional left H -modules and whose morphisms are H -linear homomorphisms. We shall show that Rep(H) has a natural structure of a monoidal category with duality.

For objects V, W of Rep(H) set

V ⊗ W = x ∈ V ⊗k W|x = ∆(1) · x ⊂ V ⊗k W —– (1)

with the obvious action of H via the comultiplication ∆ (here ⊗k denotes the usual tensor product of vector spaces). Note that ∆(1) is an idempotent and therefore V ⊗ W = ∆(1) × (V ⊗k W). The tensor product of morphisms is the restriction of usual tensor product of homomorphisms. The standard associativity isomorphisms (U ⊗ V ) ⊗ W → U ⊗ (V ⊗ W ) are functorial and satisfy the pentagon condition, since ∆ is coassociative. We will suppress these isomorphisms and write simply U ⊗ V ⊗ W.

The target counital subalgebra Ht ⊂ H has an H-module structure given by h · z = εt(hz),where h ∈ H, z ∈ Ht.

Ht is the unit object of Rep(H).

Define a k-linear homomorphism lV : Ht ⊗ V → V by lV(1(1) · z ⊗ 1(2) · v) = z · v, z ∈ Ht, v ∈ V.

This map is H-linear, since

lV h · (1(1) · z ⊗ 1(2) · v) = lV(h(1) · z ⊗ h(2) · v) = εt(h(1)z)h(2) · v = hz · v = h · lV (1(1) · z ⊗ 1(2) · v),

∀ h ∈ H. The inverse map l−1V: → Ht ⊗ V is given by V

l−1V(v) = S(1(1)) ⊗ (1(2) · v) = (1(1) · 1) ⊗ (1(2) · v)

The collection {lV}V gives a natural equivalence between the functor Ht ⊗ (·) and the identity functor. Indeed, for any H -linear homomorphism f : V → U we have:

lU ◦ (id ⊗ f)(1(1) · z ⊗ 1(2) · v) = lU 1(1) · z ⊗ 1(2) · f(v) = z · f(v) = f(z·v) = f ◦ lV(1(1) · z ⊗ 1(2) · v)

Similarly, the k-linear homomorphism rV : V ⊗ Ht → V defined by rV(1(1) · v ⊗ 1(2) · z) = S(z) · v, z ∈ Ht, v ∈ V, has the inverse r−1V(v) = 1(1) · v ⊗ 1(2) and satisfies the necessary properties.

Finally, we can check the triangle axiom idV ⊗ lW = rV ⊗ idW : V ⊗ Ht ⊗ W → V ⊗ W ∀ objects V, W of Rep(H). For v ∈ V, w ∈ W we have

(idV ⊗ lW)(1(1) · v ⊗ 1′(1)1(2) · z ⊗ 1′(2) · w)

= 1(1) · v ⊗ 1′(2)z · w) = 1(1)S(z) · v ⊗ 1(2) · w

=(rV ⊗ idW) (1′(1) · v ⊗ 1′(2) 1(1) · z ⊗ 1(2) · w),

therefore, idV ⊗ lW = rV ⊗ idW

Using the antipode S of H, we can provide Rep(H) with a duality. For any object V of Rep(H), define the action of H on V = Homk(V, k) by

(h · φ)(v) = φ S(h) · v —– (2)

where h ∈ H , v ∈ V , φ ∈ V. For any morphism f : V → W , let f: W → V be the morphism dual to f. For any V in Rep(H), we define the duality morphisms dV : V ⊗ V → Ht, bV : Ht → V ⊗ V∗ as follows. For ∑j φj ⊗ vj ∈ V* ⊗ V, set

dV(∑j φj ⊗ vj)  = ∑j φj (1(1) · vj) 1(2) —– (3)

Let {fi}i and {ξi}i be bases of V and V, respectively, dual to each other. The element ∑i fi ⊗ ξi does not depend on choice of these bases; moreover, ∀ v ∈ V, φ ∈ V one has φ = ∑i φ(fi) ξi and v = ∑i fi ξi (v). Set

bV(z) = z · (∑i fi ⊗ ξi) —– (4)

The category Rep(H) is a monoidal category with duality. We know already that Rep(H) is monoidal, it remains to prove that dV and bV are H-linear and satisfy the identities

(idV ⊗ dV)(bV ⊗ idV) = idV, (dV ⊗ idV)(idV ⊗ bV) = idV.

