Infrageometry

A spatial hypergraph $H = (V, E)$ is an undirected simple hypergraph: a set $V$ of points and a set $E$ of nonempty subsets of $V$, the edges, none repeated. An edge $\{v\}$ with one vertex is allowed. $H$ is finitary when each vertex lies in finitely many edges and each edge is finite.

The incidence space $X(H)$ is the set $V \sqcup E$ ordered by incidence, $v \le e$ when $v \in e$, with the Alexandrov topology whose open sets are the up-sets. Vertices are closed points, called positions. Edges are open points, called infinitesimal positions. The smallest open set around a position is $U_v = \{v\} \cup \operatorname{St}(v)$, the position with its star of edges.

The closed neighbourhood $N[v]$ is the set of points sharing an edge with $v$. The hop distance $d(v, w)$ is the least number of edges a motion from $v$ to $w$ crosses. The ball is $B_r(v) = \{w : d(v, w) \le r\}$, so $B_1(v) = N[v]$, and the sphere $S_r(v)$ is where $d = r$.

The link $\operatorname{Lk}(v)$ is the spatial hypergraph on the neighbours of $v$ whose edges are $e \setminus \{v\}$ for the edges $e$ through $v$ with at least two vertices. Its edges correspond to the directions at $v$: it is the discrete unit sphere at $v$. The link dimension counts recursively, $\dim_v H = 1 + \max_w \dim_w \operatorname{Lk}(v)$.

A subspace is a subset of $X(H)$ with the subspace topology. When the traces $e \cap S$ of its edges are nonempty and distinct, it is again an incidence space, that of the hypergraph of traces. A closed subspace is a sub-hypergraph. An open subspace holds every edge through each of its positions, so edges may dangle. The departures at $v$ that leave a closed subspace $S$ form its normal space, and the frontier of $S$ is where the normal space is not empty.

A tangent vector at $v$ is a departure $v \rightharpoonup e$ into an edge $e \ni v$: the infinitesimal motion from the position $v$ into the infinitesimal position $e$. The tangent space $T_vH$ is the set of departures at $v$. When $\{v\}$ is an edge, the departure $v \rightharpoonup \{v\}$ is the stationary vector, the hypergraphic zero, and nothing else supplies one. The tangent bundle $TH$ is the set of all incidences.

The motion graph $M(H)$ is the directed graph on the points of $X(H)$ with a departure $v \rightharpoonup e$ and an arrival $e \rightharpoonup v$ for each incidence. A motion is a walk in $M(H)$, $\gamma = (v_0, e_1, v_1, \dots, e_n, v_n)$, made of half-steps; two half-steps make a step. Its velocity at $v_i$ is $v_i \rightharpoonup e_{i+1}$. A geodesic is a motion with fewest steps between its ends.

A moving point is a token in a hypergraph rewriting system. At a position $x$ the token is attached to it by a directed edge; inside an edge it is one more vertex of that edge. For each edge size there are two rules, a departure that moves the token into an edge and an arrival that moves it out. The multiway evolution from one state explores every motion at once.

A step path is a word $v_0 v_1 \cdots v_n$ of positions in which each two consecutive ones share an edge: a chain of directed binary path edges. It is a motion with its infinitesimal positions forgotten. Staying, $v\,v$, needs the unary edge $\{v\}$.

Step paths are rewritten by four rules and their inverses: contract $a\,c\,b \to a\,b$ and expand $a\,b \to a\,c\,b$ when $a, b, c$ are distinct and lie in one edge of size at least three; cancel $a\,b\,a \to a$; and stall $a\,a \to a$. Two paths with the same ends are area-homotopic when a chain of rule applications, a derivation, turns one into the other. Hyperedges of size three or more act as filled areas, and the derivation is the proof of equivalence.

Two motions with the same ends are homotopic when their step paths are area-homotopic. Area homotopy is the primary notion and the homotopy of motions is derived from it. The finer relation that only cancels backtracks, $x\,y\,x \to x$, is the homotopy of the incidence space; it keeps the infinitesimal positions a motion passes and leaves hyperedges unfilled.

A barrier is a set of pairs of adjacent positions such that every triple inside an area contains an even number of them. The crossing parity of a path is the parity of its number of steps across the barrier. No rule changes it, so two paths of different crossing parity are not homotopic: the barrier is a certificate of inequivalence, checked area by area.

$\pi_1(H, v)$ is the group of closed step paths at $v$ up to area homotopy. A spanning tree of the adjacency graph gives a presentation: each step outside the tree is a generator, and each triangle inside an area is a relator $g_{ab}\,g_{bc}\,g_{ca}$. It is the edge-path group of the simplicial complex whose simplices are the subsets of edges.

A field of type $X$ on $H$ is a finite multiset of edges of that type laid on the space. Fields are partial: a field says nothing where it has no edges. Several fields live together as one multiset of typed edges, the state on which rewrite rules act.

