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Coproduct

In category theory, the coproduct, or categorical sum, is a category-theoretic construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces virginia football uniforms. The coproduct of a family of objects is essentially the “least specific” object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.

Let C be a category and let X1 and X2 be objects in that category. An object is called the coproduct of these two objects, written X1X2 or X1X2 or sometimes simply X1 + X2, if there exist morphisms i1 : X1X1X2 and i2 : X2X1X2 satisfying a universal property: for any object Y and morphisms f1 : X1 → Y and f2 : X2 → Y, there exists a unique morphism f : X1X2Y such that f1 = fi1 and f2 = fi2. That is, the following diagram commutes:

The unique arrow f making this diagram commute may be denoted f1f2 or f1f2 or f1 + f2 or [f1, f2]. The morphisms i1 and i2 are called canonical injections, although they need not be injections nor even monic.

The definition of a coproduct can be extended to an arbitrary family of objects indexed by a set J. The coproduct of the family {Xj : jJ} is an object X together with a collection of morphisms ij : XjX such that, for any object Y and any collection of morphisms fj : XjY, there exists a unique morphism f from X to Y such that fj = fij. That is, the following diagrams commute (for each jJ):

The coproduct of the family {Xj} is often denoted

or

Sometimes the morphism f may be denoted

to indicate its dependence on the individual fj.

The coproduct in the category of sets is simply the disjoint union with the maps ij being the inclusion maps. Unlike direct products, coproducts in other categories are not all obviously based on the notion for sets one liter glass water bottle, because unions don’t behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be dramatically different from each other. For example, the coproduct in the category of groups, called the free product, is quite complicated. On the other hand, in the category of abelian groups (and equally for vector spaces), the coproduct, called the direct sum, consists of the elements of the direct product which have only finitely many nonzero terms. (It therefore coincides exactly with the direct product in the case of finitely many factors.)

In the case of topological spaces coproducts are disjoint unions with their disjoint union topologies. That is, it is a disjoint union of the underlying sets, and the open sets are sets open in each of the spaces, in a rather evident sense. In the category of pointed spaces mobile pouch for running, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point).

Despite all this dissimilarity, there is still, at the heart of the whole thing, a disjoint union: the direct sum of abelian groups is the group generated by the “almost” disjoint union (disjoint union of all nonzero elements, together with a common zero), similarly for vector spaces: the space spanned by the “almost” disjoint union; the free product for groups is generated by the set of all letters from a similar “almost disjoint” union where no two elements from different sets are allowed to commute.

The coproduct construction given above is actually a special case of a colimit in category theory. The coproduct in a category C can be defined as the colimit of any functor from a discrete category J into C. Not every family {Xj} will have a coproduct in general, but if it does, then the coproduct is unique in a strong sense: if ij : XjX and kj : XjY are two coproducts of the family {Xj}, then (by the definition of coproducts) there exists a unique isomorphism f : XY such that fij = kj  for each j in J.

As with any universal property, the coproduct can be understood as a universal morphism. Let Δ: CC×C be the diagonal functor which assigns to each object X the ordered pair (X,X) and to each morphism f:XY the pair (f,f). Then the coproduct X+Y in C is given by a universal morphism to the functor Δ from the object (X,Y) in C×C.

The coproduct indexed by the empty set (that is, an empty coproduct) is the same as an initial object in C.

If J is a set such that all coproducts for families indexed with J exist, then it is possible to choose the products in a compatible fashion so that the coproduct turns into a functor CJC. The coproduct of the family {Xj} is then often denoted by ∐j Xj, and the maps ij are known as the natural injections.

Letting HomC(U,V) denote the set of all morphisms from U to V in C (that is, a hom-set in C), we have a natural isomorphism

given by the bijection which maps every tuple of morphisms

(a product in Set, the category of sets, which is the Cartesian product, so it is a tuple of morphisms) to the morphism

That this map is a surjection follows from the commutativity of the diagram: any morphism f is the coproduct of the tuple

That it is an injection follows from the universal construction which stipulates the uniqueness of such maps. The naturality of the isomorphism is also a consequence of the diagram. Thus the contravariant hom-functor changes coproducts into products. Stated another way, the hom-functor, viewed as a functor from the opposite category Cop to Set is continuous; it preserves limits (a coproduct in C is a product in Copp).

