My undergraduate thesis on category theory, physics, and the relations between the two.

• V1: Categorical Structures in Physics. 164 pages. PDF here.
• V2: Elements of Categorical Meta-Physics. 232 pages. PDF here.

This page contains a description of how each version came to be and what it covers, as well as a list of contents of Elements of Categorical Meta-physics.

Categorical Structures in Physics

I'm not great at math, but I'm good enough to breeze through an undergraduate degree without really studying or going to class. By the end of my third year (2020) I had nearly enough credit for the major, along with minors in physics and CS, but a few back-to-back course sequences held me in place for a fourth year. Since I could ace them in my sleep, I had lots of free time, which I decided to burn writing a senior thesis on the topics that I was most interested in: category theory and mathematical physics.

To be able to do it as part of an official program, with an actual thesis advisor, I had to pretend that there was a particular thing I wanted to show — some spiel like "the burgeoning language of higher category theory offers the potential to provide a more rigorous foundation for fundamental physics" or whatever — but my goal was to use this as an opportunity to spend a year trying to learn about said topics, the expected thesis itself offering the chance to create a long-term plan and the motivation to go through with it. When I want to learn something thoroughly, I find it best to try and explain it thoroughly, so what I planned to write was more like a massive set of disparate notes than an essay organized around a unifying principle.

And so I started writing. When I get really into something, I can pour effort into it indefinitely—so I spent months at a time just reading and learning and writing. When the deadline approached, around two thirds of the way into the academic year, submission seemed more like an interruption than a conclusion; I hastily finalized some parts that vaguely substantiated the thesis, throwing them together into an auxiliary document. This formed the first version of the thesis, titled "Categorical Structures in Physics". It covers three modern, high-tech applications of category theory to physics:

1. Synthetic differential geometry: Physical reasoning fundamentally relies on concepts of smoothness and infinitesimality, but the modern mathematical expressions of these concepts (topological manifolds with smooth atlases and co/tangent vectors) are unrepresentative of how we "actually" reason about them. Some mathematical objects, such as the ring of "dual numbers", $\mathbb R[\varepsilon]/(\varepsilon^2)$, do allow us to coherently speak of a number $\varepsilon$ small enough that $\varepsilon^2=0$, but they don't play along with other fundamental concepts like integration. Synthetic differential geometry is a categorical framework that allows us to introduce the notions of smoothness and infinitesimality at the level of basic logic, by constructing a topos that behaves as though it were a universe of smooth objects, and then formulating physics in its internal logic.
2. Topos quantum theory: A separate attempt to apply topos theory to physics. While the measurement of classical observables follows an essentially Aristotelian logic, the measurement of quantum observables does not—but one can formulate an internally consistent 'quantum logic' which quantum measurement does obey. If we can build a topos whose internal logic resembles this quantum logic, and then rewrite the $\mathsf{Set}$-based formalization of classical mechanics in this topos, we might be able to reformulate quantum mechanics in a consistent and parsimonious manner dependent only on our formulation of quantum logic.
3. Topological quantum field theory: Any particular quantum field theory has a pseudo-Riemannian manifold $(M, g_{\mu\nu})$ it's defined on, and a Lagrangian density ${\cal L}$ over this manifold which upon canonical quantization (or path integration) yields the dynamics of the theory. If ${\cal L}$ is independent of the metric $g_{\mu\nu}$, we call the theory a topological quantum field theory (TQFT). While TQFTs have no real dynamics of their own (their Hamiltonians vanish), they're often combined with to non-topological QFTs in order to couple the local evolution of said QFTs to the global topology of the manifold. It turns out—as first realized by Michael Atiyah—that a large class of TQFTs we'd like to understand are no more than particularly nice functors from the category of manifolds and cobordisms to the category of vector spaces and linear operators. This result doesn't seem to generalize in any sensible way until one looks at higher categories: the cobordism hypothesis of Baez and Dolan states that extended TQFTs are again just functors, but this time from a symmetric monoidal $(\infty, n)$-category of cobordisms. The problem is that these things are ridiculously tricky to define and work with, but, as is common in category theory, figuring out the right definition is three quarters of the battle—so it's hoped that we'll gain insight into physical theories by thinking about how they have to be structured from the viewpoint of higher category theory.

Elements of Categorical Meta-physics

With that, the nominal goal was accomplished. But my workflow operates on momentum more than anything; with several months of it behind me and so much undone in front of me, I could only keep going. At some point near the end of the year, though, my research led me to snap on to an actual narrative on the relation between category theory and physics.

