Cobordism hypothesis

For a symmetric monoidal ${\displaystyle (\infty ,n)}$-category ${\displaystyle {\mathcal {C}}}$ which is fully dualizable and every ${\displaystyle k}$-morphism of which is adjointable, for ${\displaystyle 1\leq k\leq n-1}$, there is a bijection between the ${\displaystyle {\mathcal {C}}}$-valued symmetric monoidal functors of the cobordism category and the objects of ${\displaystyle {\mathcal {C}}}$.

Symmetric monoidal functors from the cobordism category correspond to topological quantum field theories. The cobordism hypothesis for topological quantum field theories is the analogue of the Eilenberg-Steenrod axioms 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 ${\displaystyle {\mathcal {C}}}$-valued symmetric monoidal functors and the objects of ${\displaystyle {\mathcal {C}}}$ is uniquely defined by its value for the point.

• Cobordism

• Freed, Daniel S. (11 October 2012). "The Cobordism hypothesis". Bulletin of the American Mathematical Society. American Mathematical Society (AMS). 50 (1): 57–92. doi:10.1090/s0273-0979-2012-01393-9. ISSN 0273-0979.
• Seminar on the Cobordism Hypothesis and (Infinity,n)-Categories, 2013-04-22
• Jacob Lurie (4 May 2009). On the Classification of Topological Field Theories

• cobordism hypothesis in nLab