A simplicial homotopy becomes a polynomial one by sliding a single vertex at a time.

A.1 Algebras, homotopy and T-closed categories

An admissible category of non-unital -algebras is closed under ideals, quotients, and fibre products (§2.1); is admissible. Two homomorphisms are elementary homotopic if some has and . This relation is reflexive and symmetric; its transitive closure is polynomial homotopy. One puts . A category is T-closed if the forgetful functor has a left adjoint ; then is the universal extension (§3), and for one has .

A.2 Simplicial algebras and their spaces of homomorphisms

The ordinary simplicial algebra is , with ; the face sends to according as . For a simplicial set , . To distinguish completion from the ordinary simplicial algebra, this page writes for the subdivision-completed simplicial ind-algebra, . Theorem 2.4 identifies its Hom simplicial set with the Kan replacement of the ordinary one:

With , the boundary kernels are

A.3 The spaces and spectra K(A,B)

The maps (§4), where is the natural -linear splitting of , give the sequence and the space . By Theorem 5.1, , so these spaces form an -spectrum; Excision Theorems A and B make it excisive in both variables, Comparison Theorem A identifies with (polynomial homotopy classes), and Comparison Theorem B identifies with .

A.4 The Hauptlemma and formula (8)

For the homotopy in (1) is, for , the ring homomorphism (formula (8)),

and at , at . For the paper gives an algorithm (Steps 0 to 3, p. 1246) that moves the subdivided facet to through subposets , each isomorphic to , by moving one vertex to an adjacent vertex at a time; each step is a polynomial homotopy assembled from or on the simplices that contain , and from the constant homotopy elsewhere. The same method with the cube proves (2) (p. 1248).

A.5 The commutative model used for the pictures

Take with , and . For commutative , , so

since consists of compatible families of polynomials on the simplices of . I use two families of examples: for a polynomial map in barycentric coordinates, and polynomial maps on the cube. This is the case of the Hauptlemma where is the free commutative algebra on three generators and the ground field; the statement itself is proved for every T-closed admissible category.

B.1 The slide

The ring homomorphism is dual to a polynomial map , bilinear in . For fixed it is the affine map taking the vertex of to

so a family of -simplices in : at the facet omitting vertex , at the facet omitting vertex , and in between the vertices slide one step each along edges of . For this is a hinge about the common -face. The homotopy lives on the domain side, which is why it acts on every at once: is natural in and no choice is made in the target.

B.2 The posets

The vertices of are the nonempty subsets of ordered by inclusion, and its simplices are the chains; it is realised in with each subset at the barycentre of the corresponding face (exactly: the simplicial structure is affine on every chain). The cube has the subsets of (equivalently with the coordinatewise order) as vertices and the chains as top simplices (Kuhn triangulation). The facet is ; for the cube, the top face (last coordinate ) is and the bottom face is , the paper’s .

B.3 The sweep

A subposet isomorphic to is a PL -disc in the -ball ; its simplices are its maximal chains. An elementary slide replaces one vertex by an adjacent vertex . It is legal if are joined by an edge of and every stands in the same order relation to as to (below, above or incomparable); then . On a simplex at position the new simplex has at position if and at otherwise, and the slide from to is if and if . Geometrically the vertex slides linearly along the edge from to ; the target image is applied to the disc.

B.4 Dictionary

C.1 One simplex: a single polynomial homotopy

For the faces and of one simplex are elementarily homotopic: the facet slides across the simplex. In the model is a polynomial homotopy of curves (or surfaces) on a polynomial patch whose ends are the two boundary curves (or faces) of the patch. The page checks the formula exactly (column sums, endpoints) and the composite numerically.

C.2 What the chain demonstrates

The sweep constructs elementary homotopies on the domain, using only the simplicial structure. Its local formulas therefore work for every compatible algebra map into the polynomial algebra on that complex. The chain is the method used here to establish polynomial homotopy; it is not a proof that no single elementary homotopy exists for a particular target.

Indeed, in this page’s free commutative model, write the endpoint maps as triples and in the common source’s polynomial-function algebra. Assigning generator to extends uniquely to one elementary homotopy from to . If both triples vanish on the boundary, this interpolation does too. Thus these examples have no obstruction to a single homotopy; their mathematical content is the target-independent, simplexwise mechanism of the Hauptlemma.

