PRO (category theory)

In category theory, a PRO is a strict monoidal category whose objects are the natural numbers (including zero), and whose tensor product is given on objects by the addition on numbers.

Some examples of PROs:

the discrete category of natural numbers,

the simplex category of natural numbers and monotonic functions.

The name PRO is an abbreviation of "PROduct category". PROBs and PROPs are defined similarly with the additional requirement for the category to be braided, and to have a symmetry (that is, a permutation), respectively. All of the examples above are PROPs, except for the simplex category and Bij_Braid; the latter is a PROB but not a PROP, and the former is not even a PROB.

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.

Book of Proverbs

The Book of Proverbs (Hebrew: מִשְלֵי, Míshlê (Shlomoh), "Proverbs (of Solomon)") is the second book of the third section (called Writings) of the Hebrew Bible and a book of the Christian Old Testament. When translated into Greek and Latin, the title took on different forms: in the Greek Septuagint (LXX) it became Παροιμίαι Paroimiai ("Proverbs"); in the Latin Vulgate the title was Proverbia, from which the English name is derived.

Proverbs is not merely an anthology but a "collection of collections" relating to a pattern of life which lasted for more than a millennium. It is an example of the Biblical wisdom tradition, and raises questions of values, moral behaviour, the meaning of human life, and right conduct. The repeated theme is that "the fear of God (meaning submission to the will of God) is the beginning of wisdom". Wisdom is praised for her role in creation; God acquired her before all else, and through her he gave order to chaos; and since humans have life and prosperity by conforming to the order of creation, seeking wisdom is the essence and goal of the religious life.

2013–14 Pro12

The 2013–14 Pro12 (also known as the RaboDirect Pro12 for sponsorship reasons) was the 13th season of the Pro 12 rugby union competition originally known as the Celtic League, the fourth with its current 12-team format, and the third with RaboDirect as title sponsor.

Leinster were the defending champions, having beaten Ulster in the 2013 playoff final.

The twelve competing teams were the four Irish teams, Connacht, Leinster, Munster and Ulster; two Scottish teams, Edinburgh and Glasgow Warriors; four Welsh teams, Cardiff Blues, Newport Gwent Dragons, Ospreys and Scarlets; and two Italian teams, Benetton Treviso and Zebre.

Changes for the season

Ireland

New Zealand native Pat Lam replaces Eric Elwood as head coach of Connacht, following Elwood's decision to step down, while out-half Dan Parks takes on a coaching role with Connacht U18 Schools/Clubs while continuing to play with the senior team. Also, following changes between captains in recent seasons, the start of the season saw three players, Gavin Duffy, John Muldoon and Michael Swift, captain the side jointly. After a poor run of form culminating in a 43–10 defeat against Edinburgh, former Chiefs captain Craig Clarke was made team captain with the others continuing to "lead off the field".

Steinberg Cubase

Cubase is a music software product developed by German musical software and equipment company Steinberg for music recording, arranging and editing as part of a digital audio workstation. The first version, which ran on the Atari ST computer and recorded via MIDI only, was released in 1989.

In January 2003, Steinberg was acquired by U.S. firm Pinnacle Systems, within which it operated as an independent company before being sold to Yamaha Corporation in December, 2004.

Operation

Cubase creates projects, which allow the operator to edit MIDI files, raw audio tracks, and other associated information like lyrics, and to present them in a range of formats including musical scores, editing console, event lists, etc. The operator can also mix the various tracks down into a stereo .wav file ready to be burned to a compact disc (CD) in Red Book format, or .mp3 burned to CD or DVD as files, or to be published on the Web.

Versions

Notes

Cubase has existed in three main incarnations. Initially Cubase, which featured only MIDI, and which was available on the Atari ST, Macintosh and Windows.