Category Theory in Physics  

In General > s.a. functors; logic; Structural Realism.
* Idea (Coecke): A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof.
* Examples of applications: Unitary representations of the Lorentz group [@ Crane gq/00]; > s.a. Topos Theory.
@ Reviews and introductions: Moore IJTP(98); Coecke in(06)-a0808, Coecke & Paquette a0905; Baez & Lauda in(11)-a0908 [n-categories]; Beer et al a1811 [applications in physics, anyons].
@ General references: Thomas m.AG/00-proc; Marcinek m.QA/00 [particle interactions]; Oeckl JGP(03)ht/01 [generalized lattice gauge theory]; Coecke & Lal FP(12)-a1107 [causal structures and symmetric monoidal categories]; Lal & Teh a1404 [and physical structuralism]; Amorim & Ben-Bassat ATMP(17)-a1601 [2 -category of Lagrangians]; Tull a1602 [operational theories of physics]; Veilahti a1712 [higher theory].
@ Stacks: Sharpe ht/06-proc [and derived categories]; Benini et al CMP(18)-a1704 [stack of Yang-Mills fields on Lorentzian manifolds]; Berktav a1907 [stacky formulation of general relativity, stack of Ricci-flat 3D Lorentzian metrics].

In Gravitation
@ Classical gravity: Morava mp/04-conf [2-categories and topological gravity]; Zuo a2007 [general relativity with a cosmological constant].
@ Quantum gravity / spacetime structure: Miković & Vojinović CQG(12) [and Poincaré 2-group]; & Ko Sanders; > s.a. modified approaches to quantum gravity; quantum spacetime models.

In Other Field Theories > s.a. fiber bundles [natural bundles].
@ References: Weatherall a1505-in [rev, theoretical structure and theoretical equivalence]; Scholz a1607 [Weyl and automorphisms]; Tachikawa a1712-proc.
> Specific proposals: see theories beyond the standard model; types of gauge theories and yang-mills theories [based on Lie 2-groups].

In Quantum Theory > s.a. generalized quantum mechanics; Topos Theory [Butterfield-Isham-Döring].
@ General references: Schlesinger JMP(99); Abramsky & Coecke a0808-ch; Harding IJTP(09); Filk & von Müller AdP(10)-a0907 [framework]; Coecke & Perdrix a1004-proc [environment and classical channels]; Bergholm & Biamonte JPA(11)-a1010 [and quantum information science]; Abramsky & Heunen a1011 [H*-algebras and non-unital Frobenius algebras]; Lehmann a1012; Heunen FP(12) [complementarity]; Gogioso & Zeng a1501/ACS [representation theory]; Gogioso a1501, Bolotin a1502 [categorical semantics]; Coecke & Kissinger a1510, a1605 [overview, 2/3]; Gogioso & Genovese EPTCS(18)-a1703 [quantum field theory]; Tull a1902-PhD [general operational theories]; de Ronde & Massri a2002 [pure and mixed states]; Gogioso a1905 [diagrammatic approach]; Heunen & Vicary 19.
@ Categories of relations: Heunen & Tull EPTCS(15)-a1506 [as models]; Mehta & Zhang LMP(20)-a1907 [Frobenius objects].
@ Categorical quantum mechanics: Abramsky & Heunen a1206 [and operational theories]; Gogioso & Genovese EPTCS(17)-a1605 [infinite-dimensional], EPTCS(18)-a1703 [*Hilb]; Gogioso a1709-PhD [dynamics]; Gogioso & Genovese a1805 [and quantum field theory].
@ n-categories: Kapustin a1004-proc [and topological field theory]; Vicary a1207 [2-categorical formalism for classical information, quantum systems, and their interactions]; Benini et al a2003 [2-categorical algebraic quantum field theories].
@ Specific types of theories: Morton TAC(06)m.QA [combinatorial model for harmonic oscillator]; Coecke & Edwards a0808-proc [Spekkens' toy theory]; Stirling & Wu a0909 [braided systems]; de Ronde & Massri IJTP(18)-a1801, a1802, a1807 [logos categorical approach].
> Specific topics: see fock space; particle models; particle statistics; quantum information [and 2-categories]; quantum oscillators; spin-statistics theorem.

In Other Disciplines
@ References: Hines a1303-in [categorical linguistics and models of meaning]; Blass & Gurevich BEATCS-a1807 [computer science].

"If categories start showing up in your field, you should have left the field five years ago."
– Writing on the wall in one of the men's rooms at Perimeter Institute, 2002-2004.

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