If J is a finite set, say J = {1,…,n}, then the coproduct of objects X1,…,Xn is often denoted by X1⊕.. waterproof plastic bags.⊕Xn. Suppose all finite coproducts exist in C, coproduct functors have been chosen as above, and 0 denotes the initial object of C corresponding to the empty coproduct. We then have natural isomorphisms

These properties are formally similar to those of a commutative monoid; a category with finite coproducts is an example of a symmetric monoidal category.

If the category has a zero object Z, then we have unique morphism XZ (since Z is terminal) and thus a morphism XYZY. Since Z is also initial, we have a canonical isomorphism ZYY as in the preceding paragraph. We thus have morphisms XYX and XYY, by which we infer a canonical morphism XYX×Y. This may be extended by induction to a canonical morphism from any finite coproduct to the corresponding product. This morphism need not in general be an isomorphism; in Grp it is a proper epimorphism while in Set* (the category of pointed sets) it is a proper monomorphism. In any preadditive category, this morphism is an isomorphism and the corresponding object is known as the biproduct. A category with all finite biproducts is known as an semiadditive category.

If all families of objects indexed by J have coproducts in C, then the coproduct comprises a functor CJC. Note that, like the product, this functor is covariant.

Mario Saralegui

Mario Daniel Saralegui Iriarte (24 kwietnia 1959) – piłkarz urugwajski, pomocnik (rozgrywający). Wzrost 178 cm, waga 73 kg.

Saralegui karierę piłkarską rozpoczął w 1977 w klubie Peñarol Montevideo. W reprezentacji Urugwaju zadebiutował 31 maja 1979 w przegranym 1:5 meczu z Brazylią. Wziął udział w turnieju Copa América 1979, gdzie Urugwaj odpadł już w fazie grupowej. Saralegui zagrał tylko w dwóch meczach z Ekwadorem.

Znacznie bardziej udany by turniej Copa América 1983, gdzie Urugwaj zdobył tytuł mistrza Ameryki Południowej. Saralegui zagrał w dwóch meczach fazy grupowej – z Chile i Wenezuelą, oraz w dwóch meczach półfinałowych z Peru.

Salaregui większość swojej kariery spędził w klubie Peñarol Montevideo, z którym 6 razy zdobył mistrzostwo Urugwaju, a ponadto Copa Libertadores i Puchar Interkontynentalny.

W 1984 przeniósł się do Hiszpanii, do klubu Elche. Po roku wrócił do Peñarolu, którego barwy reprezentował w finałach mistrzostw świata w 1986 roku, gdzie Urugwaj dotarł do 1/8 finału. Mając koszulkę z numerem 16 zagrał w trzech meczach grupowych – z Niemcami, Danią i Szkocją.

Po mistrzostwach przeniósł się do Argentyny, do klubu River Plate Buenos Aires, a po roku do Estudiantes La Plata. W 1989 na rok wrócił do Peñarolu, by następnie w 1991 grać dla ekwadorskiego klubu Barcelona Guayaquil. W 1992 znów wrócił do Peñarolu, w którym zakończył karierę piłkarską.

Salaregui od 1979 do 13 czerwca 1986 rozegrał w reprezentacji Urugwaju 29 meczów i zdobył 2 bramki

Po zakończeniu kariery piłkarskiej został trenerem. Doprowadził Peñarol do mistrzostwa Urugwaju w turnieju Clausura 2008, po wygraniu 5:3 decydującego o tytule meczu z klubem River Plate Montevideo.

