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Page "Category theory" ¶ 54
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two and functors
An arrow between two functors is a natural transformation when it is subject to certain naturality or commutativity conditions.
A natural transformation is a relation between two functors.
Sometimes two quite different constructions yield " the same " result ; this is expressed by a natural isomorphism between the two functors.
The major tool one employs to describe such a situation is called equivalence of categories, which is given by appropriate functors between two categories.
Many texts and articles either use the opposite convention or completely unrelated symbols for these two functors.
These isomorphisms are " natural " in the sense that they define a natural transformation between the two involved functors.
Additionally, every pair of adjoint functors comes equipped with two natural transformations ( generally not isomorphisms ) called the unit and counit.
* 1-cells ( arrows ) between two objects and are the functors from to,
* 2-cells between two 1-cells ( functors ) and are the natural transformations from to.
The computational sledgehammer par excellence is the spectral sequence ; these are essential in the Cartan-Eilenberg and Tohoku approaches where they are needed, for instance, to compute the derived functors of a composition of two functors.
If F and G are two functors from C to Set, then the exponential F < sup > G </ sup > is the functor whose value on the object X of C is given by the set of all natural transformations from ( X ,−) × G to F.
One may therefore view an adjointness relation between two functors as a " very weak form of equivalence ".
Hence, given the information that the identity functors form an equivalence of categories, in this example one still can choose between two natural isomorphisms for each direction.
In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors.
Most constructions that can be carried out in D can also be carried out in D < sup > C </ sup > by performing them " componentwise ", separately for each object in C. For instance, if any two objects X and Y in D have a product X × Y, then any two functors F and G in D < sup > C </ sup > have a product F × G, defined by ( F × G )( c )
* Coequalisers can be large: There are exactly two functors from the category 1 having one object and one identity arrow, to the category 2 with two objects and exactly one non-identity arrow going between them.
The coequaliser of these two functors is the monoid of natural numbers under addition, considered as a one-object category.
If A and B are two abelian categories, we can consider the functor category B < sup > A </ sup > consisting of all functors from A to B.

two and F
When there are only two means to compare, the t-test and the ANOVA F-test are equivalent ; the relation between ANOVA and t is given by F = t < sup > 2 </ sup >.
Fluoride " loses " a pair of valence electrons because the electrons shared in the B — F bond are located in the region of space between the two atomic nuclei and are therefore more distant from the fluoride nucleus than they are in the lone fluoride ion.
Lt. John F. Kennedy, future U. S. President, saves all but two of his crew.
If the domain of F is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals.
The torque caused by the two opposing forces F < sub > g </ sub > and-F < sub > g </ sub > causes a change in the angular momentum L in the direction of that torque ( since torque is the time derivative of angular momentum ).
In Heavy Metal: F. A. K. K. ² and Shadow Man, this tactic was developed further, allowing the player to wield two dissimilar weapons at once, firing each one independently.
The upgrade is being carried out by Embraer and Elbit ; it includes a new avionics suite, a full glass cockpit with three MFDs, HOTAS configuration and a new multimode radar, the Italian Grifo F. The first F-5EM was handed over on September 21, 2005, and it is scheduled at a rate of two aircraft being delivered each month from that date onwards.
This situation arises for all homonuclear diatomic molecules and is particularly a problem for F < sub > 2 </ sub >, where the minimum energy of the curve with molecular orbital theory is still higher in energy than the energy of two F atoms.
The function F is called universal if the following property holds: for every computable function f of a single variable there is a string w such that for all x, F ( w x ) = f ( x ); here w x represents the concatenation of the two strings w and x.
The function F is called prefix-free if there are no two elements p, p ′ in its domain such that p ′ is a proper extension of p. This can be rephrased as: the domain of F is a prefix-free code ( instantaneous code ) on the set of finite binary strings.
The In10did method ( pronounced " intended ") is a ten-key limited chord system that places one key under each finger in order to utilize all of them, however only two are needed for any operations ( excluding the " F " keys, which require three key presses ).
* Input: two custom game controllers, hardwired to the console ( original release ) or removable ( Channel F System II )
These were accounts of two visions, one from Joseph Smith and the other from his nephew, Joseph F. Smith in 1918.
Another product rule for the cross product of two vector fields F and G in three dimensions involves the curl and reads as follows:
To form F with an associative operator, we need only do two additional compositing operations to integrate the new layer C: F = S *( C * D ).
However, associativity can allow us to rewrite F =( A * B )*( C * D ) where there are clearly two operations that do not depend on each other that may be executed in parallel.
A device on the PCI bus or USB is identified by two ID's which consist of 4 numbers and / or letters A to F. The vendor ID identifies the vendor of the device.
The foci of the ellipse are two special points F < sub > 1 </ sub > and F < sub > 2 </ sub > on the ellipse's major axis and are equidistant from the center point.

