Conical spaces

Conical spaces

Idea

A conical space is a set equipped with a notion of taking real-linear combinations with nonnegative coefficients of its elements.

Conical spaces appear prominently in microlocal analysis, where wave front sets of distributions are disjoint unions of (open-)conical spaces; also in Lorentzian geometry in the guise of open past/future causal cones and light cones.

Definition

Conical sets

A conical set is a module of the rig $\mathbb{R}^+ = {[{0,\infty}[}$: that is, the rig of nonnegative real numbers, with ordinary addition and multiplication as the rig operations.

Similarly one may consider “open conical sets” which are modules over the rig of positive real numbers.

Extended conical sets

An extended conical space is a module over the rig $\bar{\mathbb{R}}^+ = [0,\infty]$ of nonnegative extended real numbers, with $0 \cdot \infty \coloneqq 0$.

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.

Examples

General

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

Relation to vector spaces

Any rig homomorphism $A \to B$ gives a ‘restriction of scalarsfunctor $R\colon B Mod \to A Mod$ and ‘extension of scalarsfunctor $L \colon A Mod \to B Mod$. In particular, the rig homomorphism

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

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