nuclear object




Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



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


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.


  • 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.


  • 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).


  • 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.