Composition is a product of maps, and commutativity is a shuffle of coordinates.
In the motivic instance of the paper, is the category of pointed motivic spaces, , and . Then is Voevodsky's category of framed correspondences. Over , a level- correspondence from a point to a point is a finite set with a holomorphic framing vanishing exactly on (Remark 4.4). The experiments draw these, and the structures of Theorems 2.4, 3.2, 5.2 and 5.5 acting on them.
Algebraic: the displayed polynomials define genuine framed correspondences with explicit zeros. Composition, conjugation and stabilisation use Def. 2.3 and eq. (2). The formulas imply the identities; floating-point evaluations provide implementation diagnostics. Scope: composition and module experiments use simplicial degree 1; stabilisation uses uniformly labelled supports. Graphical: sampled surfaces and indexed polylines encode the data (section B). Not shown: Theorem 4.15, the Nisnevich theory of Section 6, and stable homotopy beyond the degree of the displayed classes.
Selection
Computation
Numerical diagnostics
Floating-point checks of this implementation. Algebraic justifications are given in A and C.
How to operate the experiments
The 3D view
- Drag to rotate, scroll or pinch to zoom, shift-drag (or two-finger drag) to pan. Double-click or Reset view restores the camera.
- With the view focused: arrow keys rotate, + and − zoom, 0 resets.
- Click a zero, sphere or ring to read the point of ℂⁿ it stands for. In a stack, the complete tuple is highlighted. The tuple selector also reaches points hidden by overlaps. In the first experiment a click draws the local picture of the framing.
- Expand enlarges the view; Esc closes it.
The four experiments
- One framed correspondence: choose a framing f(z), move its zeros, and read the degree.
- Composition and shuffles: the identities in simplicial degree 1, where all sphere labels are the unique nonbasepoint label; composition through Y = 𝔸¹; and the Σₙ-action. Play runs the permutation.
- Stabilisation: add σ-layers and watch equivariance (4) and bilinearity (8). Each correspondence carries one common cut word on its support; general componentwise labels are not implemented.
- Homotopy over Δ¹: zeros as braids in ℂ × [0,1]; slide s, or sweep it.
The mathematics behind the picture
Notation follows the paper. Theorem, corollary and equation numbers follow arXiv v2 (Garkusha, Correspondences and stable homotopy theory). The experiments use Sections 2 to 5 in the motivic instance of Section 4; Section 6 is discussed in D but not drawn.
A. Mathematical construction
A.1 Symmetric sequences and ring objects
Let be a symmetric monoidal category with finite colimits and zero object in which is an isomorphism. Symmetric sequences have the tensor product , whose symmetry isomorphism involves the shuffle , which moves the first slots past the last and keeps each block in order. A ring object has -objects , equivariant multiplications and a unit (Def. 2.1); it is commutative if .
A.2 Correspondences
For and full and closed under (Def. 2.3),
with composition given by the chain
and with acting by conjugation, (eq. (2)).
A.3 The spectral version
Let and let be a symmetric ring -spectrum (a ring object with a central unit , Def. 3.1). With the simplicial sphere, (Def. 5.1) is a symmetric -spectrum in . Theorem 3.2 shows that is a symmetric (commutative) ring -spectrum; Theorem 5.2 that is enriched over symmetric -spectra, the pairings being bimorphisms (the bilinearity diagram (8)); Theorem 5.5 that for commutative this category is also enriched over , by the diagram (9).
A.4 The motivic instance and holomorphic framed correspondences
Take (pointed motivic spaces), , , and , the free commutative monoid on (Section 3). This is a commutative ring object, so the commutative clauses of Theorems 2.4, 3.2(2) and 5.5 apply. Then , and the introduction notes that Voevodsky's graded category is recovered from Theorem 2.4. By Voevodsky's lemma (Remark 4.4) an element is the class of a quadruple
Here is an étale neighbourhood of the support, with its specified lift of ; the equation for the support is understood set-theoretically. Multiplicities refer to the zero scheme cut out by . In every displayed algebraic example one may take and . The framing itself need not be étale, and repeated zeros are allowed.
and for over it is the class of a holomorphic triple with finite, an open neighbourhood of and holomorphic with , up to shrinking . The canonical is the level-one correspondence .
Unwound formulas. Following the five arrows of Def. 2.3 on points, for and , is the quadruple on with framing and map
For this is the product of the two framings, with zero set . In general the zero set is the fibre product . The conjugation action (2) is , and . These are Def. 2.3 and eq. (2) written on quadruples.
A.5 Simplicial labels and the scope of the experiment
The spectrum of Def. 4.6 has levels . Each level is a simplicial set. Write , where . In degree , its nonbasepoint simplices are cut words . A general framed correspondence with this target can assign a different cut word to each connected component of its support:
The stabilisation experiment restricts to uniformly labelled supports, for which is constant. It stores one pair ; this does not describe every simplex of . For example, at , the zeros of may carry cuts respectively. That componentwise labelling is valid but is not implemented here. Uniform labels are preserved by the displayed products, permutations and bonding maps.