A vector field of type $X$ is a multiset of arcs of the motion graph: departures $X(v \rightharpoonup e)$ from vertices to edges, and arrivals $X(e \rightharpoonup v)$ from edges to vertices. A step $a \to b$ of $X$ is a departure $a \rightharpoonup e$ followed by an arrival $e \rightharpoonup b$; a stationary step $a \rightharpoonup \{a\} \rightharpoonup a$ stays on a unary edge.

An arrival is a negative direction: $X(e \rightharpoonup v)$ is the opposite of $X(v \rightharpoonup e)$, and the rule $X(v \rightharpoonup e),\ X(e \rightharpoonup v) \to \varnothing$ removes the two together. What remains is the net field, departures minus arrivals on each incidence. A step and its reverse cancel, and so does a stationary step.

A scalar field of type $f$ is a multiset of unary edges $f(v)$ on the vertices. Its value at $v$ is not given; it is inferred, as the multiplicity $|f|_v$ of $f$ at $v$. A unary edge is drawn as a disk at its vertex, and a scalar field as concentric disks, one per copy.

Two values are compared by the rule $f(a),\ f(b) \to \varnothing$, applied to working copies of the unary edges at $a$ and $b$ until one side runs out. What is left is the difference, at the higher end. Along a step $a \to b$ this is the derivative, $\uparrow$, $\circ$ or $\downarrow$, together with its size.

Saturating a state under rules that add edges applies every rule to every match, round after round, until a round adds nothing. The rounds and the edges each one adds form a trace. Because the rules only add, the result does not depend on the order in which they are applied.

A flow is integrated by closing up under two rules: a departure and an arrival of $X$ make a step, $X(a \rightharpoonup e),\ X(e \rightharpoonup b) \Rightarrow X^{+}(a \to b)$, and a further step extends it, $X^{+}(a \to b),\ X(b \rightharpoonup e),\ X(e \rightharpoonup c) \Rightarrow X^{+}(a \to c)$. Round $k$ of the saturation adds the displacements of time $k$.

From a step of $X$ from $a$ to $b$ and a step of $Y$ from $b$ to $c$, the composition rule adds $(X;Y)(a \to c)$: first the flow of $X$, then that of $Y$. It is a transitive closure across two types, defined where the edges meet and nowhere else.

Compose both ways. Where $(X;Y)(a \to c)$ and $(Y;X)(a \to c')$ end apart, $c \ne c'$, the bracket rule adds the step $c' \to c$ of type $[X, Y]$, as a departure and an arrival through the smallest edge holding both ends. Where the square closes, the fields commute there; where a composite is missing, the bracket is undefined. The orientation is the classical one: flowing along $X$ then $Y$ ends past flowing along $Y$ then $X$ by $[X, Y]$.

A map $f: H \to H'$ sends vertices to vertices and each edge to an edge or to a vertex, keeping incidence; an edge sent to a vertex collapses. Maps are the continuous maps of incidence spaces that send positions to positions. The pushforward of a departure is $f_*(v \rightharpoonup e) = f(v) \rightharpoonup f(e)$. Immersions, submersions and local isomorphisms are read from the pushforwards.

The interval $I_n$ is the path graph on $0, 1, \dots, n$. A motion with $n$ steps is a map $\gamma: I_n \to H$, and its velocity at $i$ is the pushforward of $i \rightharpoonup \{i, i+1\}$. Curves are maps from intervals, as in the continuum.

The scalar line $\mathcal{L}$ is the path on $\mathbb{Z}$ with a unary edge at each point; it has three tangent vectors at each point, $\downarrow$, $\circ$ and $\uparrow$. A scalar function is a map $f: H \to \mathcal{L}$. Its derivative along a departure is the pushforward. A point is regular, a local maximum, a local minimum or flat according to the derivatives that occur at it.

In the map version a vector field chooses a departure at some points, and a transit says, for each edge entered, by which vertex to leave it. Together they define a flow $\varphi$, one step $v \to X(v) \to \varphi(v)$, and integral curves that end, rest or cycle.

The pointed ball $(B_r(v), v)$ is the subspace induced on $B_r(v)$ with its centre marked. Given a model space $M$, a chart of $H$ at $v$ of radius $r$ is an isomorphism of pointed balls $(B_r(v), v) \to (B_r(m), m)$. $H$ is locally modelled on $M$ at radius $r$ when it has a chart at every point; the charts form an atlas, and where two balls overlap their charts give a transition map between pieces of $M$.

A hypergraph of triangles is a combinatorial surface when the link of every point is a cycle: around each point the triangles close up into a disk. This is the link reading of being a manifold; local models are the other reading.

The link dimension is purely local and purely hypergraphic. The growth dimension, the slope of $\log |B_r(v)|$ against $\log r$, measures dimension at large scale and tends to $n$ on the grid $\mathbb{Z}^n$. It uses real numbers, so it is kept outside the core as a device for comparison with the continuum, as is the linear tangent space $\mathbb{Z}[T_vH]$.