Take ∑j φj ⊗ vj ∈ V ⊗ V, z ∈ Ht, h ∈ H. Using the axioms of a quantum groupoid, we have

h · dV(∑j φj ⊗ vj) = ((∑j φj (1(1) · vj) εt(h1(2))

= (∑j φj ⊗ εs(1(1)h) · vj 1(2)j φj S(h(1))1(1)h(2) · vj 1(2)

= (∑j h(1) · φj )(1(1) · (h(2) · vj))1(2)

= (∑j dV(h(1) · φj  ⊗ h(2) · vj) = dV(h · ∑j φj ⊗ vj)

therefore, dV is H-linear. To check the H-linearity of bV we have to show that h · bV(z) =

bV (h · z), i.e., that

i h(1)z · fi ⊗ h(2) · ξi = ∑i 1(1) εt(hz) · fi ⊗ 1(2) · ξi

Since both sides of the above equality are elements of V ⊗k V, evaluating the second factor on v ∈ V, we get the equivalent condition

h(1)zS(h(2)) · v = 1(1)εt (hz)S(1(2)) · v, which is easy to check. Thus, bV is H-linear.

Using the isomorphisms lV and rV identifying Ht ⊗ V, V ⊗ Ht, and V, ∀ v ∈ V and φ ∈ V we have:

(idV ⊗ dV)(bV ⊗ idV)(v)

=(idV ⊗ dV)bV(1(1) · 1) ⊗ 1(2) · v

= (idV ⊗ dV)bV(1(2)) ⊗ S−1(1(1)) · v

= ∑i (idV ⊗ dV) 1(2) · fi ⊗ 1(3) · ξi ⊗ S−1 (1(1)) · v

= ∑1(2) · fi ⊗ 1(3) · ξi (1′(1)S-1 (1(1)) · v) 1′(2)

= 1(2) S(1(3)) 1′(1) S-1 (1(1)) · v ⊗ 1′(2) = v

(dV ⊗ idV)(idV ⊗ bV)(φ)

= (dV ⊗ idV) 1(1) · φ ⊗ bV(1(2))

= ∑i (dV ⊗ idV)(1(1) · φ ⊗ 1(2) · 1(2) · 1(3) · ξi )

= ∑i (1(1) · φ (1′(1)1(2) · fi)1′(2) ⊗ 1(3) · ξi

= 1′(2) ⊗ 1(3)1(1) S(1′(1)1(2)) · φ = φ,

QED.

# 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 → Setsfin, 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) − Setsfin, 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, Gˆ = 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, πi(X) does not seem to classify anything other than maps from Si into X!

This is abstract homotopy theory par excellence.

# Categorial Functorial Monads

Algebraic constructs (A,U), such as Vec, Grp, Mon, and Lat, can be fully described by the following data, called the monad associated with (A,U):

1. the functor T : Set → Set, where T = U ◦ F and F : Set → A is the associated free functor,

2. the natural transformation η : idSet → T formed by universal arrows, and

3. the natural transformation μ : T ◦ T → T given by the unique homomorphism μX : T(TX) → TX that extends idTX.

In these cases, there is a canonical concrete isomorphism K between (A,U) and the full concrete subcategory of Alg(T) consisting of those T-algebras TX →x X that satisfy the equations x ◦ ηX = idX and x ◦ Tx = x ◦ μX. The latter subcategory is called the Eilenberg-Moore category of the monad (T, η, μ). The above observation makes it possible, in the following four steps, to express the “degree of algebraic character” of arbitrary concrete categories that have free objects:

Step 1: With every concrete category (A,U) over X that has free objects (or, more generally, with every adjoint functor A →U X) one can associate, in an essentially unique way, an adjoint situation (η, ε) : F -|U : A → X.

Step 2: With every adjoint situation (η, ε) : F -|U : A → X one can associate a monad T = (T, η, μ) on X, where T = U ◦ F : X → X.

Step 3: With every monad T = (T, η, μ) on X one can associate a concrete subcategory of Alg(T) denoted by (XT, UT) and called the category of T-algebras.

