Weaving Space from Arrows: The Homotopy of Directed Graphs
07 May 2026, Yanjiang
Directed graphs, when interpreted through cubical homotopy, encode all topological shapes, from spheres to tori, in their arrow networks.
What if a simple arrow, drawn between two points, could encode the shape of a sphere, a torus, or even higher-dimensional spaces? This is the startling claim at the heart of a new preprint (arXiv:2605.04959) by a team led by Briony Eldridge at the Beijing Institute of Mathematical Sciences and Applications (BIMSA). The researchers develop a homotopy theory for directed graphs and prove a deep equivalence: an ∞-category built from these graphs is the same as the ∞-category of topological spaces. To say it plainly: graphs with arrows can model all the topological information of geometric shapes, provided you read the arrows correctly.
To grasp what this means, picture a city mapped as a network of one-way streets. Each intersection is a node, each street an arrow pointing in the allowed direction. Topology, the mathematics of shape, usually ignores such directional constraints—a rubber sheet can be stretched any which way, after all. But Eldridge and colleagues—including Sergei O. Ivanov, Xiaomeng Xu, Shing-Tung Yau, and Mengmeng Zhang—show that when you correctly encode direction, the graph’s hidden geometry leaps into focus. Think of it like a radio signal: the static of disconnected streets settles into a coherent picture once you tune to the correct topological frequency.
Homotopy has long been the art of reshaping without tearing. In ordinary topology, we probe a space by mapping spheres into it: a loop tells us about one-dimensional holes, a sphere’s surface about two-dimensional voids. Directed graphs, however, resist spherical probes because a sphere’s surface has no natural orientation that aligns with one-way arrows. This is not stubbornness but a structural fact: a one-way street admits no U-turn. The team turns to cubes instead. A cube’s edges can be oriented, its faces filled, all while respecting the graph’s one-way logic. The resulting cubical homotopy groups, also called A-groups or reduced GLMY groups, become the precise tools for measuring directed topological features.
Here the tapestry begins to weave. Cubical homotopy captures information that ordinary homotopy misses—features born from directionality itself. For a standard graph, an A-group might count loops that exist only because some paths are forbidden. This is not a defect; it is a new kind of topological signal. Consider a directed cycle: it looks like a loop, but it cannot be reversed. An ordinary homotopy group might miss the cycle entirely because reversal is allowed, but the cubical group sees it clearly.
To appreciate why this matters, recall the classical homotopy hypothesis: In classical topology, the homotopy hypothesis asserts that spaces—specifically CW-complexes—can be fully modeled by higher groupoids, of which the fundamental groupoid is only the first layer. That is, spaces can be modeled by combinatorial data. Eldridge’s work extends this hypothesis to directed graphs. She and her collaborators show that by localizing the category of directed graphs at those maps that induce isomorphisms on A-groups, one obtains an ∞-category, denoted DGra∞. Localizing means formally adding inverses for certain morphisms—treating them as equivalences—so that the structure becomes richer and more flexible. Their main theorem declares that DGra∞ is equivalent to the ∞-category of spaces. This equivalence is stark: it says that from the directed-graph perspective, nothing is lost about space itself. All topological information, up to homotopy, can be read from arrows and their one-way itineraries.
Why cubes, and why A-groups? Traditional homotopy theory uses spheres because a loop (a circle) is a naturally undirected probe. A directed graph, by contrast, requires something that respects its one-way arrows. Cubes fit this bill: a square has a natural direction from one corner to another, and higher cubes extend this pattern. The A-groups, therefore, are not an ad hoc invention; they are the natural homology of a world where direction matters. This is not a matter of will, but of mathematical necessity—the arrows cannot be reversed without breaking the graph’s structure. In an image that lingers, the team has shown that directed graphs form a kind of topological laboratory, where the familiar shapes of classical geometry reappear as patterns of permitted motion.
The shift to ∞-categories is not mere technical excess. In an ordinary category, you have objects and morphisms; in an ∞-category, you also have morphisms between morphisms, and so on endlessly. This infinite tower reflects the layered structure of homotopy theory, where a path can be continuously deformed into another. The team’s construction shows that directed graphs, when equipped with A-group equivalences, naturally organize into such a tower. It is like discovering that a blueprint of a building, drawn with simple rules, automatically generates the blueprint’s own meta-documents—instructions for how to modify the instructions.
The philosophical weight of this result settles as you read the paper. Topology is often called the mathematics of continuity, of the rubber-sheet universe. But directionality—one-way motion—conjures thoughts of time, of processes that cannot be reversed. The team’s work hints that spacetime itself might be understood through combinatorial data, where the one-way arrows of causality are not an added layer but the very fiber of geometry. This is not a metaphor; it is a precise mathematical statement about the interchangeability of graphs and spaces. The team does not claim to have solved the problem of quantum gravity; what they provide is a rigorous framework in which such questions can be posed.
Of course, the road from equivalence theorem to concrete applications is long. The ∞-categories here are abstract, their constructions delicate. Yet the bridge is built. Where a physicist might model a quantum spacetime as a directed graph of events, a mathematician can now import the full machinery of homotopy theory. Where a computer scientist studies a model of computation as a network of states and transitions, the same topological glasses can be worn. The team is already working on these questions, though a definitive timeline remains open.
Eldridge’s work, spanning BIMSA, the University of Southampton, and Tsinghua University, brings together ideas from directed set theory, category theory, and algebraic topology. The derivation—published as a preprint—establishes that the directed-graph description is complete, that it sees all the spaces classical topology sees. The next question is not whether graphs can encode shape, but what shape itself becomes when viewed through the lens of direction. Perhaps one day, when physicists build models of discrete spacetime, they will use these cubical homotopy tools to compute topological invariants directly from causal sets.
In that sense, this work touches something primordial. We are used to thinking of arrows as instructions: go here, not there. Now those instructions can bloom into spheres, tori, and the vast catalogue of topological forms. It is as if the mathematician has discovered that the ink on the page knows the shape of the object it depicts—not roughly, but precisely, in the language of homotopy. The philosophical implications are profound. For decades, physicists have sought a theory of everything; perhaps the mathematics of directed graphs offers a clue that the “everything” includes not just shapes but the irreversible flow that shapes inhabit.
Yanjiang is an online editor of Loom Science
References
- Briony Eldridge et al., The discrete homotopy hypothesis for directed graphs, arXiv:2605.04959