Exponential object

In mathematics, specifically in , an exponential object or map object is the generalization of a function space in set theory. with all and exponential objects are called . Categories (such as of ) without adjoined products may still have an exponential law. Let be a category, let and be of , and let have all with . An object together with a morphism is an exponential object if for any object and morphism there is a unique morphism (called the transpose of ) such that the following diagram commutes: This assignment of a unique to each establishes an isomorphism (bijection) of hom-sets, If exists for all objects in , then the functor defined on objects by and on arrows by , is a right adjoint to the product functor . For this reason, the morphisms and are sometimes called exponential adjoints of one another. Alternatively, the exponential object may be defined through equations: Existence of is guaranteed by existence of the operation . Commutativity of the diagrams above is guaranteed by the equality . Uniqueness of is guaranteed by the equality . The exponential is given by a universal morphism from the product functor to the object . This universal morphism consists of an object and a morphism . In the , an exponential object is the set of all functions . The map is just the evaluation map, which sends the pair to . For any map the map is the curried form of : A Heyting algebra is just a bounded lattice that has all exponential objects. Heyting implication, , is an alternative notation for . The above adjunction results translate to implication () being right adjoint to meet (). This adjunction can be written as , or more fully as: In the , the exponential object exists provided that is a locally compact Hausdorff space. In that case, the space is the set of all continuous functions from to together with the compact-open topology. The evaluation map is the same as in the category of sets; it is continuous with the above topology.

New Results in Integer and Lattice Programming

Christoph Hunkenschröder

An integer program (IP) is a problem of the form

\min \{f(x) : \, Ax = b, \ l \leq x \leq u, \ x \in \Z^n\}

, where

A \in \Z^{m \times n}

b \in \Z^m

l,u \in \Z^n

, and

f: \Z^n \rightarrow \Z

is a separable convex objective function. The problem o ...

EPFL2020

On compact representations of Voronoi cells of lattices

New Results in Integer and Lattice Programming

On compact representations of Voronoi cells of lattices

On compact representations of Voronoi cells of lattices

On compact representations of Voronoi cells of lattices

New Results in Integer and Lattice Programming