Fully dualizable object

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.