Falling somewhere in a category between Einstein's theory and sand fleas -- difficult to see but undeniably there, nevertheless -- is the tropical green `` city '' of Islandia, a string of offshore islands that has almost no residents, limited access and an unlimited future.

Because there is no canonical well-ordering of all sets, a construction that relies on a well-ordering may not produce a canonical result, even if a canonical result is desired ( as is often the case in category theory ).

There are several results in category theory which invoke the axiom of choice for their proof.

On the other hand, other foundational descriptions of category theory are considerably stronger, and an identical category-theoretic statement of choice may be stronger than the standard formulation, à la class theory, mentioned above.

The most general setting in which these words have meaning is an abstract branch of mathematics called category theory.

In category theory, an automorphism is an endomorphism ( i. e. a morphism from an object to itself ) which is also an isomorphism ( in the categorical sense of the word ).

This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren't necessarily sets.

His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory into its foundations.

His emphasis on the role of universal properties brought category theory into the mainstream as an important organizing principle.

During this time he had officially as students Michel Demazure ( who worked on SGA3, on group schemes ), Luc Illusie ( cotangent complex ), Michel Raynaud, Jean-Louis Verdier ( cofounder of the derived category theory ) and Pierre Deligne.

Alexander Grothendieck's work during the ` Golden Age ' period at IHÉS established several unifying themes in algebraic geometry, number theory, topology, category theory and complex analysis.

He gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed, to protest against the Vietnam War ( The Life and Work of Alexander Grothendieck, American Mathematical Monthly, vol.

The choice between the two definitions usually matters only in very formal contexts, like category theory.

In category theory, n-ary functions generalise to n-ary morphisms in a multicategory.

* Cone ( category theory ), a family of morphisms resembling a geometric cone

Notable theories falling into this category include the Holonomic brain theory of Karl Pribram and David Bohm, and the Orch-OR theory formulated by Stuart Hameroff and Roger Penrose.

In applying the elemental theory to beings that function on a cosmic scale ( e. g. Yog-Sothoth ) some authors created a separate category termed aethyr.

Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

However it is important to note that the objects of a category need not be sets nor the arrows functions ; any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category, and all the results of category theory will apply to it.

Given a concrete category ( C, U ) and a cardinal number N, let U < sup > N </ sup > be the functor C → Set determined by U < sup > N </ sup >( c ) = ( U ( c ))< sup > N </ sup >.

The class of all N-ary predicates and N-ary operations of a concrete category ( C, U ), with N ranging over the class of all cardinal numbers, forms a large signature.

Examples of bishops belonging to this category are coadjutor bishops, auxiliary bishops, bishops emeriti, vicars apostolic, nuncios, superiors of departments in the Roman Curia, and cardinal bishops of suburbicarian dioceses ( since they are not in charge of the suburbicarian dioceses ).

* The category Set of all sets has the subcategory of all cardinal numbers as a skeleton.

* The category K-Vect of all vector spaces over a fixed field has the subcategory consisting of all powers, where n is any cardinal number, as a skeleton ; the maps are exactly the n × m matrices with entries in K.

Dialling these numbers will cause the local switch to announce which carrier your calls are being routed through for a specific category of calls.

However, there is an argument that nuclear and biological weapons do not belong in the same category as chemical and " dirty bomb " radiological weapons, which have limited destructive potential ( and close to none, as far as property is concerned ), whereas nuclear and biological weapons have the unique ability to kill large numbers of people with very small amounts of material, and thus could be said to belong in a class by themselves.

Conversely, given any ring, we can form a category by taking objects A < sub > n </ sub > indexed by the set of natural numbers ( including zero ) and letting the hom-set of morphisms from to be the set of-by-matrices over, and where composition is given by matrix multiplication.

The product over the set of all prime numbers of the restriction of these functors to the category of torsion groups, is a faithful functor from the category of torsion groups to the product over all prime numbers of the categories of p-torsion groups.

