In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the cobordism hypothesis, due to
John C. Baez and James Dolan, concerns the classification of extended
topological quantum field theories
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.
Although TQFTs were invented by physicists, they are also of mathem ...
(TQFTs). In 2008,
Jacob Lurie
Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study. Lurie is a 2014 MacArthur Fellow.
Life
When he was a student in the Science, Mathematics, and Computer Science ...
outlined a proof of the cobordism hypothesis, though the details of his approach have yet to appear in the literature as of 2022.
In 2021, Daniel Grady and Dmitri Pavlov claimed a complete proof of the cobordism hypothesis, as well as a generalization to bordisms with arbitrary geometric structures.
Formulation
For a symmetric monoidal
-category
which is fully dualizable and every
-morphism of which is adjointable, for
, there is a bijection between the
-valued symmetric monoidal functors of the cobordism category and the objects of
.
Motivation
Symmetric monoidal functors from the cobordism category correspond to
topological quantum field theories
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.
Although TQFTs were invented by physicists, they are also of mathem ...
. The cobordism hypothesis for topological quantum field theories is the analogue of the
Eilenberg–Steenrod axioms
In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homo ...
for homology theories. The Eilenberg–Steenrod axioms state that a homology theory is uniquely determined by its value for the point, so analogously what the cobordism hypothesis states is that a topological quantum field theory is uniquely determined by its value for the point. In other words, the bijection between
-valued symmetric monoidal functors and the objects of
is uniquely defined by its value for the point.
See also
*
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dim ...
References
Further reading
*
Seminar on the Cobordism Hypothesis and (Infinity,n)-Categories 2013-04-22
*Jacob Lurie (4 May 2009).
On the Classification of Topological Field Theories
External links
*
Quantum field theory
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