Category, plural categories, may refer to:
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).
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.
A concurrency in a road network is an instance of one physical road bearing two or more different highway, motorway, or other route numbers. When two freeways share the same right-of-way, it is sometimes called a common section or commons. Other terminology for a concurrency includes overlap,coincidence,duplex (two concurrent routes), triplex (three concurrent routes) multiplex (any number of concurrent routes),dual routing or triple routing.
Concurrent numbering can become very common in countries that allow it. Where multiple routes must pass through a single mountain crossing, or through a major city, it is often economically and practically advantageous for them all to be accommodated on a single physical road. In some countries, however, concurrent numbering is avoided by posting only one route number on road signs; these routes disappear at the start of the concurrency and reappear when it ends. Criticism of concurrencies include environmental intrusion, as well as being considered a factor in road accidents.