SUMMARY / RELATED TOPICS

In category theory, an epimorphism is a morphism f: X → Y, right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: Y → Z, g 1 ∘ f = g 2 ∘ f ⟹ g 1 = g 2. Epimorphisms are categorical analogues of surjective functions, but it may not coincide in all contexts; the dual of an epimorphism is a monomorphism. Many authors in abstract algebra and universal algebra define an epimorphism as an onto or surjective homomorphism; every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of category theory given above. For more on this, see § Terminology below; every morphism in a concrete category whose underlying function is surjective is an epimorphism. In many concrete categories of interest the converse is true. For example, in the following categories, the epimorphisms are those morphisms that are surjective on the underlying sets: Set: sets and functions.

To prove that every epimorphism f: X → Y in Set is surjective, we compose it with both the characteristic function g1: Y → of the image f and the map g2: Y →, constant 1. Rel: sets with binary relations and relation-preserving functions. Here we can use the same proof as for Set, equipping with the full relation ×. Pos: ordered sets and monotone functions. If f: → is not surjective, pick y0 in Y \ f and let g1: Y → be the characteristic function of and g2: Y → the characteristic function of; these maps are monotone if is given the standard ordering 0 < 1. Grp: groups and group homomorphisms; the result that every epimorphism in Grp is surjective is due to Otto Schreier. FinGrp: finite groups and group homomorphisms. Due to Schreier. Ab: abelian groups and group homomorphisms. K-Vect: vector spaces over a field K and K-linear transformations. Mod-R: right modules over a ring R and module homomorphisms; this generalizes the two previous examples. Top: topological spaces and continuous functions. To prove that every epimorphism in Top is surjective, we proceed as in Set, giving the indiscrete topology, which ensures that all considered maps are continuous.

HComp: compact Hausdorff spaces and continuous functions. If f: X → Y is not surjective, let y ∈ Y − fX. Since fX is closed, by Urysohn's Lemma there is a continuous function g1:Y → such that g1 is 0 on fX and 1 on y. We compose f with both g1 and the zero function g2: Y →. However, there are many concrete categories of interest where epimorphisms fail to be surjective. A few examples are: In the category of monoids, the inclusion map N → Z is a non-surjective epimorphism. To see this, suppose that g1 and g2 are two distinct maps from Z to some monoid M. For some n in Z, g1 ≠ g2, so g1 ≠ g2. Either n or −n is in N, so the restrictions of g1 and g2 to N are unequal. In the category of algebras over commutative ring R, take R → R, where R is the group ring of the group G and the morphism is induced by the inclusion N → Z as in the previous example; this follows from the observation that 1 generates the algebra R, the inverse of the element represented by n in Z is just the element represented by −n.

Thus any homomorphism from R is uniquely determined by its value on the element represented by 1 of Z. In the category of rings, the inclusion map Z → Q is a non-surjective epimorphism. A similar argument shows that the natural ring homomorphism from any commutative ring R to any one of its localizations is an epimorphism. In the category of commutative rings, a finitely generated homomorphism of rings f: R → S is an epimorphism if and only if for all prime ideals P of R, the ideal Q generated by f is either S or is prime, if Q is not S, the induced map Frac → Frac is an isomorphism. In the category of Hausdorff spaces, the epimorphisms are the continuous functions with dense images. For example, the inclusion map; the above differs from the case of monomorphisms where it is more true that monomorphisms are those whose underlying functions are injective. As for examples of epimorphisms in non-concrete categories: If a monoid or ring is considered as a category with a single object the epimorphisms are the right-cancellable elements.

If a directed graph is considered as a category (objects are the vertices, morphisms are the paths, composition of morphisms is the concatenation

Mary Theresa Ledóchowska, was a Polish Roman Catholic Religious Sister and missionary, who founded the Missionary Sisters of St. Peter Claver, dedicated to service in Africa, she has been beatified by the Catholic Church. Mary Theresa was the eldest of seven children. Members of the Polish nobility and her siblings - including Wlodimir Ledóchowski, Superior General of the Society of Jesus, St. Ursula Ledóchowska and Ignacy Kazimierz Ledóchowski - were born in Loosdorf on the Lower Austrian estate of their parents, Count Antoni Halka-Ledóchowski and his wife, Countess Josephine Salis-Zizers; as a young girl, Ledóchowska displayed talent as a writer. She loved society life and would dress in her finest attire to attend the balls which were part of the family's social life, she was educated by the Sisters of Loreto, in her school life displayed a devotion to the Catholic faith, instilled in her by the Sisters and her family. This social life continued until both she and father contracted smallpox in 1885.

