Let wel: Die datums van die Afrikaanse wikipedia ondergaan tans heelwat veranderings. Hier is skakels na twee weergawes van die dae in geskiedenis. Die skakels in die boonste weergawe wys na die artikels wat uiteindelik gebruik sal word. Werk asb dae in die ander twee formate by sodat ons kan standardiseer. Daar is afgespreek dat 1 Junie as templaat gebruik sal word. ----

Eksterne Engelse skakels

- [http://www.on-this-day.com/ on-this-day.com]
- [http://www.historychannel.com/today/ The history channel: this day in history]
- [http://www.todayinsci.com/ Today in science] Kategorie:Lys ja:365日 ko:366일

Geskiedenis

Die geskiedenis is 'n gesistimatiseerde, chronologiese studie van die verlede met die doel om historiese gebeure so feitlik akkuraat moontlik weer te gee.
- Mousteriaans (Europa-Asië) 150,000 v.C. - 35,000 v.C.
- Middelsteentyd (Afrika) 150,000 v.C. - 35,000 v.C.
- Bo-Paleolitikum (Europa) 35,000 v.C. - 12,000 v..
- Laat-Steentyd (Afrika) 25,000 v.C - 2000 v.C.
- Mesolitikum (Europa) 12,000 v.C. 8000 v.C.
- Neolitikum (Midde-Ooste) 12,000 v.C. - 6000 v.C.
- Moderne mens6000 v.C. - 3000 v.C
- Bronstyd 3000 v.C - 750 v.C.
- Ystertyd 750 v.C.
- Antieke geskiedenis
- Middeleeue
- 15de Eeu
- Renaissance
- 16de Eeu
- 17de Eeu
- 18de Eeu
- 19de Eeu
- 20ste Eeu
- 21ste Eeu
- Eeue Artikels met plaaslike inhoud:
- Geskiedenis van Brakpan Kategorie:Geskiedenis fiu-vro:Aolugu ja:歴史 ko:역사 ms:Sejarah simple:History th:ประวัติศาสตร์ zh-min-nan:Le̍k-sú

18de eeu

Die Kaap was vir die grootste deel van die 18de eeu 'n kolonie van Nederland. Eeue

17de eeu

18de eeu

19de eeu

1700	1701	1702	1703	1704	1705	1706	1707	1708	1709
1710	1711	1712	1713	1714	1715	1716	1717	1718	1719
1720	1721	1722	1723	1724	1725	1726	1727	1728	1729
1730	1731	1732	1733	1734	1735	1736	1737	1738	1739
1740	1741	1742	1743	1744	1745	1746	1747	1748	1749
1750	1751	1752	1753	1754	1755	1756	1757	1758	1759
1760	1761	1762	1763	1764	1765	1766	1767	1768	1769
1770	1771	1772	1773	1774	1775	1776	1777	1778	1779
1780	1781	1782	1783	1784	1785	1786	1787	1788	1789
1790	1791	1792	1793	1794	1795	1796	1797	1798	1799
1800	1801	1802	1803	1804	1805	1806	1807	1808	1809

Kategorie:18de eeu ja:18世紀 ko:18세기

19de eeu

Eeue:

18de eeu

19de eeu

20ste eeu

1801	1802	1803	1804	1805	1806	1807	1808	1809	1810
1811	1812	1813	1814	1815	1816	1817	1818	1819	1820
1821	1822	1823	1824	1825	1826	1827	1828	1829	1830
1831	1832	1833	1834	1835	1836	1837	1838	1839	1840
1841	1842	1843	1844	1845	1846	1847	1848	1849	1850
1851	1852	1853	1854	1855	1856	1857	1858	1859	1860
1861	1862	1863	1864	1865	1866	1867	1868	1869	1870
1871	1872	1873	1874	1875	1876	1877	1878	1879	1880
1881	1882	1883	1884	1885	1886	1887	1888	1889	1890
1891	1892	1893	1894	1895	1896	1897	1898	1899	1900

