Category

Category, plural categories, may refer to:

Philosophy and general uses

Mathematics

Linguistics

Lusternik–Schnirelmann category

In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space is the homotopical invariant defined to be the smallest integer number such that there is an open covering of with the property that each inclusion map is nullhomotopic. For example, if is the circle, this takes the value two.

Sometimes a different normalization of the invariant is adopted, which is one less than the definition above Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below).

In general it is not easy to compute this invariant, which was initially introduced by Lazar Lyusternik and Lev Schnirelmann in connection with variational problems. It has a close connection with algebraic topology, in particular cup-length. In the modern normalization, the cup-length is a lower bound for LS category.

It was, as originally defined for the case of X a manifold, the lower bound for the number of critical points that a real-valued function on X could possess (this should be compared with the result in Morse theory that shows that the sum of the Betti numbers is a lower bound for the number of critical points of a Morse function).

Grammatical category

A grammatical category is a property of items within the grammar of a language; it has a number of possible values (sometimes called grammemes), which are normally mutually exclusive within a given category. Examples of frequently encountered grammatical categories include tense (which may take values such as present, past, etc.), number (with values such as singular, plural, and sometimes dual), and gender (with values such as masculine, feminine and neuter).

Although terminology is not always consistent, a distinction should be made between these grammatical categories (tense, number, etc.) and lexical categories, which are closely synonymous with the traditional parts of speech (noun, verb, adjective, etc.), or more generally syntactic categories. Grammatical categories are also referred to as (grammatical) features.

The name given to a grammatical category (as an uncountable noun) is generally also used (as a countable noun) to denote any of the possible values for that category. For example, the values available in a given language for the category "tense" are called "tenses", the values available for the category "gender" are called "genders", and so on.

Cave

A cave or cavern is a hollow place in the ground, especially a natural underground space large enough for a human to enter. Caves form naturally by the weathering of rock and often extend deep underground. The word cave can also refer to much smaller openings such as sea caves, rock shelters, and grottos.

Speleology is the science of exploration and study of all aspects of caves and the cave environment. Visiting or exploring caves for recreation may be called caving, potholing, or spelunking.

Types and formation

The formation and development of caves is known as speleogenesis. Caves are formed by various geologic processes and can be variable sizes. These may involve a combination of chemical processes, erosion from water, tectonic forces, microorganisms, pressure, and atmospheric influences.

It is estimated that the maximum depth of a cave cannot be more than 3,000 metres (9,800 ft) due to the pressure of overlying rocks. For karst caves the maximum depth is determined on the basis of the lower limit of karst forming processes, coinciding with the base of the soluble carbonate rocks.

Caves (beer)

Caves is a dark Belgian beer, available around the cities of Lier and Ghent.

Caves has a long history of success and decline. From the 16th century onwards, it was brewed by several breweries around the city of Lier, but when the last brewery Cuykens disappeared in 1967 it seemed that this was also the end of the beer tradition of Caves.

However, in 1976 the cultural organization 'De Heren van Lier' decided to start brewing the beer again.

Caves is a beer of high fermentation, based on grain, malt and hops. No conservatives or sugars are used during the brewing process.