The composition and module experiments use simplicial degree . Then every nonempty support has the unique cut word , so the identification
allows those experiments to omit the sphere labels. The basepoint is the empty correspondence. The sphere spectrum over additionally uses the algebraic simplices: . Its simplicial structure involves both algebraic pullback along faces and degeneracies and the sphere labels.
B. Geometric interpretation
B.1 Level one: the surface of a framing
For the framing is sampled on a rectangular grid and drawn with height and hue . The exact height function vanishes at the zeros; the triangle mesh approximates it and need not touch the plane at an off-grid zero. Zero markers use the explicit root coordinates independently of the mesh. Around a zero of multiplicity , the phase winds times. Its local degree is the winding number of on a sufficiently small circle, and .
B.2 The stack of planes
A point is stored as an indexed polyline through in parallel planes. The indexed list retains every coordinate and multiplicity. Permuting slots moves the corresponding vertices to their destination layers; the displayed products concatenate tuples. A union of unlabelled segments alone need not retain the grouping: and give the same union of segments in three layers. The tuple selector and full-polyline highlight preserve access to the stored grouping, including coincident vertices.
For composition through an extra plane for is inserted; the polyline records . The cut tokens are the common support label of the uniformly labelled examples. They move with their slots; they do not describe a geometric realisation of the simplicial sphere.
B.3 Families over the simplex
A framed correspondence over is a family with finite over the base. Its real slice is drawn as the braid of zeros in ; the faces and are the planes .
B.4 Dictionary
| Graphical element | Mathematical object | Status |
|---|---|---|
| Surface over a plane (Exp. 1) | the framing of a level-one correspondence | sampled mesh of an explicit function; height and hue are display choices |
| White spheres on the surface | zeros with multiplicity (size) | exact |
| Stack of planes, indexed polyline through it | and a point of | tuple data retained; overlaps require selection to resolve (B.2) |
| Exchange of layers (animation) | the permutation action of on | endpoints exact; intermediate frames illustrative |
| Rings versus coloured spheres | zero set of the reference route versus the raw route, carried by the permutation | exact comparison, computed |
| Green plane (through Y) | and the point | exact |
| Tokens on the left of a stack | cuts in | exact labels; no geometry of is drawn |
| Braid curves, white slice (Exp. 4) | zeros of over the real segment of ; the slice is | explicit roots at samples; polygonal curves between samples |
C. Mathematical mechanism
C.1 Composition is a fibre product of zero sets
For the composite of and is their product, so zero sets multiply, multiplicities multiply, and the degree is multiplicative: . This is the ring structure on seen through the degree. For the zero set is : each zero of the first framing contributes the zeros of the second framing over . If the fibre degenerates (for example over ) two zeros merge into one of multiplicity two.
C.2 Commutativity is a shuffle
The ring structure on is the product of framed correspondences, so commutativity (Prop. 4.7, Thm. 3.2(2)) says : exchanging the factors moves the layers of past the layers of . Each point keeps its coordinates and changes only its layer. Because carries no coefficients other than the maps themselves, nothing else is needed, and all identities of the paper hold on the nose. This is the mechanism behind the hypothesis " is commutative" in Thms. 2.4, 3.2(2) and 5.5 in this instance. For a more general ring object the multiplication would also have to commute with the exchange of the two factors, and that is what the hypothesis asserts; the page does not draw such an .
C.3 Symmetric monoidal and module structures
The interchange law of Thm. 2.4 compares , with layers , with , with layers ; they differ by , the exchange of the two middle blocks. The module law (9) compares and ; they differ by , which moves the layer of past those of . For , , and are all products; they differ for general objects.
C.4 Stabilisation
The structure map of sends and a cut to : a new layer carrying the framing , whose only zero is the origin, and the cut . The iterated map adds such layers, using ; since this framing is fixed by , the map is -equivariant (eq. (4)), with acting on the cuts only. The bilinearity diagram (8) says with : in the last route the new layer sits in front of the layers of and moves it to the end.
C.5 Homotopy and degree
The sphere spectrum is built from . A one-simplex in the algebraic-simplex direction is a family with support finite over the whole base. The page uses on , so and . Such a family supplies a homotopy between its faces after passage to the framed-motive construction.
Each valid family here is monic in , of degree respectively. Therefore
This proves finite flatness and constant fibre length over all complex values of , including collisions. Degree conservation for these families follows from monicity, independently of the sampled numerical checks. For the excluded family ,
which is not finite over . Its zero escapes as , and its fibre degree drops. Thus it cannot supply a framed correspondence over . The stable identification is supplied by Cor. 4.13 and the sphere comparison of Thm. 4.5. The degree detects the stable classes of the displayed examples; the page does not classify unstabilised framed correspondences or compute the full stable homotopy type.