1 Rodríguez • 2 Gutiérrez • 3 Acevedo • 4 Diogo • 5 Bossio • 6 Batista • 7 Alzamendi • 8 Barrios • 9 da Silva • 10 Francescoli • 11 Santín • 12 Álvez • 13 Vega • 14 Pereyra • 15 Rivero • 16 Saralegui17 Zalazar • 18 Paz • 19 Ramos • 20 Aguilera • 21 Cabrera • 22 Otero • trener: Borrás

de Luca (1932–34) · Piendibene (1934) · Velásquez (1935–40) · Piendibene (1940–41) · de Luca (1941) · Morquio (1941) · L. Fernández (1941–42) · Harley (1942) · de Luca (1942–43) · Arremón (1943) · de Hegedüs (1943) · Tejada (1944) · Suppici (1945) · Tejada (1946) · Clulow (1947) · Galloway (1948) · Hirschl (1949–51) · López (1952–55) · Máspoli i Varela (1955t· Hirschl (1956) · Bagnulo (1958–59) · Scarone&nbsp virginia football uniforms;(1959–61) · Guttmann (1962) · Anselmo (1962) · Máspoli (1963–67) · Milans (1968–69) · Brandão (1969–70) · Máspoli (1970–71) · Hohberg (1971) · Viera (1972) · Faccio (1972–73) · Bagnulo (1973–74) · Rodríguez (1974) · Bagnulo (1974–75) · Schiaffino (1975–76) · Máspoli (1976) · Sani (1977–80) · Tuane (1980) · Prais (1980t· Etchegoyen (1980) · Kistenmacher (1980t· Ghiggia (1980) · Cubilla (1981) · Bagnulo (1982–83) · Balseiro (1983) · H. Fernández (1984) · Máspoli (1985–86) · Silva (1986) · Tabárez (1987) · Morena (1988) · Mazurkiewicz (1988–89) · Roque (1989) · Fleitas (1989–90) · Menotti (1990–91) · Duarte (1991t· Ortiz (1991) · Petrović (1992) · Máspoli (1992t· Olivera (1992t· Faccio (1992) · Pérez (1993–95) · Fossati (1996) · Botello (1996) · Pérez (1997–98) · Ribas (1999–2001) · Pérez (2002) · Aguirre (2003–04) · Morena (2005) · Garisto (2006) · Saralegui (2006) · Pérez (2006–07) · Matosas (2007) · Saralegui (2008–09) · Ribas (2009) · Púa (2009t· Aguirre (2010) · Keosseian (2010) · Machín (2010t· Aguirre (2011) · Pérez (2011–12) · da Silva (2012–13) · Alonso (2013) · Gonçalves (2013–14) · Fossati (2014) · Montero (2014t· Bengoechea (2015–16) · da Silva (2016) · Curutchet (2016) · Ramos (od 2017)

Roosevelt County, Montana

Roosevelt County is a county located in the U.S. state of Montana. As of the 2010 census, the population was 10,425. Its county seat is Wolf Point. Roosevelt County was created by the Montana Legislature in 1919 from Sheridan County. The name honors former president Theodore Roosevelt, who had died shortly before the county was formed.

According to the U.S. Census Bureau, the county has a total area of 2,369 square miles (6,140 km2), of which 2,355 square miles (6,100 km2) is land and 15 square miles (39 km2) (0.6%) is water. Over 74 percent of the county’s land area lies within the Fort Peck Indian Reservation.

As of the census of 2000, there were 10,620 people, 3,581 households, and 2,614 families residing in the county. The population density was 4 people per square mile (2/km²). There were 4,044 housing units at an average density of 2 per square mile (1/km²). The racial makeup of the county was 40.93% White, 0.05% Black or African American, 55.75% Native American, 0.43% Asian, 0.05% Pacific Islander, 0.25% from other races, and 2.53% from two or more races. 1.23% of the population were Hispanic or Latino of any race. 12.6% were of Norwegian and 11.5% German ancestry. 94.8% spoke English and 3.4% Dakota as their first language.

There were 3,581 households out of which 40.50% had children under the age of 18 living with them, 47.20% were married couples living together, 18.90% had a female householder with no husband present, and 27.00% were non-families. 23.60% of all households were made up of individuals and 10.30% had someone living alone who was 65 years of age or older. The average household size was 2.89 and the average family size was 3.40.

