abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
Examples
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
In QFT and String theory
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
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 with unit is an object such that the canonical morphism with is an isomorphism.
A morphism is called nuclear if there exists a morphism such that
is commutative, where , the name of , obtains via adjointness from .
is nuclear precisely if is nuclear.
If is nuclear then the canonical map to the double dual is an isomorphism.
The full subcategory of nuclear objects is symmetric monoidal closed.
is nuclear precisely if is nuclear.
In a cartesian monoidal only is nuclear.
If is nuclear there exists a natural trace morphism .
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 .
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.