D. Connection with the theory
| Experiment | Statement in the paper | What is computed |
|---|---|---|
| One framed correspondence | Remark 4.4 (holomorphic framed correspondences); Thm. 4.5 and Cor. 4.13 for | zeros and multiplicities of explicit framings; winding numbers; invariance of the degree under contraction of the zeros |
| Composition and shuffles | Def. 2.3; Thm. 2.4 (enrichment, symmetric monoidal part); eq. (2); Prop. 4.7 and Thm. 3.2(2); Thm. 5.5, diagram (9) | explicit polynomial composition and conjugation; shuffle and module identities in simplicial degree 1; numerical diagnostics on framings and listed zeros |
| Stabilisation | Def. 4.6; Thm. 3.2(1), eq. (4); Thm. 5.2 and diagram (8) | on uniformly labelled supports; the chosen and three bilinearity routes with |
| Homotopy over | Def. 4.6 (); Prop. 4.12(3); Thm. 4.5; Cor. 4.13 | braids of zeros; conservation of degree along four explicit families, one of them not finite over |
| Not drawn | Thm. 4.15 (the Quillen equivalence ); Section 6 (Thm. 6.7, Nisnevich excisiveness; Cors. 6.8, 6.9; Thm. 6.11 and Thm. 6.12) | nothing: they need the homotopical and topological input (Levine, Wilson–Østvær, Zargar, Garkusha–Panin) that no finite picture carries |
The identities drawn here are consequences of the theorems for . The experiments therefore illustrate the theorems and test my reading and implementation of them; they do not test the theorems.
E. Limitations
Algebraic justification. The framings are explicit polynomials. Products, conjugation and bonding use the formulas in A.4, and their identities follow algebraically by associativity and reordering the relevant coordinate blocks. Monicity proves finite flatness of the valid families and of each displayed second framing over . The chosen complete-intersection zero schemes have the stated lengths.
Numerical diagnostics. The implementation compares framings at eight deterministic pseudorandom complex points in double precision and compares listed zero multisets after rounding to . Winding numbers are sampled on small circles. Passing these tests does not prove a polynomial identity, establish the completeness of an arbitrary root list, or test the neighbourhood equivalence relation. No historical test-suite count is asserted here.
Choices. The catalogue contains polynomials with roots known in closed form. Multiplicities are supplied from their algebraic structure. For the displayed fibre products the second polynomial is monic, so the local length is the first local length times the second fibre length; this is not a numerical multiplicity algorithm. Objects are a point and , and there are at most four complex-coordinate layers. Composition and module experiments use simplicial degree 1. The stabilisation experiment implements only uniformly labelled supports and nonbasepoint cut words, not the full simplicial mapping spaces or their face and degeneracy operators.
Graphical approximation. The surface is a sampled triangle mesh. The real parameter slice of a family is drawn by sampled polygonal curves. Heights, hues and colours are display choices. A permutation animation moves coordinate vertices to their destination layers; intermediate frames have no categorical meaning. Indexed tuple data are retained, while an unselected union of overlapping polylines may conceal the grouping.
Not shown. General étale neighbourhoods and their equivalence relation; the empty correspondence, which is the basepoint; general componentwise simplicial labels; the motivic topology; geometric realisation of ; Theorem 4.15; and Section 6. All displayed polynomial examples, including composition through , have algebraic representatives with the identity étale neighbourhood.
What the experiments show
- A level-one framed correspondence is a degree- local picture. The winding of the framing at each zero is its multiplicity, and the total is the degree.
- Composition factors through zero sets. For a point it is a product with multiplicities multiplying; through it is a fibre product, and degeneracies of the fibre appear as merged zeros.
- In this instance commutativity is a shuffle and nothing else. The commutativity of , the interchange law and the module law are each an exchange of blocks of layers.
- The structure maps are visible. The iterated map is equivariant because is invariant under , and the shuffle of the bilinearity diagram is the movement of the σ-layer past the layers of .
- Homotopy preserves the degree, and finiteness is why. Zeros may collide inside a finite flat family; they may not escape.
Questions the experiments suggest
- Section 6 assumes that has contractible alternating group action: an even permutation acts on as the identity up to -homotopy. In the stack picture an even permutation of layers is a determinant-one permutation matrix on , a product of elementary matrices, hence algebraically homotopic to the identity; an odd one has determinant . Can this be shown in the picture, with the explicit path acting on the stack?
- For a Thom ring spectrum such as MGL (Section 6), what replaces the zero set , and what is the analogue of the degree on ?
- Over a field other than the class of a framed correspondence lies in (Cor. 4.13), not just in . I recall that over the real points give a signed count, but I have not checked this against the literature here, and nothing on the page uses it. How much of the picture survives on real points?
Extensions
- The explicit homotopy of a 3-cycle. Draw a path of elementary matrices from the permutation matrix of a 3-cycle to the identity, acting on the stack of three layers, and contrast it with a transposition, which cannot be reached this way.
- Framed correspondences over a base curve. Take a curve, so that is a finite cover of and composition through becomes a composition of covers with monodromy.
- The spectral category of Section 6. Add an -structure (a Thom ring spectrum) and draw the objects of .