Step 4:  With every concrete category (A,U) over X that has free objects one can associate a distinguished concrete functor (A,U) →K (XT , UT) into the associated category of T-algebras called the comparison functor for (A, U).

Concrete categories that are concretely isomorphic to a category of T-algebras for some monad T have a distinct “algebraic flavor”. Such categories (A,U) and their forgetful functors U are called monadic. It turns out that a concrete category (A, U ) is monadic iff it has free objects and its associated comparison functor (A,U) →K (XT , UT) is an isomorphism. Thus, for concrete categories (A,U) that have free objects, the associated comparison functor can be considered as a means of measuring the “algebraic character” of (A,U); and the associated category of T-algebras can be considered to be the “algebraic part” of (A,U). In particular,

(a) every finitary variety is monadic,

(b) the category TopGrp, considered as a concrete category

1. over Top, is monadic,
2. over Set, is not monadic; the associated comparison functor is the forgetful functor TopGrp → Grp, so that the construct Grp may be considered as the “algebraic part” of the construct TopGrp,

(c) the construct Top is not monadic; the associated comparison functor is the forgetful functor Top → Set itself, so that the construct Set may be considered as the “algebraic part” of the construct Top; hence the construct Top may be considered as having a trivial “algebraic part”.

Among constructs, monadicity captures the idea of “algebraicness” rather well. Unfortunately, however, the behavior of monadic categories in general is far from satisfactory. Monadic functors can fail badly to reflect properties of the base category (e.g., the existence of colimits or of suitable factorization structures), and they are not closed under composition.

# Of Magnitudes, Metrization and Materiality of Abstracto-Concrete Objects.

The possibility of introducing magnitudes in a certain domain of concrete material objects is by no means immediate, granted or elementary. First of all, it is necessary to find a property of such objects that permits to compare them, so that a quasi-serial ordering be introduced in their set, that is a total linear ordering not excluding that more than one object may occupy the same position in the series. Such an ordering must then undergo a metrization, which depends on finding a fundamental measuring procedure permitting the determination of a standard sample to which the unit of measure can be bound. This also depends on the existence of an operation of physical composition, which behaves additively with respect to the quantity which we intend to measure. Only if all these conditions are satisfied will it be possible to introduce a magnitude in a proper sense, that is a function which assigns to each object of the material domain a real number. This real number represents the measure of the object with respect to the intended magnitude. This condition, by introducing an homomorphism between the domain of the material objects and that of the positive real numbers, transforms the language of analysis (that is of the concrete theory of real numbers) into a language capable of speaking faithfully and truly about those physical objects to which it is said that such a magnitude belongs.

