Two days ago, Dassault Systèmes invited Olivia Caramello to speak on "Toposes and 'bridges' for artificial general intelligence." Not another academic colloquium. A presentation to industrial engineers building simulation systems for aerospace, automotive, and manufacturing. The title mentions AGI explicitly.
The video has 45 views.
Meanwhile, analysts dissect the circular financing between NVIDIA and OpenAI. They're right about the economic fragilities: the $100 billion investment loop, the unsustainable energy consumption, the diminishing returns of hyperscaling. But they're missing the fundamental threat.
Not financial. Not energetic. Mathematical.
The architecture dominating AI today rests on vector spaces and cosine similarity. These aren't engineering limitations that more compute or better data will solve. They're ontological constraints embedded in the formalism itself. And there already exists a mathematical universe that transcends these limitations: Grothendieck toposes.
Caramello speaking at Dassault Systèmes isn't academic pageantry. It's a signal that serious industrial actors understand the next architecture won't emerge from scaling the current paradigm. It will require changing the mathematics.
I. The Circular Economy of Delusion
Let's establish what the lucid commentators already see. In September 2025, NVIDIA announced a $100 billion investment in OpenAI to fund massive data center buildouts. OpenAI will fill those data centers with NVIDIA GPUs. The arrangement was immediately flagged as "circular financing"—NVIDIA essentially funding its own future sales.
The numbers are staggering:
- NVIDIA has invested $53 billion across 170 AI deals between 2020 and 2025
- In 2025 alone: $23.7 billion across 59 deals
- NewStreet Research estimates: for every $10B NVIDIA invests in OpenAI, it will see $35B in GPU purchases—27% of its annual revenue
The CoreWeave Loop
CoreWeave, the cloud infrastructure provider, exemplifies the closed loop:
- NVIDIA owns 7% of CoreWeave (worth ~$3 billion)
- CoreWeave has purchased at least 250,000 NVIDIA GPUs at ~$30,000 each—$7.5 billion flowing back to NVIDIA
- NVIDIA committed $6.3 billion to purchase cloud capacity from CoreWeave that it can't sell elsewhere
- OpenAI has signed $22.4 billion in commitments to CoreWeave for GPU cloud services
The pattern: money rotates through a handful of companies (NVIDIA, OpenAI, Microsoft, Oracle, CoreWeave, AMD) creating the illusion of breakneck growth. Jay Goldberg of Seaport Global Securities compares it to asking your parents to co-sign your mortgage. Stacy Rasgon at Bernstein Research calls it "bubble-like behavior."
The Capital Expenditure Problem
Tech giants will invest $700 billion in AI infrastructure over the next two years—not financial investments that can be canceled with a click, but heavy capital expenditures. Free cash flow at Amazon, Google, Meta, and Microsoft is expected to shrink 43% between Q4 2024 and Q1 2026.
History provides context. During the late-1990s dot-com bubble, Cisco launched extensive vendor-financing schemes allowing customers to buy equipment they couldn't afford. Revenue roundtripping and circular deals exacerbated the crash when it came. Global Crossing went bankrupt. Executives paid large legal settlements.
The S&P 500 sits around 6,688—a 100-fold increase since its March 2009 nadir of 666, while US GDP merely doubled in the same period. The "Magnificent Seven" (Alphabet, Amazon, Apple, Meta, Microsoft, NVIDIA, Tesla) account for 37% of S&P 500 market capitalization—the highest concentration on record.
II. The Open-Source Pincer Movement
While NVIDIA and OpenAI construct their closed ecosystem, China has been building something different.
DeepSeek: The Efficiency Shock
In late December 2024, DeepSeek—a previously unknown startup born from the High-Flyer hedge fund—released DeepSeek-V3, an open-source large language model built in two months for less than $6 million using reduced-capability H800 chips (not the advanced H100s banned by US export controls).
In January 2025, DeepSeek-R1 followed. Third-party benchmarks showed it outperforming OpenAI's o1, Meta's Llama 3.1, Anthropic's Claude Sonnet 3.5, and OpenAI's GPT-4o on complex problem-solving, math, and coding—at 5% of the cost.
The release caused a geopolitical earthquake:
- NVIDIA's stock plummeted
- Analysts began discussing "the slow unwinding of the AI bet"
- Microsoft CEO Satya Nadella called it "super impressive in terms of compute efficiency"
- Perplexity CEO Aravind Srinivas: "Necessity is the mother of invention. Because they had to figure out work-arounds, they actually ended up building something a lot more efficient."
