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Lexikon


category


eory> A category K is a collection of objects, obj(K), and a collection of ef="module.php?name=Lexikon&file=search&eid=1&query=morphisms">morphisms (or "ef="module.php?name=Lexikon&file=search&eid=1&query=arrows">arrows"), mor(K) such that 1. each morphism f has a "typing" on a pair of objects A, B written f:A->B. This is read ' f is a morphism from A to B' . A is the "source" or "ef="module.php?name=Lexikon&file=search&eid=1&query=domain">domain" of f and B is its "target" or "ef="module.php?name=Lexikon&file=search&eid=1&query=co-domain">co-domain". 2. There is a ef="module.php?name=Lexikon&file=search&eid=1&query=partial function">partial function on morphisms called ef="module.php?name=Lexikon&file=search&eid=1&query=composition">composition and denoted by an ef="module.php?name=Lexikon&file=search&eid=1&query=infix">infix ring symbol, o. We may form the "composite" g o f : A -> C if we have g:B->C and f:A->B. 3. This composition is associative: h o (g o f) = (h o g) o f. 4. each object A has an identity morphism id_A:A->A associated with it. This is the identity under composition, shown by the equations id_B o f = f = f o id_A. In general, the morphisms between two objects need not form a ef="module.php?name=Lexikon&file=search&eid=1&query=set">set (to avoid problems with ef="module.php?name=Lexikon&file=search&eid=1&query=Russell' s paradox">Russell' s paradox). An example of a category is the collection of sets where the objects are sets and the morphisms are functions. Sometimes the composition ring is omitted. The use of capitals for objects and lower case letters for morphisms is widespread but not universal. Variables which refer to categories themselves are usually written in a script font. (1997-10-06)

