It is a vast generalisation of Cayley's theorem from group theory ( viewing a group as a particular kind of category with just one object ).

From another point of view, representable functors for a category C are the functors given with C. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory.

Conjecturally there is just one essential type of global L-function, with two descriptions ( coming from an algebraic variety, coming from an automorphic representation ); this would be a vast generalisation of the Taniyama – Shimura conjecture, itself a very deep and recent result () in number theory.

The main application of fibred categories is in descent theory, concerned with a vast generalisation of " glueing " techniques used in topology.

In the Dutch East Indies ( now Indonesia ) the vast majority of Dutch settlers were in fact Eurasians known as Indo-Europeans, formally belonging to the European legal class in the colony.

A well-known class of IBM PC keyboards is the Model M. Introduced in 1985 and manufactured by IBM, Lexmark and Unicomp, the vast majority of Model M keyboards feature a buckling spring key design and many have fully swappable keycaps.

Aside from the richest and most powerful of the dukes and the king himself, Lombard noblemen tended to live in cities ( unlike their Frankish counterparts ) and hold little more than twice as much in land as the merchant class ( a far cry from the provincial Frankish aristocrat who held a vast swathe of land hundreds of times larger than the nearest man beneath him ).

The period following the American Civil War brought vast fortunes to a new class of entrepreneurs in the New York area, and many built large estates in Westchester.

Due to China's enormous population growth and the body of its appointed scholar-officials being accepted in limited size ( about 20, 000 active officials during the Song period ), the larger scholarly gentry class would now take over grassroots affairs on the vast local level.

In Russia, the working class, although a small minority in a predominantly peasant based society, were organised in vast factories owned by the capitalist class, and into large working class districts.

In 1905, the working class in Russia, a generation brought together in vast factories from the relative isolation of peasant life, saw the result of its labour as a vast collective effort, and the only means of struggling against its oppression in terms of a collective effort also, forming workers councils ( soviets ), in the course of the revolution of that year.

This prediction has come true, as vast numbers of business travelers are buying airfares only in economy class for business travel.

The class Gastropoda contains a vast total of named species, second only to the insects in overall number.

The group maintained that the vast majority of people in Britain remained exploited by the ruling class and their official literature has long stated that about 75 % of the country is working-class.

Others, however, believe that there is no deeper deterministic reality in quantum mechanics — experiments have shown a vast class of hidden variable theories to be incompatible with observations.

The historian H. Warington Smyth considered the junk one of the most efficient ship designs, stating that " As an engine for carrying man and his commerce upon the high and stormy seas as well as on the vast inland waterways, it is doubtful if any class of vessel … is more suited or better adapted to its purpose than the Chinese or Indian junk, and it is certain that for flatness of sail and handiness, the Chinese rig is unsurpassed.

Other satirical vignettes involve a lowly worker who upstages his superiors by displaying his vast culinary knowledge while ordering at a gourmet French restaurant ; a housewife who rises from her deathbed to cook one last meal for her family ; and a women's etiquette class ( held by Mariko Okada ) on how to eat spaghetti properly.

The vast majority of families in the United States today are not the " middle class family with a bread-winning father and a stay-at-home mother, married to each other and raising their biological children " that has been viewed as the norm.

The Catholic Church and its orders, especially the Jesuits, were granted vast hacienda holdings, linking the interests of the church with the rest of the landholding class.

But the working class, although many were organised in vast modern western-owned enterprises, comprised no more than a small percentage of the population and " more than half the land is owned in common by the peasants ".

The West Island is an informal term for a large territory of low-density middle and upper-middle class housing, low and medium density commercial sectors and vast industrial operations roughly constituting the Western third of the Island of Montreal.

The working class, both blue-collar and white-collar, is the vast majority of society and is the key to the fight for socialism.

* this population ’ s vast internal social differentiation ( North / South, gender and class positionalities );

The Navy immediately focused on designing a new class of submarines, but the Bureau of Ships believed that the vast fleet of existing Gato, Balao, and Tench class submarines could be modified to incorporate the desired improvements.