She was nursed back to health. After his death, their uncle, Cardinal Mieczysław Halka Ledóchowski, took charge of their care. During her convalescence, Ledóchowska had begun to reflect on the meaninglessness of her life, her sister Julia claimed that Mary Theresa had made a vow of virginity during that period. From 1885 to 1890, in order to help her family, which had fallen into economic difficulties after the death of the Count, through the connections of her uncle, she obtained the position of lady-in-waiting to Princess Alice of Parma, the Grand Duchess of Tuscany, at the imperial palace in Salzburg. While living at court, she resumed her participation in the social functions there, again attending concerts and balls. At the same time, she maintained a strict commitment to the practice of the faith. Under the guidance of a Franciscan friar who served as spiritual director to both the princess and her, she was admitted to the Third Order of St. Francis, following its spirituality, with an emphasis on the Passion of Christ.

Shortly after her arrival, two members of the Franciscan Missionaries of Mary came to the court seeking financial help for their missionary work. Ledóchowska listened intently as the two Religious Sisters shared their experiences of working with lepers in Madagascar, she took no action, however. The following year two other Sisters of the same congregation arrived at the court with the same purpose; this time their accounts of work in the overseas missions sparked a desire in her to commit herself to this work. Her interest in the missions increased when she read a pamphlet on Cardinal Charles Lavigerie's anti-slavery campaign. Pope Leo XIII had entrusted the evangelization of Africa to Lavigerie, she began to publicize his cause. In 1889 Princess Alice arranged for Ledóchowska to meet Lavigerie, who encouraged her to establish committees throughout the Austrian Empire in order to combat slavery, she proceeded to do so, starting them in Salzburg, San Ippolito and Krakow. She began to use her literary talent to oppose slavery and to protest the inhuman treatment of women prevalent in Africa.

She wrote a novel entitled Zaida to show the terrible consequences of slavery for women. At the same time, she began a mission page in a Catholic periodical; these mission features, called Echo From Africa, were based on letters from missionaries serving in Africa. The page of letters evolved into a monthly magazine, which made its debut in 1889, with her as the publisher though this was still unheard of in the 19th century; the magazine soon became a full-time job, Emperor Franz Joseph released Ledóchowska from her duties at the imperial court in 1891 so that she could devote all of her time and energy to the missions. Ledóchowska left the court and took up residence with a community of the Daughters of Charity of St. Vincent de Paul in Salzburg. Struggling to find financial support for her project, she lived in near poverty, surviving on a prebend granted to her by Empress Elisabeth of Austria, who named her a canoness; as the work expanded, Ledóchowska's vision took shape gradually. She began to recruit other women as "auxiliary missionaries", whom she organized in 1894 as the Sodality of St. Peter Claver for the African Missions and the Liberation of Slaves, an association of laywomen.

She placed her work under the patronage of the Spanish Jesuit missionary, Peter Claver, who spent a lifetime in service to the enslaved African people brought to South America, which earned him the title of "Apostle to the Slaves". The society's goals were to publicize the needs of the missions in Africa and to raise funds for this work. On April 29, 1894, Pope Leo XIII formally blessed the enterprise, approving the St. Peter Claver Sodality as a Pious Association of the Faithful on 19 April 1894. Out of this society, the auxiliary missionaries developed into a religious congregation. On 8 September 1897, she and her first companions professed their permanent religious vows as Missionary Sisters of St. Peter Claver, they adopted the Jesuit Constitutions for their own use, to combine the elements of contemplation with an active life of service. The new foundress traveled throughout Europe to raise support for the missions which she and her Sisters served, addressing various conferences and international gatherings of Catholics to speak on the needs of the missions.

