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.

Pop!

Pop! was a British group consisting of Glenn Ball, Hannah Lewis, Jamie Tinkler, and Jade McGuire. The group was formed in the summer of 2003 when they were signed by Pete Waterman. They released three singles on the Sony BMG record label, but disbanded in early 2005, due to poor record sales.

The group had three hit singles, which all reached the top 40 in the UK Singles Chart during 2004 and 2005, but each performed under expectations. Their debut single "Heaven and Earth" was withdrawn from the chart because the single contained a track whose length was longer than permitted for that format. Following singles were "Can't Say Goodbye" and "Serious". A fourth single, "Xanadu" (a cover of the Olivia Newton-John and Electric Light Orchestra song), was due to be released but was cancelled when the group split. They had also recorded songs for a planned studio album. Their single "Heaven and Earth" was licensed for use in Dancing Stage Fusion by Konami in 2005.

Group members

Pop (U2 album)

Pop is the ninth studio album by Irish rock band U2. It was produced by Flood, Howie B, and Steve Osborne, and was released on 3 March 1997 on Island Records. The album was a continuation of the band's 1990s musical reinvention, as they incorporated alternative rock, techno, dance, and electronica influences into their sound. Pop employed a variety of production techniques that were relatively new to U2, including sampling, loops, programmed drum machines, and sequencing.

Recording sessions began in 1995 with various record producers, including Nellee Hooper, Flood, Howie B, and Osborne, who were introducing the band to various electronica influences. At the time, drummer Larry Mullen, Jr. was inactive due to a back injury, prompting the other band members to take different approaches to songwriting. Upon Mullen's return, the band began re-working much of their material but ultimately struggled to complete songs. After the band allowed manager Paul McGuinness to book their upcoming 1997 PopMart Tour before the record was completed, they felt rushed into delivering it. Even after delaying the album's release date from the 1996 Christmas and holiday season to March 1997, U2 ran out of time in the studio, working up to the last minute to complete songs.

Ice pop

An ice pop is a water-based frozen snack. It is also referred to as a popsicle (Canada, U.S.), freeze pop (Ireland, U.S.), ice lolly (United Kingdom, Ireland), ice block, icy pole (parts of Australia and New Zealand), or chihiro (Cayman Islands). It is made by freezing flavored liquid (such as fruit juice) around a stick, generally resembling a tongue depressor. Often, the juice is colored artificially. Once the liquid freezes solid, the stick can be used as a handle to hold the ice pop. When an ice pop does not have a stick, it is called, among other names, a freezie.

History

Frank Epperson of Oakland, California, popularized ice pops after patenting the concept of "frozen ice on a stick" in 1923. He initially called it the Epsicle. A couple of years later, Epperson sold the rights to the invention and the Popsicle brand to the Joe Lowe Company in New York City.

Epperson claimed to have first created an ice pop in 1905 at the age of 11 when he accidentally left a glass of powdered soda and water with a mixing stick in it on his porch during a cold night, a story printed on the back panel of Popscicle brand treat boxes in the 80's and 90's. However, the evidence for this is scant.