The innermost rings of this vast, multi-ringed crater are the inner and outer Montes Rook, and the outermost ring are the Montes Cordillera, 930 km in diameter.

General Lukacs himself is mortally wounded on his right temple and dies several hours later. The " Iron Ring ", el cinturón de hierro, is a vast, labyrinthine fortification around Bilbao, consisting of bunkers, tunnels, and fortified trenches in several rings, protected by artillery.

Others are confined to vast reservations, and not only does the Australian government justifiably not wish them to be viewed as exhibits in a zoo, but on their reservations they are extremely fugitive, shunning camps, coming together only for corroborees at which their strange culture comes to its highest pitch -- which is very low indeed.

Faulkner traces, in his vast and overpowering saga of Yoknapatawpha County, the gradual changes which seep into the South, building layer upon layer of minute, subtle innovation which eventually tend largely to hide the Old Way.

Our most elemental and unavoidable impressions, he says, are those of being involved in a large arena of powers which have a longer past than our own, which are interrelated in a vast movement through the present toward the future.

After scouring around a bit in the open area, I came across what proved to be tar-soaked logs which crackled and burned brightly, giving off vast rolls of smoke into the ashen sky.

Thus the government simultaneously undertook the vast burden of social security which had traditionally been privately discharged, and created a national scarcity which has engendered calamitous problems of social security.

There is a vast difference between the community of reconciliation which the New Testament describes and the community of congeniality found in the average church building.

A relatively recent role for amateur astronomers is searching for overlooked phenomena ( e. g., Kreutz Sungrazers ) in the vast libraries of digital images and other data captured by Earth and space based observatories, much of which is available over the Internet.

They include the 22 proteinogenic (" protein building ") amino acids which combine into peptide chains (" polypeptides ") to form the building blocks of a vast array of proteins.

A vast number of engraved stones are in existence, to which the name " Abrasax-stones " has long been given.

The contemporary ecclesiastics recorded with wonder many instances of the Visigoths ' clemency: Christian churches saved from ravage ; protection granted to vast multitudes both of pagans and Christians who took refuge therein ; vessels of gold and silver which were found in a private dwelling, spared because they " belonged to St. Peter "; at least one case in which a beautiful Roman matron appealed, not in vain, to the better feelings of the Gothic soldier who attempted her dishonor.

The most striking feature is the existence of two great lines of depression, due largely to the subsidence of whole segments of the Earth's crust, the lowest parts of which are occupied by vast lakes.

This trait permits a vast increase in the range of foodstuffs which can be fed on.

It has been estimated that during his entire career Aalto designed over 500 individual buildings, approximately 300 of which were built, the vast majority of which are in Finland.

To further strengthen Athens ' grip on its empire, Pericles in 450 began a policy of establishing cleruchiai — quasi-colonies that remained tied to Athens and which served as garrisons to maintain control of the League's vast territory.

His commissions included The Triumphal Arch, a vast work printed from 192 separate blocks, the symbolism of which is partly informed by Pirckheimer's translation of Horapollo's Hieroglyphica.

Berlin is situated in northeastern Germany, approximately west of the Polish border, in an area of low-lying marshy woodlands with a mainly flat topography, part of the vast Northern European Plain which stretches all the way from northern France to western Russia.

Among the vast number of different biomolecules, many are complex and large molecules ( called biopolymers ), which are composed of similar repeating subunits ( called monomers ).

The caves and bunkers were connected to a vast system throughout central Peleliu, which allowed the Japanese to evacuate or reoccupy positions as needed, and to take advantage of shrinking interior lines.

In fact, Shestov used the story of Job as a central signifier for his core philosophy ( the vast critique of the history of Western philosophy, which he saw broadly as a monumental battle between Reason and Faith, Athens and Jerusalem, secular and religious outlook ):