Kategorie:19de eeu ja:19世紀 ko:19세기 simple:19th century th:คริสต์ศตวรรษที่ 19 zh-min-nan:19 sè-kí

20ste eeu

Eeue:

19de eeu

20ste eeu

21ste eeu

1901	1902	1903	1904	1905	1906	1907	1908	1909	1910
1911	1912	1913	1914	1915	1916	1917	1918	1919	1920
1921	1922	1923	1924	1925	1926	1927	1928	1929	1930
1931	1932	1933	1934	1935	1936	1937	1938	1939	1940
1941	1942	1943	1944	1945	1946	1947	1948	1949	1950
1951	1952	1953	1954	1955	1956	1957	1958	1959	1960
1961	1962	1963	1964	1965	1966	1967	1968	1969	1970
1971	1972	1973	1974	1975	1976	1977	1978	1979	1980
1981	1982	1983	1984	1985	1986	1987	1988	1989	1990
1991	1992	1993	1994	1995	1996	1997	1998	1999	2000

Kategorie:20ste eeu ja:20世紀 ko:20세기 simple:20th century

1812

Gebeure

- Maart 26 - 'n Aardbewing verwoes Caracas, Venezuela
- April 30 - Louisiana word die 18de V.S. staat.
- Oktober-Desember - Napoleon se terugval van Moskou.

Geboortes

- Junie 9 - Johann Gottfried Galle, Duitse astronoom (+ 1910)

Sterftes

- Oktober 13 - Generaal Isaac Brock ---- Dae | Eeue | Geskiedenis

1812

Kategorie:19de eeu ko:1812년 simple:1812

1815

Gebeure

- 3 Februarie - Die eerste moderne kaasfabriek word in Switserland geopen
- 7 April - Tombora bars uit en 92 000 mense sterf.
- 20 Maart - Napoléon I keer terug uit Elba. Begin van die Honderd Dae
- 22 Junie - Napoléon I word weer gedwing af te tree
- Genève word 'n Switserse stad en die hoofstad van die nuwe gelyknamige kanton.

Geboortes

- 7 Julie - Théodore Hersart de la Villemarqué (1815-1895), Bretonse skrywer

Sterftes

- ---- Dae | Eeue | Geskiedenis

1815

Kategorie:19de eeu ko:1815년 simple:1815 th:พ.ศ. 2358

1816

Gebeure

- Argentinië verklaar sy onafhanklikheid van Spanje

Geboortes

Sterftes

- Thomas Girten -Waterverfskilder ---- Dae | Eeue | Geskiedenis

1816

Kategorie:19de eeu ko:1816년 simple:1816

1817

Gebeure

- Die element Litium word ontdek deur Johann Arfvedson.
- 22 April- Onder leiding van die profeet-stamhoof Makana val die Xhosa Grahamstad aan.

Geboortes

Sterftes

- ---- Dae | Eeue | Geskiedenis

1817

Kategorie:19de eeu ko:1817년 simple:1817

Triangulated category

In mathematics, a triangulated category is a category satisfying some axioms that are based on the properties of a derived category. Some examples are the homotopy category of spectra, and the derived category of an abelian category.

Introduction

The notion of a derived category was introduced in his thesis by Verdier, based on some ideas of Grothendieck. He also defined the notion of a triangulated category, by noting that a derived category had some special "triangles" and writing down axioms for the basic properties of these triangles.

