Subwiki:Property-theoretic categorization

Property-theoretic categorization is part of the broader property-theoretic paradigm of organization followed on some subject wikis, particularly those in mathematics.

Properties are a more precise formulation of the general idea of genus and differentia.

On a subject wiki using property-theoretic categorization, a complete listing of property-theoretic supercategories is available at the category titled Property-theoretic categories. For instance, on the Groupprops wiki, this page is Groupprops:Category:Property-theoretic categories.

Properties
A property over a collection of objects is something which every object either has (or satisfies) or does not have (or does not satisfy).

Examples from elementary mathematics:


 * Being prime is a property over the collection of natural numbers. Every natural number either has the property of being prime, or does not have the property of being prime.
 * Being positive is a property over the collection of real numbers. Every real number either is positive or is not positive.
 * Being isosceles is a property over the collection of triangles. Every triangle either is isosceles or is not isosceles.

Examples from higher mathematics:


 * Being Abelian is a property over the collection of groups. Every group either is Abelian or is not Abelian.
 * Being compact is a property over the collection of topological spaces. Every topological space either is compact or is not compact.

The collection of objects over which the property is being evaluted is here termed the context space for the property.

Categorization of properties
All the properties over the same context space are put in a category, and the category is labeled by the context space. For instance all properties over the context space of groups, are called group properties, and are put in a category titled Group properties. A generic element here is termed a group property.

There are two formats for naming the collection of properties over a given context space:


 * context space name, followed by the word properties: For instance, group properties, subgroup properties.
 * properties of, followed by context space name: For instance, properties of topological spaces.

There is no distinction between the two ways of naming, and one may be chosen over the other for reasons of greater ease of use.

The categories that list all properties over a given context space are termed property categories, and a list of all such categories is found in the category titled Properties in the subject wiki. For instance, Groupprops:Category:Properties lists all the property categories on the Group Properties Wiki.

Further subcategories of the property category
The category of all properties over a given context space may often be overwhelmingly huge and diverse. In some cases, this category is divided into subcategories. However, the following general convention is observed: all properties are listed both in the parent category and in the subcategories.

Subcategories are created in many ways:


 * Based on metaproperties: properties the properties may or may not satisfy.
 * Based on variation, analogy or opposites relationships with an existing pivotal property: For instance, Topospaces:Category:Variations of normality lists properties of topological spaces obtained by varying the property of being a normal space.
 * Based on meta-criteria like degree of importance, implementation in software packages, and historical development.

Metaproperties
Metaproperties are properties whose context space is itself a property space. In other words, they are properties evaluated for properties.

Here are some examples:


 * A property of natural numbers is termed multiplicatively closed if whenever $$a,b$$ are natural numbers satisfying the property, then $$ab$$ also satisfies the property.
 * A property of groups is termed direct product-closed if the direct product of a family of groups, each with the property, also has the property.

Categorization of metaproperties
The metaproperties over a given context space (i.e., the properties that can be evaluated for properties over that context space) are put in a category. For instance, Groupprops:Category:Group metaproperties is the category of all group metaproperties. A generic element of this category is termed a group metaproperty.

There are two formats for naming the collection of properties over a given context space:


 * context space name, followed by the word metaproperties: For instance, group metaproperties, subgroup metaproperties.
 * metaproperties of, followed by context space name: For instance, metaproperties of topological spaces.

The categories that list all metaproperties over a given context space are termed metaproperty categories and the list of all metaproperty categories is available at a category called Metaproperties. For instance, the metaproperty category for groups is Groupprops:Category:Metaproperties.