In the county the population was spread out with 34.60% under the age of 18, 7.90% from 18 to 24, 25.80% from 25 to 44, 20.20% from 45 to 64, and 11.60% who were 65 years of age or older. The median age was 32 years. For every 100 females there were 98.30 males. For every 100 females age 18 and over, there were 93.70 males.

The median income for a household in the county was $24,834, and the median income for a family was $27,833. Males had a median income of $25,177 versus $19,728 for females. The per capita income for the county was $11,347. About 27.60% of families and 32.40% of the population were below the poverty line, including 41.60% of those under age 18 and 15.10% of those age 65 or over.

As of the 2010 United States Census, there were 10,425 people, 3,553 households, and 2,548 families residing in the county. The population density was 4.4 inhabitants per square mile (1.7/km2). There were 4,063 housing units at an average density of 1.7 per square mile (0.66/km2). The racial makeup of the county was 60.4% American Indian, 35.8% white, 0.4% Asian, 0.1% black or African American, 0.2% from other races, and 3.0% from two or more races. Those of Hispanic or Latino origin made up 1.3% of the population. In terms of ancestry, 20.0% were Norwegian, 16.3% were German, 6.1% were Irish, and 1.3% were American.

Of the 3,553 households virginia football uniforms, 42.4% had children under the age of 18 living with them, 42.9% were married couples living together, 20.5% had a female householder with no husband present, 28.3% were non-families, and 24.2% of all households were made up of individuals swimming bags waterproof. The average household size was 2.88 and the average family size was 3.41. The median age was 31.6 years.

The median income for a household in the county was $37,451 and the median income for a family was $50,146. Males had a median income of $39,008 versus $34,725 for females. The per capita income for the county was $17,821. About 15.8% of families and 21.5% of the population were below the poverty line, including 28.6% of those under age 18 and 10.9% of those age 65 or over.

Coordinates:

Louis Major

Louis Karel Major (Oostende, 20 december 1902 – Antwerpen, 19 februari 1985) was een Belgisch syndicalist highest rated water bottles, volksvertegenwoordiger en minister voor de BSP.

Major was een zoon van Frans Major en van Coralie De Meere. Hij trouwde met Jeanne Van Vlem. Hij begon in het leven als elektricien. Dankzij een beurs van de socialistische Textielarbeiderscentrale kon hij studeren aan de Arbeidershogeschool in Brussel en behaalde hij het diploma van maatschappelijk assistent. Dit maakte het hem mogelijk in 1919 in dienst te treden bij de metaalbewerkersbond. Hij werd vervolgens socialistisch vakbondssecretaris voor de Belgische transportarbeidersbond in West-Vlaanderen (1925-1937).

Hij werd verkozen tot gemeenteraadslid (1933-1938) in Oostende en provincieraadslid van West-Vlaanderen (1932-1946).

In 1937 werd hij algemeen secretaris van de Havenarbeiders van Antwerpen, tot in 1940. Van 1940 tot 1945 was hij (gedeeltelijk clandestien) secretaris van het Belgisch Vakverbond.

Na de Tweede Wereldoorlog begon hij volop aan een nationale carrière zowel in de vakbond als in de politiek. Hij werd adjunct-algemeensecretaris (1945-1952) en werd (met André Renard als tegenkandidaat), verkozen tot algemeen secretaris (1952-1968) van het ABVV.

Vanaf 1946 was hij BSP-volksvertegenwoordiger (1946-1974) voor het arrondissement Antwerpen en hij verwierf zich een stevige reputatie als polemist en als ‘scheldmajoor’ virginia football uniforms, zowel in de Volksgazet die hij leidde, als in het parlement. Helemaal op het einde van zijn parlementaire carrière, in de periode december 1971-maart 1974, had hij als gevolg van het toen bestaande dubbelmandaat ook zitting in de Cultuurraad voor de Nederlandse Cultuurgemeenschap, die op 7 december 1971 werd geïnstalleerd en de verre voorloper is van het Vlaams Parlement.

In 1968 werd hij minister van arbeid en tewerkstelling en bleef dit tot in 1972, de hele tijd van de regering Gaston Eyskens-André Cools healthy drink bottles.

In 1973 werd hij benoemd tot minister van Staat.