nLab
conical space

Conical spaces

Idea

A conical space is a set equipped with a notion of taking real-linear combinations with nonnegative coefficients of its elements. We may tersely define it as a module of the rig +=[0,[\mathbb{R}^+ = {[{0,\infty}[}: that is, the rig of nonnegative real numbers, with ordinary addition and multiplication as the rig operations.

The positive cone of any ordered real vector space is a conical space.

Relation to vector spaces

Any rig homomorphism ABA \to B gives a ‘restriction of scalars’ functor R:BModAModR: B Mod \to A Mod and ‘extension of scalars’ functor L:AModBModL: A Mod \to B Mod. In particular, the rig homomorphism

+\mathbb{R}^+ \hookrightarrow \mathbb{R}

produces a pair of adjoint functors between the category of \mathbb{R}-modules (that is, real vector spaces) and +\mathbb{R}^+-modules (that is, conical spaces).

In simple English: any real vector space has an underlying conical space, and any conical space freely generates a real vector space.

Extended conical spaces

An extended conical space is a module over the rig ¯ +=[0,]\bar{\mathbb{R}}^+ = [0,\infty] of nonnegative extended real numbers, with 000 \cdot \infty \coloneqq 0. For purposes of constructive mathematics, one should take ¯ +\bar{\mathbb{R}}^+ to be the space of nonnegative lower real numbers.

Since we have a rig homomorphism +¯ +\mathbb{R}^+ \hookrightarrow \bar{\mathbb{R}}^+, every extended conical space has an underlying conical space, and any conical space freely generates an extended conical space. However, there is no direct relationship between vector spaces and extended conical spaces.

That said, the extended positive cone of any ordered real vector space is an extended conical space.

Revised on August 8, 2012 07:39:19 by Toby Bartels (98.16.182.220)