Does the success of applying mathematics in the study of the physical world mean that this world has a mathematical structure in an ontological sense, or does it simply mean that we find in mathematics nothing but a convenient practical tool for putting order in our representations of the world? Neither of the answers to this question is right, and this is because the question itself is not correctly raised. Indeed it tacitly presupposes that the endeavour of our scientific investigations consists in facing the reality of “things” as it is, so to speak, in itself. But we know that any science is uniquely concerned with a limited “cut” operated in reality by adopting a particular point of view, that is concretely manifested by adopting a restricted number of predicates in the discourse on reality. Several skilful operational manipulations are needed in order to bring about a homomorphism with the structure of the positive real numbers. It is therefore clear that the objects that are studied by an empirical theory are by no means the rough things of everyday experience, but bundles of “attributes” (that is of properties, relations and functions), introduced through suitable operational procedures having often the explicit and declared goal of determining a concrete structure as isomorphic, or at least homomorphic, to the structure of real numbers or to some other mathematical structure. But now, if the objects of an empirical theory are entities of this kind, we are fully entitled to maintain that they are actually endowed with a mathematical structure: this is simply that structure which we have introduced through our operational procedures. However, this structure is objective and real and, with respect to it, the mathematized discourse is far from having a purely conventional and pragmatic function, with the goal of keeping our ideas in order: it is a faithful description of this structure. Of course, we could never pretend that such a discourse determines the structure of reality in a full and exhaustive way, and this for two distinct reasons: In the first place, reality (both in the sense of the totality of existing things, and of the ”whole” of any single thing), is much richer than the particular “slide” that it is possible to cut out by means of our operational manipulations. In the second place, we must be aware that a scientific object, defined as a structured set of attributes, is an abstract object, is a conceptual construction that is perfectly defined just because it is totally determined by a finite list of predicates. But concrete objects are by no means so: they are endowed with a great deal of attributes of an indefinite variety, so that they can at best exemplify with an acceptable approximation certain abstract objects that are totally encoding a given set of attributes through their corresponding predicates. The reason why such an exemplification can only be partial is that the different attributes that are simultaneously present in a concrete object are, in a way, mutually limiting themselves, so that this object does never fully exemplify anyone of them. This explains the correct sense of such common and obvious remarks as: “a rigid body, a perfect gas, an adiabatic transformation, a perfect elastic recoil, etc, do not exist in reality (or in Nature)”. Sometimes this remark is intended to vehiculate the thesis that these are nothing but intellectual fictions devoid of any correspondence with reality, but instrumentally used by scientists in order to organize their ideas. This interpretation is totally wrong, and is simply due to a confusion between encoding and exemplifying: no concrete thing encodes any finite and explicit number of characteristics that, on the contrary, can be appropriately encoded in a concept. Things can exemplify several concepts, while concepts (or abstract objects) do not exemplify the attributes they encode. Going back to the distinction between sense on the one hand, and reference or denotation on the other hand, we could also say that abstract objects belong to the level of sense, while their exemplifications belong to the level of reference, and constitute what is denoted by them. It is obvious that in the case of empirical sciences we try to construct conceptual structures (abstract objects) having empirical denotations (exemplified by concrete objects). If one has well understood this elementary but important distinction, one is in the position of correctly seeing how mathematics can concern physical objects. These objects are abstract objects, are structured sets of predicates, and there is absolutely nothing surprising in the fact that they could receive a mathematical structure (for example, a structure isomorphic to that of the positive real numbers, or to that of a given group, or of an abstract mathematical space, etc.). If it happens that these abstract objects are exemplified by concrete objects within a certain degree of approximation, we are entitled to say that the corresponding mathematical structure also holds true (with the same degree of approximation) for this domain of concrete objects. Now, in the case of physics, the abstract objects are constructed by isolating certain ontological attributes of things by means of concrete operations, so that they actually refer to things, and are exemplified by the concrete objects singled out by means of such operations up to a given degree of approximation or accuracy. In conclusion, one can maintain that mathematics constitutes at the same time the most exact language for speaking of the objects of the domain under consideration, and faithfully mirrors the concrete structure (in an ontological sense) of this domain of objects. Of course, it is very reasonable to recognize that other aspects of these things (or other attributes of them) might not be treatable by means of the particular mathematical language adopted, and this may imply either that these attributes could perhaps be handled through a different available mathematical language, or even that no mathematical language found as yet could be used for handling them.

# Quantum Groupoid

A (finite) quantum groupoid over k is a finite-dimensional k-vector space H with the structures of an associative algebra (H, m, 1) with multiplication m : H ⊗k H → H and unit 1 ∈ H and a coassociative coalgebra (H, ∆, ε) with comultiplication ∆ : H → H ⊗k H and counit ε : H → k such that:

1. The comultiplication ∆ is a (not necessarily unit-preserving) homomorphism of algebras such that

(∆ ⊗ id)∆(1) = (∆(1) ⊗ 1) (1 ⊗ ∆(1)) = (1 ⊗ ∆(1)) (∆(1) ⊗ 1) —– (1)

2.  The counit is a k-linear map satisfying the identity:

ε(fgh) = ε(fg(1))ε(g(2)h) = ε(fg(2))ε(g(1)h), (2) ∀ f, g, h ∈ H —– (2)

3.   There is an algebra and coalgebra anti-homomorphism S : H → H, called an antipode, such that, ∀ h ∈ H ,

m(id ⊗ S) ∆(h) = (ε ⊗ id) ∆(1)(h ⊗ 1) —– (3)

m(S ⊗ id) ∆(h) = (id ⊗ ε)(1 ⊗ h) ∆(1) —– (4)

A quantum groupoid is a Hopf algebra iff one of the following equivalent conditions holds: (i) the comultiplication is unit preserving or (ii) the counit is a homomorphism of algebras.