Conway's surreal numbers fall into category 2.

From these wind speed numbers, qualitative descriptions of damage were made for each category of the Fujita scale, and then these descriptions were used to classify tornadoes.

Furthermore, the original wind speed numbers have since been found to be higher than the actual wind speeds required to incur the damage described at each category.

For each selected value of a and b, the numbers of coincidences in each category ( N < sub >++</ sub >, N < sub >--</ sub >, N < sub >+-</ sub > and N < sub >-+</ sub >) are recorded.

* The space R of real numbers with the usual topology, is a Baire space, and so is of second category in itself.

The rational numbers are of first category and the irrational numbers are of second category in R.

There is a triple equivalence of categories between the category of smooth projective algebraic curves over the complex numbers, the category of compact Riemann surfaces, and the category of complex algebraic function fields, so that in studying these subjects we are in a sense studying the same thing.

If K is the field of real numbers, then the category is also known as Vec.

In category theory, given any family P < sub > n </ sub > of invertible n-by-n matrices defining a similarity transformation for all rectangular matrices sending the m-by-n matrix A into P < sub > m </ sub >< sup >− 1 </ sup > AP < sub > n </ sub >, the family defines a functor that is an automorphism of the category of all matrices, having as objects the natural numbers and morphisms from n to m the m-by-n matrices composed via matrix multiplication.

Abjads differ from abugidas, another category invented by Daniels, in that in abjads, the vowel sound is implied by phonology, and where vowel marks exist for the system, such as nikkud for Hebrew and harakāt for Arabic, their use is optional and not the dominant ( or literate ) form.

The class of all unital associative R-algebras together with algebra homomorphisms between them form a category, sometimes denoted R-Alg.

Sources disagree, as may differing states ' laws, as to what category of plea the Alford plea falls under: Some sources state that the Alford guilty plea is a form of nolo contendere, where the defendant in the case states " no contest " to the factual matter of the case as given in the charges outlined by the prosecution.

The collection of all functors C → D form the objects of a category: the functor category.

Fats form a category of lipid, distinguished from other lipids by their chemical structure and physical properties.

Being a form of visual art, it might also be said that many graffiti artists still fall in the category of the introverted archetypal artist.

Later, the music industry merged hillbilly music, Western Swing, and Cowboy music, to form the current category C & W, Country and Western.

" As mentioned previously, Vigoda-Gadot categorizes these behaviors in a separate category of OCBs as " compulsory OCBs " or CCBs, which he suggests are a form of " abusive supervision " and will result in poorer organizational performance, similar to what has been seen in other research on abusive supervision and coercive persuasion.

Modern instruments with 15 or fewer strings are generally more closely modeled on traditional shapes, and form a category of instrument known as small kantele, in contrast to the modern concert kantele.

Though this was the form of music most commonly considered " ragtime " in its day, many people today prefer to put it in the " popular music " category.

If the appliance is one whose purpose is for light or heat ( such as an incandescent lightbulb or electric oven ) then the lighting or heating elements may be considered as a type of fire ; if so, then turning them on constitutes both " lighting a fire " ( category 37 ) and " cooking " ( a form of baking, category 11 ), and turning them off would be " extinguishing a fire " ( category 36 ).

In Ferdydurke ( his first novel, published in autumn 1937, the date on the cover 1938 ) discusses form as a universal category which was understood both in the philosophical, sociological, and aesthetic sense.

Topological groups, together with their homomorphisms, form a category.

In this context, the term lexical category applies only to those parts of speech and their phrasal counterparts that form open classes and have full semantic content.

One then restricts the maps to only those maps that commute with these isomorphism ( restricts to the naturalizer of η ), in other words, restrict to the maps that do not change the bilinear form: The resulting category, with objects finite-dimensional vector spaces with a nondegenerate bilinear form, and maps linear transforms that respect the bilinear form, by construction has a natural isomorphism from the identity to the dual ( each space has an isomorphism to its dual, and the maps in the category are required to commute ).