The Broader Chinese Strategy
DeepSeek isn't alone:
- Kai-Fu Lee's startup 01.ai trained its model using only $3 million
- ByteDance (TikTok's parent) released an update claiming to outperform OpenAI's o1
- In June 2025, Baidu announced it would make its Ernie generative AI model open-source—described by one analyst as "throwing a Molotov into the AI world"
- Alibaba cut the cost of its Qwen-VL model by over 85% in late 2024
The Chinese government may be subsidizing these companies to undercut Western competitors. China's vice-premier Zhang Guoqing at the Paris AI Summit positioned this as ensuring AI is not "a game of rich countries"—a strategy to reshape international AI governance while the US fixates on protecting market dominance through proprietary models and export controls.
The pattern is clear: while American companies pour hundreds of billions into circular financing schemes built on expensive infrastructure, Chinese firms achieve comparable or superior performance at a fraction of the cost using open-source models, algorithmic improvements, and hardware efficiencies.
III. The Problem Everyone Misses: Ontological Limits
But here's what both the circular-financing critics and the open-source-efficiency advocates fail to address: the current AI architecture—regardless of whether it's implemented by OpenAI's proprietary models or DeepSeek's open-source alternatives—rests on a mathematical foundation with fundamental limitations.
Syntax Without Semantics
Large language models manipulate tokens in high-dimensional vector spaces. Semantic relationships are encoded as geometric relationships—typically measured by cosine similarity between embedding vectors. This works remarkably well for statistical pattern matching. It produces coherent outputs at the syntactic level.
But it has no semantic grounding.
Vector spaces and cosine similarity cannot represent the invariant structures that enable human understanding. They cannot model the relationships between different knowledge representations. They cannot capture what remains constant when we translate between different languages or different theoretical frameworks.
To put it in the language of category theory: they confuse objects with morphisms.
Vector spaces treat concepts as points in space and relationships as distances or angles between them. But in any sufficiently rich domain, the relationships are primitive—not derived from the objects, but constitutive of them.
This isn't a technical problem that more dimensions or better embeddings will solve. It's an ontological limitation of the formalism itself. You cannot represent semantic structure as vector space geometry any more than you can represent musical structure as temperature measurements. The mathematics is categorically wrong.
The RAG Problem: Knowledge as Fragments
Consider how current systems handle knowledge integration. Retrieval-Augmented Generation (RAG) systems chop documentation into chunks, embed them as vectors, retrieve similar chunks based on cosine similarity, and feed them to LLMs.
But the chunking destroys the logical progression of the original text:
- The "as mentioned above" references break
- Cross-references evaporate
- Context that was implicit in sequential reading becomes lost in fragment retrieval
The problem isn't the chunking strategy. The problem is treating knowledge as a collection of vectors to be retrieved rather than as a structured space of relationships to be queried.
IV. Einstein's Lesson: When Physics Needed New Geometry
Consider the crisis in physics at the turn of the 20th century. Newtonian mechanics worked beautifully for most phenomena. But certain observations—the orbit of Mercury, the behavior of light, the constancy of the speed of light—didn't fit.
Physicists tried to patch the theory. Add epicycles. Adjust constants. Propose ad hoc mechanisms. None of it worked.
Einstein's insight: the problem wasn't with the physics. It was with the geometry.
Newtonian physics assumed Euclidean space—flat, absolute, unchanging. Einstein replaced it with Riemannian geometry—curved spacetime where geometry itself is dynamic.
The mathematics changed. Everything else followed.
AI's Analogous Crisis
AI faces an analogous crisis. The current architecture works beautifully for statistical pattern matching. But certain capabilities—genuine semantic understanding, systematic knowledge integration, reasoning about abstract invariants—don't fit.
Companies try to patch the theory:
- More parameters
- More data
- More compute
- Better prompting
- Fine-tuning
- RAG
None of it addresses the fundamental limitation.
The problem isn't with the engineering. It's with the mathematics.
Current AI assumes vector space geometry—points and distances, embeddings and similarity. What's needed is categorical structure—objects and morphisms, theories and their invariants, toposes and their multiple representations.
Change the mathematics. Everything else follows.
V. Grothendieck's Mathematical Universe
In the early 1960s, Alexander Grothendieck introduced the concept of topos—mathematical structures that support an infinite number of invariants. These invariants represent different points of view on the same underlying reality, enabling what Grothendieck called "vision": the integration of all different perspectives, languages, and knowledge representations that we can have on a given piece of reality.
Not Just Another Category
A topos is not merely a category with additional structure. It's a mathematical universe in which one can do essentially all of mathematics—but in a way that makes explicit the relationships between different ways of describing the same phenomena.
The crucial property: toposes admit multiple representations. The same topos can be constructed from different "sites"—different collections of objects and relationships. This multiplicity of representations is not a bug; it's the essential feature that enables toposes to function as "bridges" between different mathematical theories.
