nLab
nuclear object

Contents

Context

Duality

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

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 kk is an object AA such that the canonical morphism ϕ A:AA *[A,A]\phi _A:A\otimes A^*\to [ A, A] with A *:=[A,k]A^*:=[A,k] is an isomorphism.

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

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

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

Properties

  • AA is nuclear precisely if id Aid_A is nuclear.

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

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

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

  • In a cartesian monoidal \mathcal{M} only 11 is nuclear.

  • If AA is nuclear there exists a natural trace morphism t A:[A,A]AA *A *Akt_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 2\leq 2.

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

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)

Last revised on September 19, 2014 at 18:44:06. See the history of this page for a list of all contributions to it.