In both mathematics and physics, theories have a strange kind of multiple realizability: many different structures end up giving you the same observed dynamics, often when you least expect it. Replace the imaginary time $it/\hbar$ in a quantum-mechanical path integral with thermodynamic beta $\beta=1/k_BT$, and you're suddenly doing classical statistical mechanics; gauge theory and bundle theory correspond closely enough to merit a dictionaryIn a quote from that page, the physicist C. N. Yang discusses this correspondence with the mathematician Shiing-Shen Chern: "I said I found it amazing that gauge fields are exactly connections on fiber bundles, which the mathematicians developed without reference to the physical world. I added 'this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere.' He immediately protested, 'No, no. These concepts were not dreamed up. They were natural and real.'". There are so many mathematical models that reproduce general relativity or the Standard Model. This suggests to me that something more fundamental remains to be uncovered: a meta-physical structure of physical structure. What I had found about the applications of category theory to physics simpliciter was unfortunately hopeless—over-complex models that'll never go anywhere, curiosities that just fall out of larger non-categorical frameworks, and other depressing trivialities—but here was a way to really dive deeply into the underlying principles.

Thus, a rebranding—not Categorical Physics but Categorical Meta-Physics, notes in an attempt to see how the the basic relational structures underlying physics could be captured by categorical notions. This was doomed to fail, since I hadn't yet pieced together what such "basic relational structures" might look like, how I might distinguish them from whatever internal schemas I've learned for organizing conceptual content, or whether there's any a priori reason to expect category theory in particular to (be able to) get the leading role. (It was only a year later that I'd piece together the conceptive framework through which such questions could be properly asked and answered, coincidentally in another massive document abbreviated ECM). Thus, I kinda had to just keep learning, which I did until summer began and preparing for grad school became a priority.

What follows is a list of topics covered in ECM. Those in gray are merely skeletons in the PDF above (many of them are written in scattered documents I'm not going to go through the trouble of proofing and integrating).

Contents

1. Category theory
1. Categories: definitions, kinds of morphisms, natural transformations, faithful/full functors, free/forgetful functors, co/limits and universal properties, functoriality, equivalence of categories, presheaves, Yoneda's lemma, adjunctions, Heyting algebras, co/units, monads, algebras.
2. Monoidal categories: general theory of structures on categories, kinds of products on categories, tensor bifunctors, monoidal functors, polygon identities, (cartesian) closed categories, categorical dependent products and sums, enriched categories, category theory over a base monoidal category, basic operations in 2-category theory, 2-functors, 2-natural transformations, modifications, dinatural transformations and wedges, co/ends, Kan extensions and lifts, accessibility and presentability, set theoretic issues.
2. Topos Theory
1. ToposesThe thesis uses "topoi" as the plural ('topos' is Greek for 'place', and Greek pluralizes with -i) because it sounded nicer to me back then, but nowadays I intentionally try to predictably use ordinary English grammar whenever I feel I have a choice (as with 'toposes' but not with 'desiderata').: categories of sheaves, global section and constant sheaf functors, Grothendieck toposes, geometric morphisms, elementary toposes, logical structures in a topos, power objects and transpositions, co/diagonals, equality maps, images, the Mitchell-Benabou language of a topos, geometric first-order formulas, Kripke-Joyal semantics, theories of objects in toposes, Weil algebras.
2. Classifying toposes: flat functors, theories, kinds of logical formulas, classifying toposes, the theory of objects.
3. The Language of Higher Categories
1. Simplices: the simplex category, face/degeneracy maps, other shapes, siimplicial sets, boundaries/horns, simplicial maps and functors, homotopy categories, (co)skeleta, barycentric subdivision, Kan fibrant replacement.
2. Model category theory: homotopical categories, model structures, weak equivalences, homotopy categories, lifting problems, saturated, lifting properties, factorization systems, model categories, (co)fibrations, modle structures, Quillen adjunctions and equivalences.
3. Simplicial objects: cosmoses, simplicially enriched categories, homotopy nerves, model structures for simplicial categories, enriched model categories, model pushouts and pullbacks, (co)tensors, two-variable adjunctions, quasi-categories, Kan complexes, functors on simplicial sets, lists of model categories.
4. Models of higher category theory: $\infty$-cosmoses, isofibrations, $\infty$-categories, $\infty$-functors, cosmological biequivalence, homotopy 2-categories, limits (in $\infty$-categories), stable $\infty$-categories, loop space and suspension objects, prespectrum objects, ${\mathbb E}_\infty$ rings.
5. $\infty$-toposes: Grothendieck-Rezk-Lurie definition, the $\infty$-topos of $\infty$-groupoids as higher version of ${\mathsf{Set}}$, the homotopy hypothesis, $n$-truncation, connection, locality, and discreteness in $\infty$-toposes, cohesion, shape, flat, and sharp modalities, elasticity, (co)reduction, solidity.
4. Homotopy type theory
1. Type theory:
2. Homotopy type theory
5. Modality
1. Modal logic: idea, classical logic, modal logic, modal axioms
2. $\infty$-monads: 2-categorical adjunctions, walking adjunctions, homotopy coherent adjunctions, homotopy coherent monads
3. Modal homotopy types:
6. Synthetic Differential Geometry
1. Infinitesimals: the Kock-Lawvere axiom, undecidability, the synthetic Taylor's theorem, spectra of $R$-algebras, formal infinitesimals, smooth toposes, differentiation, Euclidean $R$-modules, microlinear spaces, (synthetic) Lie groups, tangent vectors, exterior derivatives, formal manifolds, smooth algebras, finitely generatd and presented ideals, smooth reals, the integration axiom, topos models of SDG, field axioms
2. Physical models: connections and curvature, parallel transport, Riemann curvature tensors and Christoffel symbols, the synthetic Einstein equations, closed and exact differential forms, presymplectic smooth sets, symplectomorphisms, correspondences, equivalences, smooth groupoids, smooth homotopy quotients, microlinear Lie groups, synthetic quantum mechanics
7. Cohesive Toposes
1. Diffeologies: smooth sets, diffeological spaces
8. Noncommutative Geometry
9. Structured Spaces
1. Scheme theory: ring spectra, local rings, residue fields, Zariski topologies, structure sheaves, pushforwards, affine and ordinary schemes, properties of schemes
3. Fractured $\infty$-Toposes
4. Pregeometry
5. Derived Algebraic Geometry
10. The Phenomenology of Space
1. Transcendental Aesthetic
2. Relativity
3. Compactification
4. Emergent Spacetime
5. Isbell Duality
11. All About $\otimes$
1. String Diagrams
2. Hilbert Spaces
3. States and Measurements
4. Categorical Quantum Mechanics
12. Ontology in Topoi
1. Topos Quantum Theory
2. Behavior Types
13. Process Theories
1. Graphical Calculi
2. Lenses
3. Profunctor Optics
4. The Topos of Dynamical Systems

