Category

Category, plural categories, may refer to:

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Mathematics

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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).

Higher category theory

In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.

Strict higher categories

An ordinary category has objects and morphisms. A 2-category generalizes this by also including 2-morphisms between the 1-morphisms. Continuing this up to n-morphisms between (n-1)-morphisms gives an n-category.

Just as the category known as Cat, which is the category of small categories and functors is actually a 2-category with natural transformations as its 2-morphisms, the category n-Cat of (small) n-categories is actually an n+1-category.

An n-category is defined by induction on n by:

So a 1-category is just a (locally small) category.

The monoidal structure of Set is the one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite products can be given a monoidal structure. The recursive construction of n-Cat works fine because if a category C has finite products, the category of C-enriched categories has finite products too.

God

In monotheism and henotheism, God is conceived of as the Supreme Being and principal object of faith. The concept of God as described by theologians commonly includes the attributes of omniscience (infinite knowledge), omnipotence (unlimited power), omnipresence (present everywhere), omnibenevolence (perfect goodness), divine simplicity, and eternal and necessary existence.

God is also usually defined as a non-corporeal being without any human biological gender, but his role as a creator has caused some religions to give him the metaphorical name of "Father". Because God is concieved as not being a corporeal being, he cannot (some say should not) be portrayed in a literal visual image; some religious groups use a man to symbolize God because of his role as the "father" of the universe and his deed of creating man's mind in the image of his own.

In theism, God is the creator and sustainer of the universe, while in deism, God is the creator, but not the sustainer, of the universe. Monotheism is the belief in the existence of one God or in the oneness of God. In pantheism, God is the universe itself. In atheism, God does not exist, while God is deemed unknown or unknowable within the context of agnosticism. God has also been conceived as being incorporeal (immaterial), a personal being, the source of all moral obligation, and the "greatest conceivable existent". Many notable philosophers have developed arguments for and against the existence of God.

God (Rip Rig + Panic album)

God is the debut album of the Post-punk band Rip Rig + Panic, released in 1981 through Virgin Records.

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God (Tori Amos song)

"God" is a song by American singer-songwriter and musician Tori Amos. It was released as the second single from her second studio album Under the Pink. It was released on February 3, 1994 by Atlantic Records in North America and on October 3 by EastWest Records in the UK.

The song reached number 44 on the UK Singles Chart. as well as #1 on the US Modern Rock Chart.

Releases

The B-sides to the American release included Amos' reworking of "Home on the Range", with new lyrics, as well as a two-song instrumental piano suite. An American cassette single featured the b-side "Sister Janet".

A completely different single was released in Europe on CD, 12" and 7" vinyl single, and cassette. The 7" single was a glossy dual sided picture disc. The various formats featured ambient and jungle house remixes of the track by CJ Bolland, Carl Craig and The Joy.

Track listing