Infinity behaves like a stubborn giant in the math room: it refuses to stay neatly finite, yet it loves to reveal patterns when you tilt your perspective just right. Devon Stockall, a researcher at the Centre for Quantum Mathematics at the University of Southern Denmark, has written a paper that tries to tame that giant by giving it a language. The work sits at the intersection of higher category theory and theoretical physics, a place where condensation is not a weather event but a precise mathematical operation that compresses complexity into something tractable. It is not about empty abstractions, it is about building tools that could one day describe the kind of symmetries that govern quantum matter and the information we can extract from it.
Condensation in math is a bit like reducing a messy tapestry to a single elegant motif. You start with a symmetry object living in an enriched infinity category and you form a new object that remembers the old one but behaves in a simpler way. The math is gentle but strict, and Stockall shows how to carry that procedure across a larger universe than the usual fusion categories by using enriched infinity categories. The door opens to modeling not just finite pockets of symmetry but continuous and derived structures that come up in physics.
Condensation here is not a lab trick but a mathematical language for turning messy symmetries into something you can study. A bold claim in the paper is that this condensation can be iterated, yielding a tower of descriptions that keep the essential data while peeling away the rest. The result is a framework that can handle symmetries of any dimension and codimension as long as the ambient world has the right kind of colimits. Stockall also introduces a trimmed version that mirrors how physicists sometimes prune details to see the forest rather than the trees. The upshot is a robust method for reshaping symmetries across layers of abstraction without losing the thread that ties them together.
A language for hidden symmetries across dimensions
In everyday math a category tracks objects and arrows between them. An infinity category adds layer after layer of arrows between arrows. Enriching these structures over a monoidal category V gives us a way to measure the arrows themselves inside V, and to compose with the flavor of V. Stockall writes about monads and algebras inside this enriched setting, and asks what it means to condense those structures when the ambient world is large and flexible.
One keystone is the Eilenberg-Moore object, a universal way to talk about modules over a monad. A monad is a rule that takes an object and returns another object along with a way to compose those rules. The EM object is the best universal home for all modules over that rule, and the paper builds two monoidal functors that take a monad to its module category. This is the algebraic backbone that makes condensation meaningful in this setting.
Condensation, in this enriched universe, becomes a precise operation you can perform again and again. You start with a symmetry object living in an enriched infinity category and you form a new object that encodes the action of that symmetry. The method works for all enriched monoidal infinity categories that have the right colimits, not just the small and friendly ones. There is also a truncated variant that mirrors how physicists sometimes dump tiny details to get the big picture. Taken together, these moves give a layered language for rearranging symmetries across multiple dimensions without losing track of where they came from.
To anchor the theory, Stockall develops the enriched Day convolution a tool that lets you carry monoidal structure through a panorama of enriched categories. This ensures that as you condense a monad into a module category and then pass to a new level of enrichment, the combinatorics stay aligned. The result is a scalable recipe for modeling how symmetries behave when you push them through several layers of abstraction.
In practical terms the framework operates in a universe where you can talk about monads in monoidal (infinity) categories, Eilenberg-Moore objects, and modules in an enriched setting. It is a language that can describe not just finite, rigid structures but continuous and derived ones that real physics often demands. The author shows how these ideas can be organized into an iterative condensation process that respects the enrichment at every step.
Why this matters for physics and computation
Condensation in physics is a familiar chorus from the lab bench: when a crowd of particles cools into a single quantum state, the whole ensemble moves as one. In the topological world of modular tensor categories and fusion categories, mathematicians have a clean version of that chorus for a particular finite class of objects. Stockall extends the chorus beyond finite frames to the grand stage of enriched infinity categories, so the same melody can describe continuous, derived, or non semisimple systems. The practical payoff is a language that could capture the complexity of quantum matter more faithfully than older models.
This matters because not all symmetries fit into the neat boxes of invertible transformations. In higher dimensional quantum field theories, symmetries appear as higher dimensional objects and can be non invertible. The paper shows you can condense such symmetries in a controlled way, and carry the same ideas to derivative and non semisimple contexts. The approach provides a unified framework to discuss when you gauge a symmetry or when you pass to a dual description by condensing it away. This matters for constructing consistent theories that describe real materials or exotic quantum phases.
