By H. L. Bentley (auth.), Klaus Heiner Kamps, Dieter Pumplün, Walter Tholen (eds.)
Read Online or Download Category Theory: Applications to Algebra, Logic and Topology Proceedings of the International Conference Held at Gummersbach, July 6–10, 1981 PDF
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Additional info for Category Theory: Applications to Algebra, Logic and Topology Proceedings of the International Conference Held at Gummersbach, July 6–10, 1981
F~netors. Let V be a sy~maetric mono~dal closed category, complete and cocomplete. e. : V-functors ~ : C°p @ B + Y) , there is a Composition of profunctors. r The composite ~ ® ¢ is defined by the coends : ~ @ ~(C, A) = J that are the cokernels of the maps B ~(C,B) ® ~(B,A) dO , d I : J~- ~(C, B)®B(B, B')®¢(B', A) B,B' do ) ~ ( C , a~ > B B)®¢(B, A) where d O and d I are respectively induced by : ~BB®~(B ',A) and ~(C,B) ® eBB ~(c,B) ~ ~(B,B') ~ ¢(B',A) ~ ~(C,B') ~ ¢(B',A) ~(C,B) ® ~(B,B') ~ ~(B',A) > ~(C,B) ~ ¢(~',A) the maps ~BB' and and ~ .
22 (1981), 283-286. C. Walters, Sheaves on a site as Cauchy-complete categories, to appear in Journal of Pure and Applied Algebra. ON ALGEBRAIC LOCALIZATIONS by Francis BORCEUX Universit@ Catholique de Louvain 1348-Louvain-La-Neuve - Belgium Throughout this paper, • denotes a topos of sheaves on a locale ~ and T denotes a finitary algebraic theory internally defined with respect to this topos ~. e. a sheaf of finitary algebraic theories on ~). e. the full exact reflective subcate- gories o f ~ T ) .
More details will be given in the proof of the next proposition. Remark. I choose this proof for sake of simplicity and quickness. But it is a very general result that (as in the case of V-functors  ) right Kan extensions of profunctors are equipped with an action of the codensity monad (which always exists since we deal with profunctors) so that they can be factorized through the Kleisli category [ ]6 ] of that codensity monad of profunctors, which in our case has only one object and so is a mono~d.