nLab nuclear object

Idea

A nuclear object is a categorical generalization of the concept of a finite-dimensional vector space, or more generally, a finite-dimensional object.

Definition

A nuclear objectin a symmetric monoidal closed category $\mathcal{M}$ with unit $k$ is an object $A$ such that the canonical morphism $\phi _A:A\otimes A^*\to [ A, A]$ with $A^*:=[A,k]$ is an isomorphism.

A morphism $f:A\to B$ is called nuclear if there exists a morphism $p:k\to B\otimes A^*$ such that

$$\array{k & \overset{p}{\to} & B\otimes A^*\\ n(f)\searrow && \swarrow\phi\\ & [A,B] & }$$

is commutative, where $n(f)$, the name of $f$, obtains via adjointness from $k\otimes A\simeq A\overset{f}{\to} B$.

Properties

• $A$ is nuclear precisely if $id_A$ is nuclear.

• If $A$ is nuclear then the canonical map to the double dual $\theta _A: A\to A^{**}$ is an isomorphism.

• The full subcategory $nuc(\mathcal{M})$ of nuclear objects is symmetric monoidal closed.

• $A$ is nuclear precisely if $A^*$ is nuclear.

• In a cartesian monoidal $\mathcal{M}$ only $1$ is nuclear.

• If $A$ is nuclear there exists a natural trace morphism $t_A:[A,A]\simeq A\otimes A^*\simeq A^*\otimes A \to k$.

Examples

• Nuclear objects in the category of complete semilattices are precisely the completely distributive lattices, or in absence of the axiom of choice, more generally, the constructively completely distributive lattices (Higgs&Rowe 1989, Rosebrugh&Wood 1994).

• Nuclear objects in the category of pointed sets are precisely the pointed sets with cardinality $\leq 2$.

• Nuclear objects in the category of Banach spaces with morphisms the bounded maps are the nuclear spaces (Rowe 1988).

Related concepts

• nucleus of a profunctor

• completely distributive lattice

• nuclear space

• self-dual object

• dualizable object

• trace

References

• D. A. Higgs, K. A. Rowe, Nuclearity in the category of complete semilattices , JPAA 57 no.1 (1989) pp.67-78.

• R. Rosebrugh, R. J. Wood, Constructive complete distributivity IV , App. Cat. Struc. 2 (1994) pp.119-144. (preprint)

• K. A. Rowe, Nuclearity , Canad.Math.Bull. 31 no.2 (1988) pp.227-235. (pdf)

• G. N. Raney, Tight Galois Connections and Complete Distributivity , Trans.Amer.Math.Soc 97 (1960) pp.418-426. (pdf)