One need of which she became aware was the lack of printed resources in the native African languages. Her publishing house began to produce books to answer this need, ranging from Bibles and dictionaries

Marianne Eigenheer was a Swiss artist. She was active both as an academic and as a working artist who displayed works in Europe and the United States, her work was done on small and large canvasses, including some wall drawings. She resided in London. Before she turned her attention to art, Eigenheer was set on a career path to become a pianist, she began receiving piano lessons as a child. However, she herself wanted to become a composer, not possible at the time. Instead, she began painting after finishing school. In 1964, she completed her teaching certification in Aarau. In 1970, she completed an art education diploma at the Lucerne School of Art and Design and began working as an artist. From 1973–1976, she studied art history and psychology at the Zurich University. From 1971–1988, she worked as a research assistant at the Museum of Art Lucerne with Jean-Christophe Ammann and with Martin Kunz. In 1987, she was granted an artist residency in Tokyo and, in 2001/2002, the residency of the Landis & Gyr Foundation in London.

She was a lecturer and art professor at various art academies and colleges:1994–96 a teaching position at the Kunstpädagogisches Institut at Goethe University Frankfurt. From 2003 to 2009 she was Professor and Director of the Institute for Curatorship and Education at the Edinburgh College of Art and was granted an honorary professorship at ECA in 2009. From 2011 to 2013, she was a tutor at the Royal College of Art in London. Marianne Eigenheer lived in London. Marianne Eigenheer’s work has its roots in drawing, in which the moving line precedes the flat, pictorial dimension, her drawings are gesticulating, free line works on paper, in which the spontaneous, subconscious action is combined with conscious formal and content-related decision-making. In the 1980s, the important series Bilder zur Lage, of postcard-sized drawings, was created, they are semiabstract forms and “mergers of different beings,” which attain comic-like, erotic associations, symbolic elements and a great dynamic. At this time, she painted large canvases, as part of the series Misere des Herzens, with animal silhouettes, human figures and hybrid beings, who seem to float or lie before monochrome backgrounds.

Eigenheer describes the creation of these pictures as follows: At one point in my life, still in Lucerne, I had a big studio and painted large pictures, out of lines, out of this tangle of lines, there first, animals emerged, to my own surprise. But these animals, or rather animal forms, were not important to me as animals, but rather, they represented my physical state, and so, as animals always are: A reflection of one’s own condition. Came works on canvas and wall drawings, which were characterized by the use of red and gold, consisting of borders and amorphous, semiabstract forms and form developments; these include the wall drawings Das Buch der 5 Ringe von Mushahi, 1991, at Kiel Central Station or Les Guédés dansent toujours, 2012. Her works always develop in "blocks," as she calls them. If a series is exhausted, she begins with materiality or a different format. Since 2011, she lays several pages next to one another and allows the pen to wander across the boundaries of the paper. In these direct drawings, the associations to the body, the body “language”, the intuitive motion in space become important.

Eigenheer views herself in continuity to other female artistic positions, as she emphasizes:These are the effects of my music studies, the motion in space and time. I can make things visible. In that, I feel related to Louise Bourgeois or Nancy Spero. Maybe this is. I anchor myself to myself again through these drawings. Parallel to drawing and painting, photography has a major importance in her works, although these images were used for the personal archive to “fix the perspective" and shown. Since 2015, Eigenheer has produced photographs that reference a collection of lace and embroidery patterns from her grandmother’s embroidery store in Lucerne. 2012: Marianne Eigenheer, sic!–Raum für Kunst, Lucerne January 7, 2011 – February 11, 2012. 2012: Das Esszimmer, Bonn 1977: Marianne Eigenheer, Kunstmuseum Luzern, March 27 –May 1, 1977. Jean-Christophe Ammann: Marianne Eigenheer, Ausstellungskatalog, Kunstmuseum Luzern, March 27 – May 1, 1977. Marianne Eigenheer: Journal Galerie E+F Schneider, Le Landeron, No 26, 1981.