Caramello's Bridges
Olivia Caramello has spent two decades developing the systematic methodology for exploiting this property. Her "theory of topos-theoretic bridges" consists of methods and techniques for transferring information between distinct mathematical theories by using toposes as universal translators.
The process works like this:
- Two theories T and T' may appear completely different—written in different languages, axiomatizing different kinds of structures, living in different domains of mathematics
- But if they're "Morita-equivalent" (sharing the same classifying topos), then that topos acts as a bridge
- Any topos-theoretic invariant manifests itself differently in the context of each theory, creating a systematic correspondence between seemingly unrelated properties
This isn't metaphor. It's a precise mathematical technique with computational implementations. Caramello has used it to establish dualities between algebraic and topological theories, to prove completeness theorems for various logics, to construct new spectra for mathematical structures.
VI. Why This Matters for AI Architecture
When Caramello speaks about AGI at Dassault Systèmes, she's not engaging in speculative philosophy. She's describing a concrete mathematical architecture that addresses the fundamental limitations of vector space AI.
In her presentation, she emphasizes three key points relevant to artificial intelligence:
1. Semantic Invariance
Current AI systems have syntax without semantics. They manipulate token sequences according to statistical patterns without accessing what those patterns represent.
Toposes provide a mathematical framework for extracting and encoding invariant semantic structures—the aspects of meaning that remain constant across different representations, different languages, different contexts.
2. Knowledge Integration
When you build the classifying topos of a theory, you're performing a completion process analogous to how the human brain extracts meaning from partial information.
The topos makes explicit everything implicit in the theory. It adds all the "imaginaries" that are implied but not stated. This completion is what enables different knowledge representations to be systematically related.
3. Hierarchical Learning
Caramello's work on relative toposes—toposes defined over other toposes—provides a formal framework for the kind of hierarchical learning that characterizes human intelligence:
- Learning a language: first the alphabet (level zero), then vocabulary, then grammar, then prose, then style
- Learning chess: first the rules, then tactics, then strategy, then personal style
- Learning mathematics: first propositional logic (order zero), then first-order theories, then higher-order logics
Current AI systems flatten this hierarchy. Everything happens in the same vector space. There's no formal distinction between levels of abstraction.
Relative topos theory provides the mathematical structure for modeling learning that unfolds across different levels—where each level is formalized as a topos defined over the previous one.
VII. The Computational Reality
The natural objection: this sounds impossibly abstract. Can it actually be implemented? Is it computationally tractable?
The honest answer: we don't know yet.
Category theory and topos theory have developed substantial computational machinery. The "bridge technique" can be mechanized for well-defined scenarios. Classifying toposes can be constructed algorithmically for geometric theories.
But translating this into performant AI systems remains an open research problem.
Current Exploratory Directions
There are exploratory directions:
- Vector Symbolic Architectures that use high-dimensional vectors to encode symbolic structure
- Knowledge graphs built on neo4j that attempt to preserve relational semantics
- Experiments with categorical constraints on neural architectures
None of this is production-ready. None of this has proven scalability. The researchers working on these approaches (and I count myself among them) are engaged in speculative engineering. We're betting that topos-theoretic principles can be computationally realized, but we haven't proven it.
What We Can Say
What we can say: the current architecture demonstrably fails at:
- Semantic grounding
- Knowledge integration across representations
- Hierarchical metalearning
The mathematical framework exists that could, in principle, address these failures. Whether it can be implemented efficiently is the empirical question.
This uncertainty doesn't weaken the argument. It strengthens it. Because the strategic question isn't "will topos-theoretic AI definitely work?" It's "what happens when someone figures out how to make it work while everyone else has $700 billion sunk into vector space infrastructure?"
VIII. The Real Moat
Here's why this matters strategically. The current AI race assumes the moat is proprietary: proprietary models, proprietary data, proprietary compute infrastructure. NVIDIA and OpenAI build their circular financing on this assumption. China's open-source strategy attacks it directly by making equivalent capabilities freely available.
But both strategies miss the deeper point.
The real moat isn't proprietary versus open-source. It's mathematical.
Whoever first implements an AI architecture based on topos-theoretic principles—with genuine semantic grounding, systematic knowledge integration, and hierarchical metalearning—will have an advantage that no amount of scaling or efficiency optimization can overcome.
Because they'll be working in a richer mathematical universe. One where:
- Invariants can be represented explicitly
- Different knowledge representations can be systematically related
- Learning can accumulate across levels of abstraction instead of being flattened into a single vector space
The circular financing between NVIDIA and OpenAI will crash. It has to—the mathematics don't work; the energy doesn't scale; the revenues can't materialize fast enough. The Chinese open-source efficiency will continue pressuring costs downward, commoditizing vector space LLMs.