A morphism of quantum groupoids is a map between them which is both an algebra and a coalgebra morphism preserving unit and counit and commuting with the antipode. The image of such a morphism is clearly a quantum groupoid. The tensor product of two quantum groupoids is defined in an obvious way.

The set of axioms is self-dual. This allows to define a natural quantum groupoid  structure on the dual vector space H’ = Homk (H, k) by “reversing the arrows”:

⟨h,φ ψ⟩ = ∆(h), φ ⊗ ψ —– (5)

⟨g ⊗ h, ∆'(φ)⟩ = ⟨gh, φ⟩ —– (6)

⟨h, S'(φ)⟩ = ⟨S(h), φ⟩ —– (7)

∀ φ, ψ ∈ H’, g, h ∈ H. The unit 1ˆ ∈ H’ is ε and counit ε’ is φ → ⟨φ,1⟩. The linear endomorphisms of H defined by

h → m(id ⊗ S) ∆(h), h → m(S ⊗ id) ∆(h) —– (8)

are called the target and source counital maps and denoted εt and εs, respectively.

From axioms (3) and (4),

εt(h) = (ε ⊗ id) ∆(1)(h ⊗ 1), εs(h) = (id ⊗ ε) (1 ⊗ h)∆(1) . (9)

In the Hopf algebra case εt(h) = εs(h) = ε(h)1.

We have S ◦ εs = εt ◦ S and εs ◦ S = S ◦ εt. The images of these maps εt and εs

Ht = εt (H) = {h ∈ H | ∆(h) =∆(1)(h ⊗ 1)} —– (10)

Hs = εs (H) = {h ∈ H | ∆(h) = (1⊗h) ∆(1)} —– (11)

are subalgebras of H, called the target (respectively source) counital subalgebras. They play the role of ground algebras for H. They commute with each other and

Ht = {(φ ⊗ id) ∆(1)|φ ∈ H’,

Hs = (id ⊗ φ) ∆(1)| φ ∈ H’,

i.e., Ht (respectively Hs) is generated by the right (respectively left) tensorands of ∆(1). The restriction of S defines an algebra anti-isomorphism between Ht and Hs. Any morphism H → K of quantum groupoids preserves counital subalgebras, i.e., Ht ≅ Kt and Hs ≅ Ks.

In what follows we will use the Sweedler arrows, writing ∀ h ∈ H , φ ∈ H’:

h ⇀ φ = φ(1)⟨h, φ(2)⟩,

φ ↼ h = ⟨h, φ(1)⟩φ(2) —– (12)

∀ h ∈ H, φ ∈ H’. Then the map z → (z ⇀ ε) is an algebra isomorphism between Ht and H. Similarly, the map y → (ε ↼ y) is an algebra isomorphism between H and H’t. Thus, the counital subalgebras of H’ are canonically anti-isomorphic to those of H. A quantum groupoid H is called connected if Hs ∩ Z(H) = k, or, equivalently, Ht ∩ Z(H ) = k, where Z(H) denotes the center of H. A k-algebra A is separable if the multiplication epimorphism m : A ⊗k A → A has a right inverse as an A − A bimodule homomorphism. When the characteristic of k is 0, this is equivalent to the existence of a separability element e ∈ A ⊗k A such that m(e) = 1 and (a ⊗ 1)e = e(1 ⊗ a), (1 ⊗ a)e = e(a ⊗ 1) ∀ a ∈ A. The counital subalgebras Ht and Hs are separable, with separability elements et = (S ⊗ id)∆(1) and es = (id ⊗S)∆(1), respectively. Observe that the adjoint actions of 1 ∈ H give rise to non-trivial maps

H → H : h → 1(1)hS(1(2)) = Adl1(h), h → S(1(1))h1(2) = Adr1(h), h ∈ H —– (13) …….

