Mathematicians sometimes talk about structures that feel almost like language forged in geometry: a place where arrows, squares, and the ways they fit together don’t just exist, they cooperate. The latest work by Aaron David Fairbanks and Michael Shulman steps into one of the most stubborn corners of this landscape. It tackles double categories — a field where you have two kinds of one-dimensional arrows, horizontal and vertical, and a two-dimensional world of squares that bind them — and then asks what happens when the rules for composing those arrows aren’t strict, but only hold up to a harmonious orchestra of isomorphisms. In plain English: what if the way you combine things in two directions can bend, twist, and still make sense as long as everything agrees up to a coherent translation? The answer Fairbanks and Shulman propose is not a tweak, but a reimagining, with a new vocabulary that feels both rigorous and surprisingly intuitive.
For readers who follow the frontiers of category theory, this work lands at the intersection of structure and meaning. It’s not just about making a prettier diagram; it’s about asking how complex systems can be built from simpler pieces when the rules themselves are allowed to wobble in a controlled, trackable way. The authors frame their project around the notion of doubly weak double categories, a concept that extends the familiar idea of a bicategory into a two-dimensional twilight zone where every direction can be “weak” in its composition. The project is deeply algebraic, but its punchline matters beyond pure math: if we want to reason about processes that unfold in two (and only two) intertwined directions — think of concurrent computations, layered physical processes, or two-way interactions in social networks — we need a theory that can tolerate gentle nonlinearity without collapsing into chaos. The paper is a rigorous architectural blueprint for that theory, and a reminder that mathematical language evolves to keep pace with the phenomena it aims to describe.
The authors, Aaron David Fairbanks and Michael Shulman, present a program that many researchers in higher category theory have long whispered about: what is the most natural, fully algebraic way to capture a double category whose two directions both admit weak composition? Their answer is twofold: first, they introduce implicit double categories — double computads with all potential compositions on 2-cells but without fixing any composition of 1-cells. Then they impose a representability condition that yields what they call doubly weak double categories. The leap is as conceptually simple as it is technically delicate: instead of trying to force both horizontal and vertical compositions to be strictly associative and unital, they allow these compositions to exist in a web of coherent isomorphisms. The payoff is a flexible, finite framework that still locks together into a coherent whole, with multiple equivalent presentations and finite axiomatizations. The work is rooted in rigorous category-theoretic ideas and leans on centuries of thinking about how to tame infinity with structure. It is a collaboration that sits at the very edge of abstract mathematics, and yet its implicit goal is practical: to give a robust language for compositionality in two interacting dimensions.
Weakness in both directions
The heart of the paper is a puzzle about coherence. In a traditional double category, you have object-level 0-cells, two kinds of 1-cells (horizontal and vertical), and 2-cells shaped like squares that witness how a horizontal and a vertical 1-cell interact. When you demand strict composition in both directions, you land in familiar territory: double categories that mirror strict 2-categories inside their internal structure. But the world we actually want to model is messier. In many natural contexts, the horizontal composition of 1-cells isn’t strictly associative or unital once you leave the clean, small-scale examples. It’s only associative up to coherent isomorphism, and the same holds for vertical composition. The classical approaches to this issue — Verity’s double bicategories and Garner’s cubical bicategories — give powerful tools, but they each leave something behind: one framework may not capture how the horizontal and vertical layers glue together via all the higher shapes you’d like to consider, while the other may lose a handle on how 2-cells inside the horizontal and vertical world relate back to the actual squares that live in the double category. Fairbanks and Shulman want a definition that preserves the best of both worlds without forcing a rigidity that makes natural examples slip away from the theory.
