= 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>.<On the Classification of Topological Field Theories by Jacob Lurie>[section 2.3, especially Definitions 2.3.21–2.3.22 and Remark 2.3.24; Proposition 4.2.3]{ref}
In the ordinary symmetric monoidal <category of vector spaces> over $k$, a finite-dimensional <vector space> $V$ has dual $V^*$, evaluation $v\otimes\lambda\mapsto\lambda(v)$ and coevaluation
$$
1\longmapsto\sum_i e^i\otimes e_i.
$$
The 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.<On the Classification of Topological Field Theories by Jacob Lurie>[Example 2.3.4]{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.
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