Roman Empire

The Roman Empire (Latin: Imperium Rōmānum; Classical Latin: [ɪmˈpɛ.ri.ũː roːˈmaː.nũː] Ancient and Medieval Greek: Βασιλεία τῶν Ῥωμαίων, tr. Basileia tōn Rhōmaiōn) was the post-Roman Republic period of the ancient Roman civilization, characterized by government headed by emperors and large territorial holdings around the Mediterranean Sea in Europe, Africa and Asia. The extended city of Rome was the largest city in the world c. 100 BC – c. 400 AD, with Constantinople (New Rome) becoming the largest around 500 AD, and the Empire's populace grew to an estimated 50 to 90 million inhabitants (roughly 20% of the world's population at the time). The 500-year-old republic which preceded it was severely destabilized in a series of civil wars and political conflict, during which Julius Caesar was appointed as perpetual dictator and then assassinated in 44 BC. Civil wars and executions continued, culminating in the victory of Octavian, Caesar's adopted son, over Mark Antony and Cleopatra at the Battle of Actium in 31 BC and the annexation of Egypt. Octavian's power was now unassailable and in 27 BC the Roman Senate formally granted him overarching power and the new title Augustus, effectively marking the end of the Roman Republic.

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.

Greek (play)

Greek is a play by Steven Berkoff.

It was first performed at the Half Moon Theatre in London on 11 February 1980, in a production directed by the author. The cast was:

It is a retelling of Sophocles' Oedipus Rex. Berkoff wrote:

The play was used as the basis for a well-received opera of the same name composed by Mark-Anthony Turnage and first performed in 1988.

References

Greek (opera)

Greek is an opera in two acts composed by Mark-Anthony Turnage to a libretto adapted by Turnage and Jonathan Moore from Steven Berkoff's 1980 verse play Greek. The play and the opera are a re-telling of Sophocles's Greek tragedy Oedipus the King with the setting changed to the East End of London in the 1980s. The opera was first performed on 17 June 1988 in the Carl-Orff-Saal, Munich, in a co-production by the Munich Biennale, the Edinburgh International Festival and the BBC.

Background and performance history

Turnage composed Greek between 1986 and 1988 as a commission from the City of Munich for one of five new operas to premiere at the first Münchener Biennale. The commission was suggested by the Biennale's founder, German composer Hans Werner Henze, who had taught Turnage at the Tanglewood Music Center and admired his work. Turnage adapted Berkoff's play for the libretto with the help of Jonathan Moore who would be the stage director of the premiere production. The adaptation involved shortening the play, re-ordering some of its text, and the addition of a police riot scene not in Berkoff's original.

Greek and Coptic

Greek and Coptic is the Unicode block for representing modern (monotonic) Greek. It was originally used for writing Coptic, using the similar Greek letters, in addition to the uniquely Coptic additions. Beginning with version 4.1 of the Unicode Standard, a separate Coptic block has been included in Unicode, allowing for mixed Greek/Coptic text that is stylistically contrastive, as is convention in scholarly works. Writing polytonic Greek requires the use of combining characters or the precomposed vowel + tone characters in the Greek Extended character block.

See also

References