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I agree. I'm a mathematician working in quantum topology, and this article seems to equal parts vague and incoherent. As for this "Wow! Topology impacts phys

I agree. I'm a mathematician working in quantum topology, and this article seems to equal parts vague and incoherent. As for this "Wow! Topology impacts physics! Useful for the first time!" is total bullshit. The interaction between topology and physics goes back a century, or centuries. The work of Poincare, the father of modern topology, for example, was motivated by studying PDEs and the behavior of dynamical systems. More spectacular is the work of the last 30 years, starting with the work of Donaldson in the 80s, and later work by Witten, between gauge theory (the mathematical formalism of the standard model) and the topology of 4-dimensional spaces. It is beautiful story, much like the origins of calculus or the application of ideas from differential geometry in the foundations of general relativity.

runT1ME 14y ago
Random question, is category theory helpful in working out equations with quantum topology?
- cottonseed 14y ago
  Short answer: Yes. Long answer: I'm not quite sure what you mean by "working out equations". Category theory is useful. Categories were introduced by Eilenberg and MacLane to formalize certain structures were studying in the context of algebraic topology (homology, cohomology and homotopy are all functors from topological to algebraic categories, and there are lots of examples of natural transformations on these functors). Algebraic topology is a foundation for quantum/low-dimensional topology, so categories naturally arise in using the tools from algebraic topology. In quantum topology, many of the natural invariants one studies are in some sense TQFTs: Topological Quantum Field Theories. The modern formulation of TQFTs is due to Atiyah. In this formulation, a TQFT is a functor from a certain kind of topological category called a cobordism category (in analogy with physics, one can think of the objects as spacetimes and the morphisms as spacetime evolution) to certain algebraic categories (which one can think of as quantum state spaces and evolution operators). These TQFTs are often not just functors, but n-functors: the categories in question are not just regular categories, but higher categories or infinity categories.

runT1ME 14y ago

Random question, is category theory helpful in working out equations with quantum topology?

cottonseed 14y ago
Short answer: Yes. Long answer: I'm not quite sure what you mean by "working out equations". Category theory is useful. Categories were introduced by Eilenberg and MacLane to formalize certain structures were studying in the context of algebraic topology (homology, cohomology and homotopy are all functors from topological to algebraic categories, and there are lots of examples of natural transformations on these functors). Algebraic topology is a foundation for quantum/low-dimensional topology, so categories naturally arise in using the tools from algebraic topology. In quantum topology, many of the natural invariants one studies are in some sense TQFTs: Topological Quantum Field Theories. The modern formulation of TQFTs is due to Atiyah. In this formulation, a TQFT is a functor from a certain kind of topological category called a cobordism category (in analogy with physics, one can think of the objects as spacetimes and the morphisms as spacetime evolution) to certain algebraic categories (which one can think of as quantum state spaces and evolution operators). These TQFTs are often not just functors, but n-functors: the categories in question are not just regular categories, but higher categories or infinity categories.