Marianne Eigenheer. Kunstverein Schaffhausen, Museum zu Allerheiligen Schaffhausen, January 12– February 10, 1985 Armin Wildermuth: Images of Change. Marianne Eigenheer's Recent Work. In: Marianne Eigenheer. Kunstverein Schaffhausen, Museum zu Allerheiligen Schaffhausen, 1985. Annemarie Monteil: Lebensspur und Farbwildwechsel. in: Künstler. Kritisches Lexikon der Gegenwartskunst. Weltkunst, Munich 1990. Marianne Eigenheer, Stephan Berg, Kunstverein Freiburg e. V.: Marianne Eigenheer. Wandarbeiten 1991/92. Waldkircher Verlagsgesellschaft, Freiburg 1991. Galerie Marianne Grob, Berlin: Marianne Eigenheer. Berlin 1996. Marianne Eigenheer, Hans Ulrich Obrist: Gespräch Marianne Eigenheer und Hans Ulrich Obrist. In: sic! Raum für Kunst Luzern: LACK. Fliegende Tiere, Körper und Sterne am Himmel. No 3, Maniac Press, Lucerne 2012. Yasimin

Hazelbury Bryan is a village and civil parish in the county of Dorset in southern England. It is situated in the Blackmore Vale 5 miles southwest of the small town of Sturminster Newton; the parish includes the hamlets of Droop, Parkgate, Pleck and Woodrow. In the 2011 census the parish had 480 dwellings, 454 households and a population of 1,059. In 1201 the village name was spelled Hasebere; the name is derived from the Old English hæsel and bearu, meaning a hazel grove or wood, plus the manorial name of the Bryene or de Bryan family. The original settlement in the village is the hamlet of Droop, the location of the parish church; the church dates from the 15th century, though it is the third building to have existed on the site. The other hamlets in the village are believed to have originated as a result of the Black Death twice afflicting the original settlement, the villagers responding by burning it and rebuilding several smaller settlements on higher ground nearby; the geology of the parish consists of Oxford clay in the northwest, a band of Corallian limestone and sand running from southwest to northeast, Kimmeridge clay in the southeast.

Drainage consists of several small streams flowing northwest and north into the River Lydden and northeast into the River Stour. Hazelbury Bryan civil parish is the most populous parish within the electoral ward of Lydden Vale, which extends from Fifehead Neville parish in the north to Mappowder in the south and Glanvilles Wootton in the west; the population of the ward at the 2011 census was 1,967. Hazelbury Bryan Parish Council

Wilhelm Herget was a German Luftwaffe military aviator during World War II, a night fighter ace credited with 73—15 daytime and 58 nighttime—enemy aircraft shot down in over 700 combat missions. The majority of his victories were claimed over the Western Front in Defense of the Reich missions against the Royal Air Force's Bomber Command. Born in Stuttgart, Herget grew up in the grew up in the German Empire, Weimar Republic and Nazi Germany. Following graduation from school and a vocational education in printing, he joined the military service in the Luftwaffe. Herget flew his first combat missions in the 1939 Invasion of Poland and in 1940, in the Battle of France and Britain. In May 1941, he participated in the Anglo-Iraqi War. In November 1941, Herget transferred to the night fighter force serving with Nachtjagdgeschwader 1. In September 1942, Herget became group commander of I. Gruppe of Nachtjagdgeschwader 4, a position he held until December 1944. Following his 63rd aerial victory, he was awarded the Knight's Cross of the Iron Cross with Oak Leaves on 11 April 1944.

The Knight's Cross, its variants were the highest awards in the military and paramilitary forces of Nazi Germany during World War II. Herget flew his last combat missions with Jagdverband 44, a Messerschmitt Me 262 jet fighter unit, in 1945. After the war, he worked in publishing. Herget died on 27 March 1974 in Stuttgart. Herget was born on 30 June 1910 in Stuttgart in the Kingdom of Württemberg of the German Empire, the son of a printer. After graduation from school, he completed his Meister training. Herget served in the Sturmabteilung as Rottenführer. In parallel, he served in the military reserve force with an Aufklärungsgruppe. In August 1939, Herget was posted to 6. Staffel of Zerstörergeschwader 76 flying a Messerschmitt Bf 110 heavy fighter. On Friday 1 September 1939, German forces invaded Poland starting World War II in Europe. Herget flew his first combat mission with ZG 76 during the invasion and was promoted to Leutnant der Reserve on 25 October 1939. In May 1940, he fought in the Battle of France and that year in the Battle of Britain.