# The Ubiquity of Self-Predicative Universality of Adjoint Functors. Note Quote.

One of the most important and beautiful notions in category theory is the notion of a pair of adjoint functors. The developers of category theory, Saunders MacLane and Samuel Eilenberg, famously said that categories were defined in order to define functors, and functors were defined in order to define natural transformations. Adjoints were defined more than a decade later by Daniel Kan but the realization of their ubiquity (“Adjoint functors arise everywhere” (MacLane) and their foundational importance has steadily increased over time (Lawvere). Now it would perhaps not be too much of an exaggeration to see categories, functors, and natural transformations as the prelude to defining adjoint functors. The notion of adjoint functors includes all the instances of self-predicative universal mapping properties discussed above. As Steven Awodey (179) put it:

The notion of adjoint functor applies everything that we have learned up to now to unify and subsume all the different universal mapping properties that we have encountered, from free groups to limits to exponentials. But more importantly, it also captures an important mathematical phenomenon that is invisible without the lens of category theory. Indeed, I will make the admittedly provocative claim that adjointness is a concept of fundamental logical and mathematical importance that is not captured elsewhere in mathematics.

“The isolation and explication of the notion of adjointness is perhaps the most profound contribution that category theory has made to the history of general mathematical ideas.” (Goldblatt)

How do the ubiquitous and important adjoint functors relate to theme of self- predicative universals? MacLane and Birkhoff succinctly state the idea of the self-predicative universals of category theory and note that adjunctions can be analyzed in terms of those universals. The construction of a new algebraic object will often solve a specific problem in a universal way, in the sense that every other solution of the given problem is obtained from this one by a unique homomorphism. The basic idea of an adjoint functor arises from the analysis of such universals. (MacLane and Birkhoff)

We will use a specific novel treatment of adjunctions (Ellerman) that shows they arise by gluing together in a certain way two universal constructions or self-predicative universals (“semi-adjunctions”). But for illustration, we will stay within the methodological restriction of using examples from partial orders (where adjunctions are called “Galois connections”).

We have been working within the inclusion partial order on the set of subsets ζ(U) of a universe set U. Consider the set of all ordered pairs of subsets <a,b> from the Cartesian product ζ(U) x ζ(U) where the partial order (using the same symbol) is defined by pairwise inclusion. That is, given the two ordered pairs <a’, b’> and <a,b>, we define

<a’,b’> ⊆ <a,b> if a ⊆’  a and b ⊆’  b.

Order-preserving maps can be defined each way between these two partial orders. From ζ(U) to ζ(U) x ζ(U), there is the diagonal map Δ(x) = <x,x>, and from ζ(U) x ζ(U) to ζ(U), there is the meet map ∩(<a,b>)  = a ∩ b. Consider now the following “adjointness relation” between the two partial orders:

Δ(c) ⊆ <a,b> iff c ⊆ ∩ (<a,b>) Adjointness Equivalence

for sets a, b, and c in ζ(U). It has a certain symmetry that can be exploited. If we fix <a,b>, then we have the previous universality condition for the meet of a and b: for any c in ζ(U), c ⊆ a ∩ b iff Δ(c) ⊆ <a,b> Universality Condition for Meet of Sets a and b.

The defining property on elements c of ζ(U) is that Δ(c) ⊆ <a,b>. But using the symmetry, we could fix c and have another universality condition using the reverse inclusion in ζ(U) x ζ(U) as the participation relation: for any <a,b> in ζ(U) x ζ(U), <a,b> ⊇ Δ(c) iff c ⊆ a ∩  b. Universality Condition for Δ(c). Here the defining property on elements <a,b> of ζ(U) x ζ(U) is that “the meet of a and b is a superset of the given set c.” The self-predicative universal for that property is the image of c under the diagonal map Δ(c) = <c,c>, just as the self-predicative universal for the other property defined given <a,b> was the image of <a,b> under the meet map ∩(<a,b>) = a ∩ b.

Thus in this adjoint situation between the two categories ζ(U) and ζ(U) x ζ(U), we have a pair of maps (“adjoint functors”) going each way between the categories such that each element in a category defines a certain property in the other category and the map carries the element to the self-predicative universal for that property.

Δ: ζ(U) → ζ(U) x ζ(U) and ∩: ζ(U) x ζ(U) → ζ(U) Example of Adjoint Functors Between Partial Orders

The notion of a pair of adjoint functors is ubiquitous; it is one of the main tools that highlights self-predicative universals throughout modern mathematics.