There is a gentle but powerful implication for quantum information The idea of iterated condensation means you can imagine building up a hierarchy of effective theories where each level remembers the previous one. Such a perspective could help in designing fault tolerant schemes that rely on robust, higher dimensional symmetries. The concept of centers and centralizers is not just mathematical garnish; it encodes what survives after you impose a symmetry and what can still be controlled or measured.
There is also a practical thread for computation. The Day convolution and the friendly language of enrichment provide a stable platform for manipulating highly structured algebraic data. In time, these ideas could seed software tools that help researchers model and test complex quantum systems. It is early days, but the promise is a more flexible toolkit for reasoning about quantum matter and the information it carries.
In this world the author hints at a future where the theory could intersect with vertex operator algebras and conformal field theories, hinting at bridges between two dimensional quantum field theories and higher dimensional analogues. The cross talk between these areas is where new physics often hides, and Stockall helps sketch the path forward.
The study sits squarely in the mathematical physics ecosystem, anchored in rigorous category theory but speaking with a physicist eye for phases and dualities. The author, Devon Stockall, writes from the Centre for Quantum Mathematics at the University of Southern Denmark, and the project is supported by the VILLUM FONDEN grants that fund his group. It is a reminder that deep math often grows in the soil where math and physics cohabit, where abstractions and experiments learn to talk the same language.
What’s surprising and where it could go next
Perhaps the most striking move is the scale. The condensation theory is not limited to tiny categories but can handle large condensates in the enriched setting. Yet the author shows that dualizable condensates behave nicely and yield dualities that mirror the original symmetry. That means you can condense and later decondense without losing the sense that you started with a real symmetry. It is a kind of mathematical reversibility that feels almost physical.
The paper also makes a precise link between monads and their Eilenberg-Moore objects and how these sit inside enriched categories. The upshot is a pair of functors that translate monads into the module categories that remember them. Those functors are monoidal in the enriched sense, which means the translation respects how pieces are glued together. In turn you get a robust way to talk about relative tensor products of modules and about internal centers — objects that capture the essence of symmetry inside a given theory. The technical machinery behind this is dense, but the narrative payoff is clear a principled way to manipulate symmetries that are ubiquitous in physics but stubborn to formalize.
Another elegant thread is truncated condensation. In physics you often care about the big picture the fully dualizable robust description without getting bogged down in every minute detail. The truncated version is a way to capture the right kind of dualizability at a coarser scale. The author shows that every large condensate carries a natural truncated condensation, a structure that is enough to guarantee dualizable objects under the right assumptions. That result is more than a technical nicety; it is a hint that the whole edifice could be used to build physically meaningful models that survive certain idealizations or approximations, which matters when bridging theory and experiment.
The work also girds itself with the modern folklore in physics about Morita equivalence. In the finite setup, condensates tied to dualizable modules end up Morita equivalent to their starting point. Stockall extends this idea to the enriched world, giving a robust sense in which different condensates describe the same physics. The upshot is a duality passport that lets theorists move between algebraic languages without losing physical content. This is the kind of flexibility that could matter when you are classifying phases of matter or designing quantum devices that rely on symmetry in clever ways.
Finally Stockall points toward a future where condensation could describe continuous gauging of symmetries and even connect with the algebraic structure of vertex operator algebras and conformal field theories. The idea that you could scale condensation to interact with smooth geometry or with physically realistic models is tantalizing. It is a reminder that the frontier of mathematics and physics often travels through shared concepts and that enriched infinity categories may be one of the most adaptable bridges yet built for that climb.
In summary Stockall shows that condensation theory can be pushed beyond the tidy corner of fusion categories into a broader and more flexible universe. The move matters not just for elegance but for the kinds of questions physics keeps throwing at us: how do complex symmetries organize themselves, how can we relate different theories across scales, and how might we leverage these ideas to better understand topological phases and quantum information. The answer is not a single theorem but a new language that promises to make the messy, multi layered reality a little more navigable.
Lead author Devon Stockall of the Centre for Quantum Mathematics at the University of Southern Denmark offers a bold, forward looking framework. If his program holds up under concrete examples and broader collaboration, we may be looking at a toolset that helps translate some of the universe’s most intricate symmetries into something that physicists and computer scientists can actually work with. The road from abstract infinity to tangible insight is long, but this paper hands us a map and a compass for the climb.