To achieve this, they start from a central idea: implicit structures. A bicategory, for instance, can be viewed as a “weak” version of a strict 2-category where equalities between composites are replaced by isomorphisms, and those isomorphisms satisfy coherence conditions. The authors push this perspective further by showing that a bicategory can be seen as a representable, implicit 2-category — a strict 2-category whose underlying 1-category is freely generated, with composites that are only defined up to isomorphism. The import is subtle but powerful: if you can describe a structure by saying “we have generating 1-cells and 2-cells that can be composed in many ways, but all the ways are identified by coherence,” you’ve packaged the complexity into a neat, algebraic theory. Their insight for doubles mirrors this: one can define a double computad (a “double graph” with both horizontal and vertical 1-cells and 2-cells that border along strings of 1-cells) and then endow it with all the possible ways to compose 2-cells along all boundary shapes, subject to coherence. The catch is that 1-cells themselves should not be obliged to compose strictly; their composites, when they exist, are determined by isomorphisms. The result is a new, doubly weak double category notion that remains anchored to a finite, algebraic presentation.
The novelty isn’t just in loosening the screws. The authors show that their doubly weak double categories are equivalent to several other well-mannered formulations when you impose a “tidiness” condition: you can recover the finite presentation from different perspectives, and you can connect to Garner’s and Verity’s frameworks in precise ways. Tidiness means a particular bijectivity condition around the interaction of 2-cells with identity squares, which makes the whole theory easier to check by hand in concrete examples. It also allows a finite axiomatization that is robust enough to be used in practice, not merely as an abstract existence claim. The upshot is a framework that is both conceptually elegant and practically usable for reasoning about two-directional weak composition in a controlled, structured way.
One of the striking aspects of the paper is its insistence on equivalence rather than strict equality. If you’re trying to model two interacting layers of structure, there will inevitably be moves that are only isomorphisms rather than equalities. By emphasizing representability, coherence, and the ability to pass to strictifications without losing the essence of the original weak structure, Fairbanks and Shulman give us a way to talk about the same mathematical reality from multiple angles — and to translate between them cleanly. This is not just an aesthetic preference: it is a practical toolkit for mathematicians who want to apply double-categorical ideas to areas like topology, algebraic geometry, and theoretical computer science where two directions of composition naturally arise and must be reconciled.
From implicit to computad to tidy
The technical engine behind the program is a chain of equivalent viewpoints that transform a difficult problem into a series of manageable steps. The authors begin with double computads, which are free-building blocks for double categories: a set of 0-cells, horizontal and vertical 1-cells, and a collection of 2-cells with their boundaries. A double computad by itself doesn’t demand any particular composition laws. It is the stage on which the actors (the 1-cells and 2-cells) improvise. From there, they introduce the notion of an implicit double category: a double computad equipped with composition operations on 2-cells — but still no fixed way to compose 1-cells. The moment you demand that every string of compatible 1-cells has a composite — that the compositions exist coherently — you land in what they call a doubly weak double category, as long as you also respect a representability condition. The chain looks like this: double computads → implicit double categories (no 1-cell composition) → implicit double categories that are representable (every string of 1-cells has a composite) → doubly weak double categories (the algebras of a finitary monad on double computads). It’s a tight, well-guided path from raw data to a robust algebraic theory.
One of the elegant moves in the paper is the articulation of two different, but compatible, finite presentations of the same idea. First, the authors show that doubly weak double categories can be described as tidy double bicategories: a double bicategory with a particular square-to-bigon conversion that is bijective on each boundary. In practice, tidy double bicategories are a finite, checkable blueprint: they package the interactions between horizontal and vertical bigons, squares, and their compositions into a finite set of operations and identities. Second, they show an equivalent, monadic presentation that uses a double computad with a finite cadre of shorthands for composing 2-cells shaped like grids. The key payoff is that you can go from a ridiculously large landscape of possible compositions to a well-posed, finite theory that still captures all the essential freedom of weak composition in both directions.
The paper then pulls this thread into the broader ecosystem of higher-category theory. They connect their construction to Verity’s double bicategories and Garner’s cubical bicategories, showing how each framework can be obtained from the same underlying structure by choosing which data to keep and which to forget. In particular, the authors demonstrate that tidiness makes the equivalence precise: tidy doubly weak double categories and tidy double bicategories line up as two faces of the same coin, with a clean passage between the double-graph data and the bicategorical data. They also show that Garner’s cubical bicategories can be recovered as a different presentation, but that the fully general, non-tidy doubly weak double categories carry strictly more data and can fail to be monadic over double graphs. The upshot is a clear map of how these competing approaches relate, and a demonstration that a single, carefully designed framework can unify them under a common umbrella.
