abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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 $k$ 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
is commutative, where $n(f)$, the name of $f$, obtains via adjointness from $k\otimes A\simeq A\overset{f}{\to} B$.
$A$ 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 $t_A:[A,A]\simeq A\otimes A^*\simeq A^*\otimes A \to k$.
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).
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.