two and G
Although there are seven other types of annual awards presented by the Academy ( the Irving G. Thalberg Memorial Award, the Jean Hersholt Humanitarian Award, the Gordon E. Sawyer Award, the Scientific and Engineering Award, the Technical Achievement Award, the John A. Bonner Medal of Commendation, and the Student Academy Award ) plus two awards that are not presented annually ( the Special Achievement Award in the form of an Oscar statuette and the Honorary Award that may or may not be in the form of an Oscar statuette ), the best known one is the Academy Award of Merit more popularly known as the Oscar statuette.
Gettysburg was his finest hour, but his relief by Maj. Gen. George G. Meade caused lasting enmity between the two men.
The AVL tree is named after its two Soviet inventors, G. M. Adelson-Velskii and E. M. Landis, who published it in their 1962 paper " An algorithm for the organization of information.
In organic chemistry, compounds composed of carbon and hydrogen are divided into two classes: aromatic compounds, which contain benzene or similar rings of atoms, and aliphatic compounds (; G. aleiphar, fat, oil ), which do not contain those rings.
The questions range from counting ( e. g., the number of graphs on n vertices with k edges ) to structural ( e. g., which graphs contain Hamiltonian cycles ) to algebraic questions ( e. g., given a graph G and two numbers x and y, does the Tutte polynomial T < sub > G </ sub >( x, y ) have a combinatorial interpretation ?).
The commutator of two elements, g and h, of a group G, is the element
Dutch physicists Hendrik B. G. Casimir and Dirk Polder at Philips Research Labs proposed the existence of a force between two polarizable atoms and between such an atom and a conducting plate in 1947, and, after a conversation with Niels Bohr who suggested it had something to do with zero-point energy, Casimir alone formulated the theory predicting a force between neutral conducting plates in 1948 ; the former is called the Casimir-Polder force while the latter is the Casimir effect in the narrow sense.
Cells in G < sub > 0 </ sub > and G < sub > 1 </ sub > usually contain two compiole.
Together, these two Chianti zones produce the largest volume of DOC / G wines in Italy.
The nucleobases are classified into two types: the purines, A and G, being fused five-and six-membered heterocyclic compounds, and the pyrimidines, the six-membered rings C and T. A fifth pyrimidine nucleobase, uracil ( U ), usually takes the place of thymine in RNA and differs from thymine by lacking a methyl group on its ring.
Here, purines form hydrogen bonds to pyrimidines, with A bonding only to T, and C bonding only to G. This arrangement of two nucleotides binding together across the double helix is called a base pair.
The 1977 edition also lists Peter G. Dowdeswell of Earls Barton for drinking two pints of beer from a single vessel in 2. 3 seconds on June 11, 1975 and two litres in 6. 0 seconds on 7 February 1975.
Joseph Smith, Jr. and Frederick G. Williams, two of the Presiding Elders on the committee, were absent, but Oliver Cowdery and Sidney Rigdon were present.
:* Turkish uses a G with a breve ( Ğ ), two letters with an umlaut ( Ö and Ü, representing two rounded front vowels ), two letters with a cedilla ( Ç and Ş, representing the affricate and the fricative ), and also possesses a dotted capital İ ( and a dotless lowercase ı representing a high unrounded back vowel ).
Particularly influential in the development of distributist theory were Catholic authors G. K. Chesterton and Hilaire Belloc, two of distributism's earliest and strongest proponents.
The top of the instrument's fingerboard range is typically near the D two octaves and a fifth above the open pitch of the G string ( G4 ) as shown in the range illustration found at the head of this article.
groups ( G, *) and ( H, ●), denoted by G × H. For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by.

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