Herget, due to his short built, had to fly a customized Bf 110 with wooden blocks attached to the rudder pedals in order to reach them. He claimed three Supermarine Spitfire fighters shot down in May 1940 and a Curtiss P-36 Hawk fighter in June and was awarded the Iron Cross 2nd Class. On 30 August 1940, Herget claimed a Spitfire on the next day. On 1 September, he claimed another on 2 September. In May 1941, Herget was transferred to Sonderkommando Junck referred to as Fliegerführer Irak, a Luftwaffe task force under the command of Oberst Werner Junck which participated in the Anglo-Iraqi War. Herget was promoted to Oberleutnant der Reserve on 1 November 1941 and transferred to the night fighter force. There he was posted to 7. Staffel of Nachtjagdgeschwader 3. On 15 January 1942, 7./NJG 3 was redesignated and became the 4. Staffel of Nachtjagdgeschwader 1. Herget was awarded the German Cross in Gold on 7 February 1942. Herget claimed his first nocturnal victory on the night of 5/6 April 1942. On 1 May 1942, Herget was appointed Staffelkapitän of 9.

Staffel of Nachtjagdgeschwader 4 and promoted to Hauptmann der Reserve on 1 October 1942. In October 1942, he became Gruppenkommandeur of I. Gruppe NJG 4 and served in this position until December 1944. Herget received the Knight's Cross of the Iron Cross on 20 June 1943 for 31 aerial victories and the destruction of five ground targets; the presentation was made by General der Flieger Josef Kammhuber. Herget was promoted to Major der Reserve on 1 October 1943. On the night of 20/21 December 1943, Herget was credited with the destruction of five Halifax and three Lancaster bombers within 45 minutes, making him an "ace-in-a-day". Following his 63rd aerial victory, Herget was awarded the Knight's Cross of the Iron Cross with Oak Leaves on 11 April 1944, the 451st soldier to receive this distinction; the presentation was made by Adolf Hitler at the Berghof, Hitler's residence in the Obersalzberg of the Bavarian Alps, on 5 May 1944. On 15 June 1944 he was shot down by Branse Burbridge. Herget and his crew bailed out and the Junkers Ju 88 G-1 crashed south-west of Nivelles.

The crash site was excavated in the summer of 2008. According to Boiten and Obermaier, Herget claimed his last aerial victory as a night fighter, a de Havilland Mosquito fighter-bomber, on the night 14/15 June 1944; this claim is not documented by Foreman and Parry, authors of Luftwaffe Night Fighter Claims 1939 – 1945. In January 1945, Herget underwent conversion training and learned to fly the new Messerschmitt Me 262 jet fighter, he served with Sonderkommission Kleinrath, a specialized task force named after Generalleutnant Kurt Kleinrath. This task force of the Reichsluftfahrtministerium main objective was to optimize test-flying and delivery schedules of newly manufactured aircraft. In this func

The Lloyd LT 500 was a compact van produced and sold by the German automaker Borgward Groups's Lloyd Motoren Werke GmbH in Bremen, Germany between 1953 and 1957. A six seater minivan version was offered by April 1954, its appearance resembles a van version of the Fiat 500. The LT 500 featured a two cylinder two stroke engine of 386 cc, which claimed a power output of 13 hp, it had a 6V starter electric. Its fuel consumption was specified to 5.7 l/100 km. A road test of the time recorded that loaded the six seater minivan had a weight to power ratio of 95 kg per unit of horse power, which accounted for a top speed of 60 km/h, elsewhere a top speed - for a less loaded version - of 75 km/h has been quoted; the steering, springing and, in particular, the three speed gear box without synchromesh on any ratio attracted criticism by the standards of 1953. Interior comfort was let down by insufficient heating and lack of ventilation possibilities attracted criticism; the ease with which seats in the minivan could be removed to convert the vehicle to a load carrier impressed the journalists, however.855 Pickups, 357 panel vans, 8,688 minibus versions had been produced by the time the LT 500 was replaced by the 596 cc LT 600.

The Lloyd LT 600 was an engine upgraded Lloyd LT 500, built between 1955 and 1961. The larger four stroke engine and the unsynchronized transmission were used in the passenger car Lloyd 600, a variation of the modell Alexander making the van take 6 l/100 km of fuel consumption. 11,249 vans and 3519 other body variations have been produced until 1961, the year as Borgward disappeared with all its brands from the markets. DKW Schnellaster Tempo Matador