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The cobordism hypothesis classifies fully extended framed topological quantum field theories by the object assigned to a point. Precisely, for a suitable symmetric monoidal $(\infty ,n)$-category $\mathcal{C}$, evaluating a framed theory at a positively framed point identifies its space of theories with the space of fully dualizable objects of $\mathcal{C}$. The word “fully” records duals and adjoints at the lower-dimensional levels needed to cut and glue manifolds.^[ref]

In one dimension, a finite-dimensional vector space $V$ has a dual $V^{\ast }$. Evaluation and coevaluation describe interval pieces; closing them into a circle gives the trace of the identity, $Z(S^1)=\dim V$. This is the simplest glimpse of how assigning data to a point controls closed-manifold invariants.

The framing hypothesis matters. Passing to oriented or other tangentially structured theories requires corresponding extra compatibility data, often expressed as a homotopy-fixed-point condition. Nor does the statement classify arbitrary unextended field theories by their point values: those theories do not include all the lower-dimensional data in the first place.

A fully dualizable object in a symmetric monoidal $(\infty ,n)$-category has a categorical dual together with adjoints for the evaluation and coevaluation data, recursively through the required higher levels. The meaning depends on $n$. For $n=1$, full dualizability is just ordinary dualizability. For $n=2$, the evaluation morphism and coevaluation morphism must have both left and right adjoints. This is a finiteness condition expressed through composition and the monoidal tensor product.^[ref]

In the ordinary symmetric monoidal category of vector spaces over $k$, a finite-dimensional vector space $V$ has dual $V^{\ast }$, evaluation $v\otimes \lambda \mapsto \lambda (v)$ and coevaluationThe dual-basis identities make the two resulting zigzag composites the identity. Conversely, any algebraic tensor used as a coevaluation is a finite sum, so these identities force the whole space to be spanned by finitely many vectors. Infinite-dimensional vector spaces have algebraic duals but are not dualizable in this monoidal category.^[ref]

The cobordism hypothesis identifies fully dualizable objects as the possible values on a point of fully extended framed topological quantum field theories. The higher adjoints supply the compatibility needed when bordisms are cut and glued in multiple dimensions.

Jacob Lurie is a mathematician whose work develops higher category theory and its applications to algebraic geometry and homotopy theory. His 2009 paper sketches a proof of the cobordism hypothesis, a classification of extended topological quantum field theories.^[ref]

The organizing idea is that the value on a point, with the appropriate duality properties, can determine the field theory on higher-dimensional manifolds and their cobordisms. Cutting and gluing become composition laws in a higher category. This explains why passing from whole manifolds to pieces of every dimension changes the algebraic framework. His chromatic lecture notes give another bridge between algebra and topology, constructing Morava K-theory from structured topological spectra.^[ref]