C.3 Adjacent-face sweeps and canonical parametrisations

For adjacent indices , let . First move by decreasing size, then by increasing size. This reproduces both worked examples (Δ² and Δ³, pp. 1246–1248) and uses slides per adjacent segment. It replaces label by , fixes the common face, and, because the indices are adjacent, gives the canonical ordered face identification.

For nonadjacent indices, the same direct substitution reaches the correct facet as a set but changes its parametrisation. This page instead follows , concatenating adjacent sweeps. The prescribed route has slides. Source vertices are tracked throughout; the endpoint is accepted only if their images agree with . For instance, from to in Δ² the required vertex transport is ; directly replacing by would instead reverse the target parameter.

Computed, with a specified scope. In the unparametrised graph of discs and legal Hasse-edge slides, from an adjacent pair of simplex facets, the shortest set-to-set lengths are with shortest state paths for . For top-to-bottom cube faces, without an additional relative condition, the lengths are with paths. These counts forget the source labels. Free-mode hints use a separate search in the labelled embedding graph, with the relative-boundary test when enabled. These are finite combinatorial computations, not minima among arbitrary polynomial homotopies.

C.4 The cube, the relative boundary, and based maps

The paper’s route moves in a linear extension of the Boolean order on . The boundary tracks stay in . For , the map vanishes on that lateral boundary, so every intermediate map descends to a based map from . The amber image is the image of this based map; it need not be an embedded sphere or even be homeomorphic to one.

In based free mode, preserving the induced poset alone is insufficient. For every original boundary simplex, the checker takes its transported vertices together with the new vertex when the simplex contains the moving vertex. It requires some coordinate among the first to be constantly or constantly on all these vertices. The whole affine track then lies in one lateral face for every value of the polynomial parameter. Only moves passing this additional test are offered. Changing the boundary condition starts a new free exploration.

The curve examples use and , so the correct nodal cubic relation is . Over the affine pinching of has coordinate ring , with pointed ideal . The drawn triples are deformations built from ; their coefficients and target projection can change the appearance.

C.5 What is not drawn: parts (3) and the role of the sweeps

Part (3) converts a polynomial homotopy back into a simplicial one at the price of replacing by in ; its proof shows that is a path space of in the Brown category (the Hauptsublemma), using the simplicial map of Proposition 4.1 (the functor , all other pairs , that is, ). Parts (1) and (2) are used in Comparison Theorem A to make well defined, and (3) for its injectivity (p. 1252).

• The elementary homotopy is geometric. The paper’s formula (8) is the dual of an honest map that slides vertices along edges, and it acts on every homomorphism at once.
• After subdivision, the homotopy is a sweep. A chain of elementary slides moves an -disc through the -ball one vertex at a time; admissible orders are constrained by the poset relations and, in the based cube case, by the lateral boundary.
• Parametrisations matter. A disc occupying the target facet need not induce the canonical face map. The source transport table makes this visible; hints use labelled embeddings.
• The cube sweep is a linear extension. Its admissible orders are the linear extensions of a Boolean lattice, and with the boundary condition it is a path in a loop space.
• Apparent misprints in the Δ² example. Recomputed with formula (8), is , not , because interpolates between and rather than between the two edges of that step; and the last set is labelled where is meant. The other entries agree. These are for the author to confirm.

1. For adjacent simplex facets in the unparametrised Hasse-edge state graph, is the shortest set-to-set length always , with exactly minimal sweeps? I have the values for only.
2. What replaces the two-phase rule for , , and how many elementary slides does it need? The paper’s Steps 0 to 3 allow many choices there.
3. For nonadjacent canonical face embeddings, how does the shortest labelled route compare with the adjacent-segment construction? Which obstructions to a single elementary homotopy survive for algebras that are not free? The present free model has none (C.2).
4. The two-phase structure, up into the interior and then down onto the far facet, resembles a collapse of the ball from one facet to the other. Is there a discrete-Morse description of the sweeps?

1. Higher subdivisions. Implement by generalising the rule, and compare the number of slides with the paper’s informal algorithm.
2. Part (3) and the path object. Draw the Hauptsublemma: the fibre product , the map , and a polynomial homotopy converted into a simplicial one.
3. Theorem 5.1. The cycle permutes the coordinates of ; on it is a rotation about the main diagonal. Draw how loops of loops are re-ordered in the identification .