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).

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.

Game (disambiguation)

A game is a recreational activity with a set of rules.

Game or games may also refer to:

Animals

In arts and entertainment

Film and television

Music

Games (film)

Games is a 1967 psychological thriller, directed by Curtis Harrington and starring Katharine Ross, James Caan, and Simone Signoret.

Plot

Paul Montgomery (James Caan) and his wife Jennifer (Katharine Ross) are a pair of wealthy but blasé Upper East Side New York socialites with an attitude sometimes accompanying the overprivileged: a propensity to amuse themselves in a bizarre, chic, and upscale fashion, in this case playing socialite games for their peers, and occasionally revealing what appears to be a playfully sadistic streak.

Lisa Schindler (Simone Signoret), an older woman from Germany, arrives at their door one day selling cosmetics; the couple invite her in, and when the conversation reveals that Lisa is believed to have psychic abilities, Paul and Jennifer ask her to arrange some "games" for their amusement. Lisa proceeds to set up several situations of simulated domestic discord that the couple can react to.

Things turn deadly when an acquaintance is accidentally killed during a so-called "game." Paul has to go to enormous lengths to conceal any evidence. He fears being blackmailed by Lisa, whose psychic activities continue and intensify during her stay. Ultimately, at the request of Paul, Lisa leaves, for the welfare of an ever more anxious Jennifer.

Games (seaQuest DSV)

"Games" is the fourth episode of the first season of seaQuest DSV. It originally aired on October 3, 1993.

Plot synopsis

The seaQuest evacuates an icy prison whose population consists of a warden and his lone prisoner, the biochemist and war criminal Dr. Rubin Zellar. Zellar is (supposedly) being kept cryogenically frozen during transport, while the warden is shown around the ship and begins to get along with Dr. Westphalen.

Crew members soon discover that the body in the stasis chamber is the warden, who was killed by Dr. Zellar. Zellar is captured easily, but escapes and holds the crew hostage with a biological agent he smuggled aboard. He threatens to release the agent unless Captain Bridger and Commander Ford destroy the UEO headquarters at Pearl Harbor.

Meanwhile, Lucas Wolenczak has been trying to access the UEO's files on Zellar, at the request of Bridger. He discovers that Dr. Westphalen's brother was among the many people murdered by Zellar. Bridger and Ford fire the missiles, but since they had removed the warheads earlier, no damage was done. Before they can arrest Zellar, Westphalen walks in, pointing a weapon at Zellar. After exchanging a few words with Zellar she pulls out a vial filled with a liquid and tells him that he deserves to die in the same way that he killed. She throws it on him, but it turns out the liquid was non-toxic.

Years (by One Thousand Fingertips)

Years (by One Thousand Fingertips) is the second studio album by Canadian folk rock band Attack in Black, released on March 10, 2009 on Dine Alone Records. The album was released both on CD and on one thousand 12" vinyl records. Singles released from the album are "Beasts" (February 24, 2009) and "Liberties" (July 2009). The layout and photography present in both CD and vinyl versions were by Daniel Romano and Ian Kehoe.

Track listing

Personnel

References