But neither of these dynamics addresses the fundamental limitation. Both are fighting over who can best exploit a mathematical framework that's categorically insufficient for the task.
IX. What Comes Next
Caramello's appearance at Dassault Systèmes signals that sophisticated industrial actors are beginning to understand this. They're investing in simulation systems, digital twins, model-based engineering—domains where the relationship between different representations is central. Where the invariants matter more than the particular encoding.
The Immediate Practical Steps
The immediate practical steps are clear, if not easy:
1. Build hybrid architectures. Current vector space LLMs aren't going away tomorrow. But they can be augmented with categorical/topos-theoretic structures that provide semantic constraints. Knowledge graphs that aren't just vector stores but genuine relational databases. Reasoning systems that operate on logical structure, not statistical similarity.
2. Make the implicit explicit. As Caramello emphasizes in her talk: "the aim is to go from the implicit to the explicit." Current systems accumulate massive amounts of implicit information through statistical analysis. The next generation must be able to articulate that information symbolically, using formal vocabularies and logical systems.
3. Implement hierarchical learning. Stop flattening everything into a single vector space. Build systems that learn at different levels of abstraction, where higher levels are formally defined over lower levels. Where the alphabet is distinguished from vocabulary, vocabulary from grammar, grammar from style.
4. Focus on invariants. Instead of optimizing for performance on particular benchmarks with particular encodings, optimize for extracting invariant structures that remain constant across different representations. This requires different evaluation criteria, different training objectives, different architectural principles.
Who's Already Working on This
The researchers who understand this aren't waiting for NVIDIA's next chip or OpenAI's next model. They're studying category theory, topos theory, knowledge representation. They're reading Grothendieck, not just transformer papers. They're building systems that manipulate structured relationships, not just vectors.
Because they understand that the current AI boom—whether it's the proprietary hyperscaling of American giants or the open-source efficiency of Chinese competitors—is fighting over who gets to be the best at doing something that's categorically insufficient.
The mathematical threat isn't coming. It's already here. It's just that most people are still looking at the wrong kind of mathematics.
X. Coda: Philosophy is All You Need
The irony is that this isn't even new. Grothendieck developed topos theory in the 1960s. Category theory emerged in the 1940s. The philosophical insight that relationships are primitive—not derived from objects but constitutive of them—dates back at least to Leibniz.
What's new is the computational scale. We now have the hardware to implement these abstract structures. We have the data to test them. We have the practical problems (language understanding, knowledge integration, common-sense reasoning) that vector space AI demonstrably cannot solve.
And we have, for the first time, serious industrial interest in mathematical frameworks that transcend the limitations of statistical learning in high-dimensional vector spaces.
Olivia Caramello at Dassault Systèmes—watched by 45 people—isn't the starting gun. It's a signal that the race has already begun.
While everyone else debates circular financing and open-source efficiency, a small number of researchers and engineers are building the mathematical architecture for what comes after.
When the current boom crashes—and it will crash, for all the economic reasons the lucid commentators correctly identify—the question won't be who survives the crash. It will be who already built the alternative.
Philosophy is all you need.
References
Key Sources on Circular Financing
- Fortune (September 2025): "Nvidia's $100 billion investment in OpenAI has analysts asking about 'circular financing'"
- Bloomberg (October 2025): "OpenAI's Nvidia, AMD Deals Boost $1 Trillion AI Boom With Circular Deals"
- The Register (November 2025): "Nvidia, OpenAI, and the trillion-dollar loop"
- Yahoo Finance (November 2025): "Nvidia's $24B AI deal blitz has Wall Street asking questions about 'murky' circular investments"
Key Sources on DeepSeek and Chinese AI
- Foreign Affairs (March 2025): "The Real Threat of Chinese AI"
- Carnegie Endowment (July 2025): "China's AI Policy at the Crossroads: Balancing Development and Control in the DeepSeek Era"
- CNBC (January 2025): "How China's new AI model DeepSeek is threatening U.S. dominance"
- The Conversation (February 2025): "DeepSeek: how China's embrace of open-source AI caused a geopolitical earthquake"
Topos Theory and Categorical Foundations
- Olivia Caramello, Theories, Sites, Toposes, Oxford University Press, 2018
- Olivia Caramello, "The unification of Mathematics via Topos Theory", arXiv:1006.3930, 2010
- Alexander Grothendieck, Récoltes et Semailles (autobiographical reflections)
- Saunders Mac Lane & Ieke Moerdijk, Sheaves in Geometry and Logic, Springer, 1992
Video Source
- Olivia Caramello, "Toposes and 'bridges' for artificial general intelligence", Dassault Systèmes, November 19, 2025 (45 views at time of writing)