Definition

A translation functor on a category D is an automorphism T from D to D. The image of X under Tⁿ is usually written as X[n]. A triangle (X, Y, Z, u, v, w) is a set of 3 objects X, Y, and Z, together with morphisms u from X to Y, v from Y to Z and w from Z to X[1]. If (X, Y, Z, u, v, w) is a triangle then the rotated triangle is (Z[−1],X, Y, −w[−1], u, v). A triangulated category is an additive category D with a translation functor and a class of distinguished triangles, satisfying the following properties.
- Any triangle isomorphic to a distinguished triangle is distinguished.
- The rotation of a distinguished triangle is distinguished.
- Any morphism can be completed to a distinguished triangle. (The third object in the triangle is called a mapping cone of the morphism.)
- The identity morphism of an object can be completed to a distinguished triangle with the third object 0.
- Given a map between two morphisms, there is a morphism between their mapping cones that makes "everything commute". So far all the axioms are reasonably natural and obvious. The final axiom, sometimes called the octahedral axiom, is notorious for being incomprehensible.
- Suppose we have morphisms from X to Y and Y to Z, so that we also have a composed morphism from X to Z. Form distinguished triangles for each of these three morphisms. The octahedral axiom states (roughly) that the three mapping cones can be made into the vertices of a distinguished triangle so that "everything commutes".

Comments on the axioms

The axioms above have seemed rather artificial. It is strongly suspected by experts that triangulated categories are not really the "correct" concept. They do however seem to work adequately in practice; and there is no current and convincing replacement. The last axiom is called the octahedral axiom, because drawing all the objects and morphisms gives the skeleton of an octahedron, four of whose faces are distinguished triangles. There seems to be no really satisfactory way to draw everything in two dimensions (see the book of Kashiwara and Schapira for details). The axioms above are not independent. In particular, the axiom implying the existence of a morphism between mapping cones can be deduced from the others. The mapping cone of a morphism is unique up to a non-unique isomorphism. This non-uniqueness is a potential source of errors. In particular the mapping cone of a morphism does not in general depend functorially on the morphism. Pierre Deligne has found further axioms that could be added, which are generalizations of (and even more complicated than) the octahedral axiom.

Examples

If A is an abelian category, then the category Kom(A) has as objects all complexes of objects of A, and as morphisms the homotopy classes of morphisms of complexes. Then Kom(A) is a triangulated category, where the distinguished triangles consist of triangles isomorphic to a morphism with its mapping cone (in the sense of chain complexes). Variations: use complexes that are bounded on the left, or on the right, or on both sides. A localization of a triangulated category is also triangulated. In particular the derived category of A, which is a localization of Kom(A), is triangulated.

t-structure and cores

In the derived category D of an Abelian category A, there are natural subcategories

D^

and

D^

, consisting of complexes whose cohomology vanishes in degrees larger then n or smaller than m. These have the following properties:
-

D^=D^[-n]

D^=D^[-n]

Hom(D^,D^)=0

D^\subset D^

D^\subset D^

- Every object Y can be embedded in a distinguished triangle (X, Y, Z, u, v, w) with $X\in D^$ and $Z\in D^$ . A t-structure on a triangulated category consists of full subcategories

D^

and

D^

satisfying the conditions above. The letter t stands for "truncation". The core of a t-structure is the category

D^\cap D^

. It is an abelian category. (A triangulated category is additive but is not usually abelian). The core of a t-structure of the derived category of A can be thought of as a sort of twisted version of A, which sometimes has better properties. For example, the category of perverse sheaves is the core of a certain (quite complicated) t-structure on the derived category of the category of sheaves. Over a space with singularities, the category of perverse sheaves is similar to the category of sheaves but behaves better.

References

Part of Verdier's thesis is reprinted in
- [http://modular.fas.harvard.edu/sga/sga/4.5/index.html SGA 4 1/2] ISBN 038708066X. Two textbooks that discuss triangulated categories are:
- Methods of Homological Algebra by Sergei I. Gelfand, Yuri I. Manin ISBN 3540435832
- Homological Algebra by S. I. Gelfand, Yu. I. Manin ISBN 3540653783 Another standard reference is:
- Faisceaux pervers, Beilinson, Bernstein, and Deligne. Astérisque 100.
- Sheaves on Manifolds (1990) M. Kashiwara and P. Schapira (concise introduction, and applications) Category:Homological algebra

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