The cobordism hypothesis classifies fully extended framed topological quantum field theories by the object assigned to a point. Precisely, for a suitable symmetric monoidal -category , evaluating a framed theory at a positively framed point identifies its space of theories with the space of fully dualizable objects of . The word “fully” records duals and adjoints at the lower-dimensional levels needed to cut and glue manifolds.[ref]
On the Classification of Topological Field Theories by Jacob Lurie. Section 2, formulation and fully dualizable objects
In one dimension, a finite-dimensional vector space has a dual . Evaluation and coevaluation describe interval pieces; closing them into a circle gives the trace of the identity, . 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.
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The Cobordism Hypothesis is a concept in the field of higher category theory, particularly in the study of topological and geometric aspects of homotopy theory. It can be loosely described as a relationship between the notion of cobordism in topology and the structure of higher categorical objects.