Why this matters today
Why should a general audience care about a theory of two-dimensional composition? Because in many real-world and theoretical systems, the process you use to combine things matters as much as the things you’re combining. In computer science, for example, you often have two layers of wiring: a horizontal flow of data through pipelines and a vertical flow of control or metadata that traverses those pipelines. In physics, you might imagine two kinds of propagating processes that intersect and interact: spatial processes and temporal processes, each with its own compositional logic. In such settings, insisting that both directions compose strictly can distort what actually happens in practice. A doubly weak double category gives you a way to model those two dimensions of interaction without forcing a rigid algebraic cage around them. The structure remains strong enough to reason about coherence, but flexible enough to accommodate the natural ambiguities of real systems.
Beyond modeling two-directional processes, this line of work helps sharpen the foundations of category theory itself. The field has spent decades chasing a balance between strict laws and flexible, human-friendly equivalences. The notion of implicit structures, representability, and the monadic viewpoints in Fairbanks and Shulman’s paper offers a template for how to build powerful theories without over-prescribing how every composite must look. The payoff is not simply a new gadget for mathematicians to wave at diagrams; it’s a sturdier, more portable language for reasoning about complex, multi-directional processes. That carries with it potential spillovers into areas like formal verification, where you want to model concurrent operations without getting tangled in a forest of edge cases, or into mathematical physics, where two types of morphisms can represent distinct kinds of symmetries or dualities that still need to talk to one another consistently.
As a result, the work feels both deeply abstract and strikingly timely. It’s a reminder that the most profound advances in mathematics often arrive not by discovering a new object, but by rethinking the rules that govern how objects interact. The idea of weak composition in two directions is as old as the notion of a category itself, but giving it a coherent, finite, and workable algebraic avatar is a leap that could influence how researchers think about composition in two dimensions for years to come. The broader message is hopeful: even in highly abstract landscapes, we can unlock practical formalisms that help us design, analyze, and reason about the intricate dances of interaction that define complex systems—whether they’re built from logic gates, spacetimes, or social networks.
The study is the work of Aaron David Fairbanks and Michael Shulman, two prominent figures in higher category theory, and the preprint is housed in the arXiv repository. While the prompt here doesn’t enumerate a single institutional home, the authors operate within the mathematics community that advances these ideas through university-based research and collaboration.
In the end, the project weaves a story about two kinds of motion: the way horizontal and vertical arrows move, bend, and meet, and the way our explanations of those moves bend and twist in response to new examples. It’s a story about structure that doesn’t demand rigidity, about coherence that doesn’t collapse into chaos, and about a language that can carry us toward a deeper, more flexible understanding of how complex systems truly work when two axes of interaction are present at once. If you’re curious about the way advanced math trains its gaze on the most abstract corners of logic, this is a map worth tracing — not as a final answer, but as a bold invitation to maneuver through a rich terrain where the squares truly do learn to walk.
As with many theoretical breakthroughs, the immediate applications may be conceptual rather than computational. Still, there is a practical deja-vu in the air: a more robust way to talk about multi-directional composition could filter into computer-science formalisms, into the semantics of programming languages that treat processes as two-dimensional flows, and into quantum or topological models where different kinds of morphisms interact in subtle, but crucial, ways. The authors’ framework doesn’t solve every problem in these domains, but it provides a precise, workable language for describing the sorts of interactions that real systems demand when they refuse to be neatly one-directional.
So if you’re listening for the next big tempo in higher category theory, Fairbanks and Shulman’s doubly weak double categories might just be the rhythm you hear: not the loudest beat in the room, but the one that helps the whole structure hold together when two kinds of movement happen at once. It’s a reminder that mathematics, at its best, suggests elegant universes in which complexity does not vanish but coheres — one where squares can be squares and still be true about how they border, bend, and connect the two directions that define their world.
Tags: category theory, double categories, higher mathematics, abstract algebra, foundations