1. Homotopy and Cohomology
1. Homotopy theory: homotopy equivalence, pointed spaces, wedge products, categories of topological spaces, loop/path spaces, reduced supensions/cylinders, composite loops, fundamental groups, weak equivalences, CW complexes, graphs
2. Homological algebra: additive categories, biproducts, zero objects/morphisms, (co)kernels, (co)images, abelian categories, (co)chain complexes, (co)homology, boundaries, cycles, morphisms and categories of chain complexes, quasi-isomorphism, bounding, split/exact chain complexes, null-homotopies, chain homotopy, left/right exact functors, projective/injective objects, left/right/projective/injective resolutions, left/right derived functors, $\operatorname{Tor}$ and $\operatorname{Ext}$, extension, splitting
3. Cohomology Theories: generalized (co)homology theories, spectra, the suspension and sphere spectra, the stable homotopy category, stable cohomotopy, higher algebra, singular cohomology, coefficients, the universal coefficient theorem, Eilenberg-MacLane spaces, Lie algebras, commutators, Lie algebra homomorphisms, adjoint representations, ideals, modules, tensor algebras, Clifford algebras, universal enveloping algebra, the Poincare-Birkhoff-Witt theorem, (co)invariant submodules, Lie algebra (co)homology, the Chevalley-Eilenberg complex, augmentation maps and ideals, derivations, Hochschild cohomology, Kahler differentials
4. K-Theory: Principal bundles, local trivializations, pullback bundles, classifying spaces, delooping, vector bundles, line bundles, Chern and Stiefel-Whitney classes
2. Some Physics
1. Banach Spaces: norms, operators, bounded operators, operator norms, $L^p$ spaces, $\ell^p$ spaces, inner products, polarization and parallelogram laws, orthogonality, Schauder bases, Banach spaces, completions, Banach algebras, duals, contracting maps, fixed points, the contraction mapping principle, Hilbert spaces, separability, the Riesz representation theorem, the Gelfand-Naimark theorem, Hermitian operators, commutators, positive and projection operators, meets and joins, diagonalizability, trace classes, trace norms, states, bra-ket notation, unitary groups, the time-independent Schrodinger equation, von Neumann algebras, commutants, factors, outer and inner supports, dimension functions, minimal and finite projections, normal states, density operators, Genfand spectra and transforms, measure theory, quantum probability spaces
2. Quantum mechanics: Lagrangians, functional derivatives, action, the Euler-Lagrange equations, inertial frames, Galilean relativity, Hamiltonians, measurements, commutators, compatibility, dispersion, translation operators, uncertainty principles, time evolution, the Schrodinger and Heisenberg principles
3. Relativity: intervals, space/light/timelike intervals, proper time, Lorentz transformations, four-vectors, the equivalence principle, pseudo-Riemannian manifold theory, the Einstein tensor, the stress-energy tensor, the Einstein field equations, the Einstein-Hilbert action.
3. Quantum Field Theory: The Poincare and Lorentz groups, Lie group and algebra representations, universal covering groups, spin, gamma matrices, chiral representations, left-/right-handed and Dirac spinors,the spin and pin groups, frames, spin structures and manifolds, spin representations, spinor bundles and fields, the Klein-Gordon equation, electromagnetic tensor, vacuum states, creation and annihilation operators, correlation functions, Green's functions, operator orderings, Feynman and free propagators, perturbation expansions, the $S$-matrix, the Dirac equation, chirality operators, the Weyl equation, covariant derivatives and curvature, gauge fields, quantum electrodynamics.