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Functors were first considered in algebraic topology, where algebraic objects ( like the fundamental group ) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps.
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Functors and like
Functors like these, which " forget " some structure, are termed forgetful functors.
Functors like these are called representable functors.
Functors belong to the most important categories in logical grammar ( alongside with basic categories like sentence and individual name ): a functor can be regarded as an " incomplete " expression with argument places to fill in.
Functors and are
Functors and natural transformations (' naturality ') are the key concepts in category theory.
Functors are structure-preserving maps between categories.
Functors are often defined by universal properties ; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.
Functors for which this assumption does not hold are called intensional.
Functors and homomorphisms
Functors often describe " natural constructions " and natural transformations then describe " natural homomorphisms " between two such constructions.
Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories.
Functors between one-object categories correspond to monoid homomorphisms.
Functors and .
Functors that forget the extra sets need not be faithful ; distinct morphisms respecting the structure of those extra sets may be indistinguishable on the underlying set.
Categories, Functors, and Natural Transformations.
were and first
At first they were only feathers and dark indistinguishable faces and bodies, hunched over their horses' heads.
Two of the new hands, a Mexican named Jose Amado and a kid known only as Laredo, were picked for the first trick of riding night herd.
After the first two murders, the warning notes were rarely ignored.
Even the knowledge that she was losing another boy, as a mother always does when a marriage is made, did not prevent her from having the first carefree, dreamless sleep that she had known since they dropped down the canyon and into Bear Valley, way, way back there when they were crossing those other mountains.
Out of compulsion to say something cheery, Ben Prime blurted, `` Well, we were lucky to be on soft ground when the first floodheads hit.
His political opponent and lifetime friend, Thomas Jefferson, achieved immortality through his authorship of the Declaration of Independence, but equally notable were the legal and constitutional reforms he instituted in his native Virginia, his role as father of our territorial system, and his acquisition of the Louisiana Territory during his first term as President.
This is the primary function of the imagination operating in the absence of the original experiential stimulus by which the images were first appropriated.
Before being daughter, wife, or mother, before being cultured ( a word now bereft both socially and politically of the sheen you children of frontiersmen bestowed on it ), before being sorry for the poor, progressive about public health, and prettily if somewhat imprecisely humanitarian, indeed first and foremost, you were a lady.
Even the first wave of homesickness had passed, although there were moments when Captain Heard pointed out on his compass the direction of Bradford that she felt a little twinge at her heart.
When they were first written, there was evidently no thought of their being published, and those which refer to the writer's love for Mrs. Meynell particularly have the ring of truth.
The evening of our first day we drove with Christopher and Judy Sakellariadis, who were friends and patients of Norton, to dine at a restaurant on the shores of the Aegean.
So persistent were these attacks that in March of the following year, Woodruff was finally moved to action, and Pike was to learn his first lesson in frontier politics, the subtle art of diversion.
In the next few months of comparative silence, Pike waited patiently until conditions were perfect for a new attack, and then, displaying a remarkable grasp of the subtleties of political infighting, gained from his first bout with Woodruff, he used these changed conditions to excellent advantage.
Although this kind of wholesale objection came at first from some men who were not technically Puritans, still, once the Puritans gained power, they climaxed the affair by passing the infamous ordinance of 1642 which decreed that all `` public stage-plays shall cease and be forborne ''.
Our first impression of the data was that the students were surprisingly orthodox and religiously involved.
But all the reports of this first embassy show that the two Savoyards were the heads of it, for they were the only ones who were empowered to swear for the king that he would abide by the pope's decision and who were allowed to appoint deputies in the event that one was unavoidably absent.
the Low Countries, where the Middle Ages were to last for another two centuries and die out only when Charles the Bold of Burgundy met his first defeat in the fields and forests below the walls of Grandson.
It was the first American war in which the death rate from disease was lower than that from battle, due to the provision of trained medical personnel ( of the 200,000 officers, 42,000 were physicians ), compulsory vaccination, rigorous camp sanitation, and adequate hospital facilities.
There were several men of ninety or more whom I knew first or last, all of whom were still productive and most of whom knew one another as if they had naturally come together at the apex of their lives.
It met a serious rebuff in New Orleans, where the two schools selected for the first moves toward integration were boycotted by white parents.
These microfossils indicate the swamp was `` formed during the Lower Cretaceous period when dinosaurs were at their heyday and when the first flowering plants were just appearing.
were and considered
He considered himself handsome and seemed to think all the girls were after him.
`` As an independent American I considered all who were not for us, and you amongst the rest, as against us, yet be assured that John Jay never ceased to be the friend of Peter Van Schaack ''.
Cried Wright, showing that automobiles were considered to be a danger as early as the 1920's.
He had failed to realize that the Burmese were not really treating him as the important visitor he considered himself.
They had lost twice with the radical Bryan, and were having no part of Hearst, whom they considered more radical than Bryan.
At one time, to most Americans, unless they were fortunate enough to live near a body of navigable water, boats were considered the sole concern of fishermen, rich people, and the United States Navy.
Francesca and Herbert considered themselves violently nonconformist and showed the world they were by filling their Colonial house with contemporary furniture and paintings and other art objects ( expensive, but not necessarily valuable, contemporary things ).
Children whose reading scores were at least one standard deviation below the regression line of each total third-grade school population were considered under-achievers for the purposes of sample selection.
There is also the question of whether the respondent based his answers on factual information and carefully considered judgment, or whether his answers were casual guesses.
In the preparation of the questionnaire the problems noted above were carefully considered, and the structure and phraseology used were designed to minimize the effects of these limitations.
( Lauri Vaska, E. M. Sloane, J. W. DiLuzio ) In the absence of direct evidence to the contrary, decomposition of solvent alcohol and coordination of its fragments to the metal were not considered, following the above heretofore-accepted assumption in preparative coordination chemistry.
If you look at a reading meant for someone else, you will probably see that many of the items could be considered as applicable to you, even when you were not in the picture at all!!
They were a vagabond lot and considered to be shady and undesirable characters.
It seems to be indispensable to the national self-esteem that the Negro be considered either as a kind of ward ( in which case we are told how many Negroes, comparatively, bought Cadillacs last year and how few, comparatively, were lynched ), or as a victim ( in which case we are promised that he will never vote in our assemblies or go to school with our kids ).
We have already seen that odd numbers were considered as being Yang, while the even numbers were Yin, so that the eight outer numbers of the Lo Shu represented these two principles in balanced equilibrium around the axial center.
`` And also, sir, two articles which were considered souvenirs now must be regarded in another light entirely.
But jolly old Uncle Donald would tell her no more than that Bobbie had certainly been considered for the job, but there were factors in a large company which outsiders and even some insiders couldn't understand.
We have considered several of your professionals because they were lamechians and above suspicion.
Lincoln " was remarkably fond of children ", and the Lincolns were not considered to be strict with their children.
Socio-cultural anthropology is considered anthropology proper in most of Europe, and in the parts of the world that were influenced by the European tradition.
The Greek sculptors considered the senses more important, and the proportions were used to unite the sensible with the intellectual.
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