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.

Buddhism

Buddhism /ˈbudɪzəm/ is a nontheistic religion or philosophy (Sanskrit: धर्म dharma; Pali: धम्म dhamma) that encompasses a variety of traditions, beliefs and spiritual practices largely based on teachings attributed to Gautama Buddha, commonly known as the Buddha ("the awakened one"). According to Buddhist tradition, the Buddha lived and taught in the eastern part of the Indian subcontinent,present-day Nepal sometime between the 6th and 4th centuries BCE. He is recognized by Buddhists as an awakened or enlightened teacher who shared his insights to help sentient beings end their suffering through the elimination of ignorance and craving. Buddhists believe that this is accomplished through the direct understanding and perception of dependent origination and the Four Noble Truths.

Two major extant branches of Buddhism are generally recognized by scholars: Theravada ("The School of the Elders") and Mahayana ("The Great Vehicle"). Vajrayana, a body of teachings attributed to Indian siddhas, may be viewed as a third branch or merely a part of Mahayana. Theravada has a widespread following in Sri Lanka and Southeast Asia. Mahayana which includes the traditions of Pure Land, Zen, Nichiren Buddhism, Shingon, and Tiantai (Tendai) is found throughout East Asia. Tibetan Buddhism, which preserves the Vajrayana teachings of eighth century India, is practiced in regions surrounding the Himalayas, Mongolia and Kalmykia. Buddhists number between an estimated 488 million and 535 million, making it one of the world's major religions.