e="border-width:thin; border-color:#333333; border-style:dashed; padding:5px;" align="left">In addition suitable contents:
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ef="module.php?name=Lexikon&op=content&tid=1695">by ] [ ef="module.php?name=Lexikon&op=content&tid=1708">C ] [ ef="module.php?name=Lexikon&op=content&tid=1724">ca ] [ ef="module.php?name=Lexikon&op=content&tid=1844">case ] [ ef="module.php?name=Lexikon&op=content&tid=1863">cat ] [ ef="module.php?name=Lexikon&op=content&tid=2001">ch ] [ ef="module.php?name=Lexikon&op=content&tid=2099">ci ] [ ef="module.php?name=Lexikon&op=content&tid=2247">co ] [ ef="module.php?name=Lexikon&op=content&tid=2330">com ] [ ef="module.php?name=Lexikon&op=content&tid=2463">composite ] [ ef="module.php?name=Lexikon&op=content&tid=2464">composition ] [ ef="module.php?name=Lexikon&op=content&tid=2791">cr ] [ ef="module.php?name=Lexikon&op=content&tid=3151">de ] [ ef="module.php?name=Lexikon&op=content&tid=3565">do ] [ ef="module.php?name=Lexikon&op=content&tid=3595">domain ] [ ef="module.php?name=Lexikon&op=content&tid=3752">du ] [ ef="module.php?name=Lexikon&op=content&tid=3834">e ] [ ef="module.php?name=Lexikon&op=content&tid=3865">ec ] [ ef="module.php?name=Lexikon&op=content&tid=3896">ed ] [ ef="module.php?name=Lexikon&op=content&tid=3929">ee ] [ ef="module.php?name=Lexikon&op=content&tid=3946">eg ] [ ef="module.php?name=Lexikon&op=content&tid=4148">er ] [ ef="module.php?name=Lexikon&op=content&tid=4150">era ] [ ef="module.php?name=Lexikon&op=content&tid=4171">es ] [ ef="module.php?name=Lexikon&op=content&tid=4199">et ] [ ef="module.php?name=Lexikon&op=content&tid=4497">fi ] [ ef="module.php?name=Lexikon&op=content&tid=4520">file ] [ ef="module.php?name=Lexikon&op=content&tid=4595">fix ] [ ef="module.php?name=Lexikon&op=content&tid=4700">fo ] [ ef="module.php?name=Lexikon&op=content&tid=4715">font ] [ ef="module.php?name=Lexikon&op=content&tid=4727">for ] [ ef="module.php?name=Lexikon&op=content&tid=4828">fr ] [ ef="module.php?name=Lexikon&op=content&tid=4940">function ] [ ef="module.php?name=Lexikon&op=content&tid=5057">ge ] [ ef="module.php?name=Lexikon&op=content&tid=5070">gen ] [ ef="module.php?name=Lexikon&op=content&tid=5434">h ] [ ef="module.php?name=Lexikon&op=content&tid=5540">hat ] [ ef="module.php?name=Lexikon&op=content&tid=5768">hr ] [ ef="module.php?name=Lexikon&op=content&tid=5931">id ] [ ef="module.php?name=Lexikon&op=content&tid=5956">ie ] [ ef="module.php?name=Lexikon&op=content&tid=6013">il ] [ ef="module.php?name=Lexikon&op=content&tid=6064">in ] [ ef="module.php?name=Lexikon&op=content&tid=6413">io ] [ ef="module.php?name=Lexikon&op=content&tid=6449">ir ] [ ef="module.php?name=Lexikon&op=content&tid=6482">is ] [ ef="module.php?name=Lexikon&op=content&tid=6558">it ] [ ef="module.php?name=Lexikon&op=content&tid=6760">K ] [ ef="module.php?name=Lexikon&op=content&tid=7091">Lex ] [ ef="module.php?name=Lexikon&op=content&tid=7399">ls ] [ ef="module.php?name=Lexikon&op=content&tid=7437">lv ] [ ef="module.php?name=Lexikon&op=content&tid=7441">ly ] [ ef="module.php?name=Lexikon&op=content&tid=7463">ma ] [ ef="module.php?name=Lexikon&op=content&tid=8032">mo ] [ ef="module.php?name=Lexikon&op=content&tid=8040">mod ] [ ef="module.php?name=Lexikon&op=content&tid=8079">module ] [ ef="module.php?name=Lexikon&op=content&tid=8167">mp ] [ ef="module.php?name=Lexikon&op=content&tid=8228">ms ] [ ef="module.php?name=Lexikon&op=content&tid=8386">na ] [ ef="module.php?name=Lexikon&op=content&tid=8460">nc ] [ ef="module.php?name=Lexikon&op=content&tid=8472">ne ] [ ef="module.php?name=Lexikon&op=content&tid=8622">nf ] [ ef="module.php?name=Lexikon&op=content&tid=8627">ng ] [ ef="module.php?name=Lexikon&op=content&tid=8630">ni ] [ ef="module.php?name=Lexikon&op=content&tid=8675">no ] [ ef="module.php?name=Lexikon&op=content&tid=8760">ns ] [ ef="module.php?name=Lexikon&op=content&tid=8838">object ] [ ef="module.php?name=Lexikon&op=content&tid=8964">om ] [ ef="module.php?name=Lexikon&op=content&tid=9204">pa ] [ ef="module.php?name=Lexikon&op=content&tid=9270">paradox ] [ ef="module.php?name=Lexikon&op=content&tid=9332">partial function ] [ ef="module.php?name=Lexikon&op=content&tid=9550">ph ] [ ef="module.php?name=Lexikon&op=content&tid=9616">ping ] [ ef="module.php?name=Lexikon&op=content&tid=9651">pl ] [ ef="module.php?name=Lexikon&op=content&tid=9908">pr ] [ ef="module.php?name=Lexikon&op=content&tid=10144">pt ] [ ef="module.php?name=Lexikon&op=content&tid=10253">query ] [ ef="module.php?name=Lexikon&op=content&tid=10364">rc ] [ ef="module.php?name=Lexikon&op=content&tid=10385">re ] [ ef="module.php?name=Lexikon&op=content&tid=10767">ro ] [ ef="module.php?name=Lexikon&op=content&tid=10820">row ] [ ef="module.php?name=Lexikon&op=content&tid=10907">Russell ] [ ef="module.php?name=Lexikon&op=content&tid=10918">S ] [ ef="module.php?name=Lexikon&op=content&tid=10922">sa ] [ ef="module.php?name=Lexikon&op=content&tid=11010">sc ] [ ef="module.php?name=Lexikon&op=content&tid=11102">script ] [ ef="module.php?name=Lexikon&op=content&tid=11150">se ] [ ef="module.php?name=Lexikon&op=content&tid=11281">set ] [ ef="module.php?name=Lexikon&op=content&tid=11314">sh ] [ ef="module.php?name=Lexikon&op=content&tid=11376">si ] [ ef="module.php?name=Lexikon&op=content&tid=11506">sit ] [ ef="module.php?name=Lexikon&op=content&tid=11556">sm ] [ ef="module.php?name=Lexikon&op=content&tid=11651">so ] [ ef="module.php?name=Lexikon&op=content&tid=11745">source ] [ ef="module.php?name=Lexikon&op=content&tid=12133">su ] [ ef="module.php?name=Lexikon&op=content&tid=12246">sy ] [ ef="module.php?name=Lexikon&op=content&tid=12359">T ] [ ef="module.php?name=Lexikon&op=content&tid=12416">tar ] [ ef="module.php?name=Lexikon&op=content&tid=12421">target ] [ ef="module.php?name=Lexikon&op=content&tid=12588">th ] [ ef="module.php?name=Lexikon&op=content&tid=12602">theory ] [ ef="module.php?name=Lexikon&op=content&tid=12721">to ] [ ef="module.php?name=Lexikon&op=content&tid=12896">tt ] [ ef="module.php?name=Lexikon&op=content&tid=12939">tw ] [ ef="module.php?name=Lexikon&op=content&tid=12986">ua ] [ ef="module.php?name=Lexikon&op=content&tid=13175">us ] [ ef="module.php?name=Lexikon&op=content&tid=13229">V ] [ ef="module.php?name=Lexikon&op=content&tid=13310">ve ] [ ef="module.php?name=Lexikon&op=content&tid=13877">ws ]






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