Then a presheaf on X is a contravariant functor from O ( X ) to the category of sets, and a sheaf is a presheaf which satisfies the gluing axiom.

Then is an additive category, and equals the-fold power.

Then a C-valued presheaf on X is the same as a contravariant functor from O ( X ) to C. This definition can be generalized to the case when the source category is not of the form O ( X ) for any X ; see presheaf ( category theory ).

Then, the localization of R at S is the initial object of this category.

Then a representation of Q is just a covariant functor from this category to the category of finite dimensional vector spaces.

Then we define the category of pure Chow motives by

Second, suppose η: F → G is a natural transformation from the left exact functor F to the left exact functor G. Then natural transformations R < sup > i </ sup > η: R < sup > i </ sup > F → R < sup > i </ sup > G are induced, and indeed R < sup > i </ sup > becomes a functor from the functor category of all left exact functors from A to B to the full functor category of all functors from A to B.

Then a functor u from C to another category C ' is left ( resp.

Then a finer statement of the s-cobordism theorem is that the isomorphism classes of this category ( up to Cat-isomorphism of h-cobordisms ) are torsors for the respective < ref > Note that identifying the Whitehead groups of the various manifolds requires that one choose base points and a path in W connecting them .</ ref > Whitehead groups Wh ( π ), where

Then T is a right exact functor from Mod-R to the category of abelian groups Ab ( in the case when R is commutative, it is a right exact functor from Mod-R to Mod-R ) and its left derived functors L < sub > n </ sub > T are defined.

Then, in 1963 three municipal districts in the area that were growing in size were merged and promoted to the category of si, meaning city, by order of Law No 1176.

Then, at the WMC 2009, the band picked up a gold standard award, and finished 7th out of 20 bands in their category, ranking them the highest out of the Irish bands to participate in the competition.

Then, the player rolls both the ordinary die and a customized eight-sided " category die " to see how far they move, and what challenge they face.

Then theoretically it may not be clear which category is meant.

Then, leading up to the 2002 Commonwealth Games in Manchester, England, it was thought that Kamil-Yusof was likely to win the bronze medal in the duet category.

Then K ( A ) is a triangulated category ; the distinguished triangles consist of triangles isomorphic to a morphism with its mapping cone ( in the sense of chain complexes ).

Then in 1991, he won the Indian dance competition, followed by the fourth prize in the Modern Jazz category in 1992 World Dance championship, London, at the age of 21.

Then, on 27 February 2004 a law was passed giving French Polynesia the particular designation of overseas country while recalling that it belongs to the category of overseas collectivity.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Then he was appointed Assistant Chief of Staff in charge of the new Operations Division ( which replaced WPD ) under Chief of Staff General George C. Marshall, who spotted talent and promoted accordingly.

Then, once this claim ( expressed in the previous sentence ) is proved, it will suffice to prove " φ is either refutable or satisfiable " only for φ's belonging to the class C. Note also that if φ is provably equivalent to ψ ( i. e., ( φ ≡ ψ ) is provable ), then it is indeed the case that " ψ is either refutable or satisfiable " → " φ is either refutable or satisfiable " ( the soundness theorem is needed to show this ).

Then suppose definition B contains 32 instances of A ; C contains 32 instances of B ; D contains 32 instances of C ; and E contains 32 instances of D. The design now contains 5 definitions ( A through E ) and 128 instances.

Then A is dense in C ( X, R ) if and only if it separates points.

Let U be a functor from D to C, and let X be an object of C. Then the following statements are equivalent:

Then we have a universal morphism for each object in C and D:

* Douglas K. Smith, Robert C. Alexander, Fumbling the Future: How Xerox Invented, Then Ignored, the First Personal Computer ( William Morrow and Company, New York, 1988 ) ISBN 1-58348-266-0

Then in 1952 the Barbados Advocate newspaper polled several prominent Barbadian politicians, lawyers, businessmen, the Speaker of the Barbados House of Assembly and later as first President of the Senate, Sir Theodore Branker, Q. C.

Then, after the medical team has made sure they are on the right track, the tip of the electrode is advanced to the plane of the cingulate where it is heated to 75-90 C. Once the first lesion is created it serves as a center around which several other lesions are created.

Then FF. C followed with Simeia ton kairon, and TXC ( now Artemis / Efthimis ) with their single Terror X Crew in 1995.

Then F and G can be restricted to D < sub > 1 </ sub > and C < sub > 1 </ sub > and yield inverse equivalences of these subcategories.

Then C ( R ) is a subspace of R < sup > R </ sup >.

Then is an open cover of S, but any finite subcollection of has the form of C discussed previously, and thus cannot be an open subcover of S. This contradicts the compactness of S. Hence, every accumulation point of S is in S, so S is closed.

Let K be a closed subset of a compact set T in R < sup > n </ sup > and let C < sub > K </ sub > be an open cover of K. Then

Introduction " The One, The Only R. C., Then Now and Forever " by Peter Straub.

Then that formula must be copied into all of the cells in column C, making sure to change the reference to A1 to a new reference for A2, etc.

Then, the markedness of the ruler comes into play: it is " anchored " at point A, and slided and rotated until one mark is at point C, and one at point D, i. e., CD = AB.

Then around four hundred million B. C., a small continent that was long and thin, collided with proto North America.

Then, in January 1849, Michael C. Nye and William Foster Nye, brothers-in-law to Covillaud's wife, Mary Murphy, bought the other half of the Cordua ranch.

Then in 1943 he received orders transferring him to the Secret Intelligence Branch of the Office of Strategic Services ( OSS ) in Washington, D. C.

Then, the bill must be approved by the Congress, a special joint session of both houses ; alternatively, the bill can be submitted to a referendum.

Then the centroids are recalculated for the new clusters, and algorithm repeated by alternate application of these two steps until convergence, which is obtained when the points no longer switch clusters ( or alternatively centroids are no longer changed ).

Then the centroids are recalculated for the new clusters, and algorithm repeated by alternate application of these two steps until convergence, which is obtained when the points no longer switch clusters ( or alternatively centroids are no longer changed ).