Quotient completion for preordered fibrations G. Rosolini joint work with Maria Emilia Maietti Category Theory 2008, Calais, 22-29 June 2008
� � � � � � � � � � � A regular category from a fibration P A fibration with finite products in B , preordered fibers, and fibred finite limits ∧ , ⊤ T B left adjoints ∃ f ⊣ f ∗ satisfying Beck-Chevalley and Frobenius ⊤ P full comprehension P ⊤ h ������ γ P �������� ��������� � P ∼ → ∃ γ P ⊤ P f ⊤ X ⊤ T ⊣ ⊣ (–) P ���� h B � � � � � � T ( γ P ) ���� T ( f ) X � B R S | | Rel ( T ) : B � C � D with T ( R ) = B × C , T ( S ) = C × D . | ∃ � π 1 ,π 3 � [ � π 1 ,π 2 � ∗ R ∧� π 2 ,π 3 � ∗ S ] Rel ( T ) is a preordered cartesian bicategory. All objects are discrete and Frobenius. � Mono ( Map ( Rel ( T ))) P Moreover Map ( Rel ( T )) is regular and is a change of base. T cod � Map ( Rel ( T )) B R.F.C. Walters, R.J. Wood , Frobenius objects in cartesian bicategories , T.A.C. 20 (2008)
� The only Xx example that comes to mind An B with finite limits and a stable factorization ( E , M ) = M → P Take T = cod B Then Rel (cod) is Rel ( B , E , M ) as in G.M. Kelly , A note on relations relative to a factorization system , CT’90 In particular, � Map ( Rel ( B , E , M )) B is the reflection from the 2-category of categories with a stable factorization system to that of regular categories, inverting precisely the monos in E .
� � � � � � � � � � � Manufacturing other examples fibration with fibration of Vert ( p ) E fibred fin.lim’s vertical arrows p c( p ) ⊣ id ⊣ dom comprehension is full A E is the left biadjoint to forgetting comprehension in fibrations with fibred finite limits. Moreover • if A has finite products, then E has finite products Vert ( p ) E p c( p ) • if has left adjoints to reindexing with BCC and FR, then so does E A Vert ( p ) E p c( p ) • if is preordered, then is preordered E A
� � � � Manufacturing other examples, 2 T p • a tripos as in Set J. Hyland, P. Johnstone, A. Pitts , Tripos Theory , Math.Proc.Camb.Phil.Soc. 88 (1980) WFF T p • for a geometric theory T , the fibration of formulas as in Sort T M. Makkai, G. Reyes , First Order Categorical Logic , LNM 611, 1977 ( A → ) po • for a category A with fin.products and weak equalizers, the fibred preordered reflection cod A essentially as in A. Carboni, E. Vitale , Regular and exact completions , J.Pure Appl.Alg. 125 (1998) ( pb C ( F )) po • for a left exact functor F : A → C , a suitable fibred preordered reflection as in cod A P. Hofstra , Relative completions , J.Pure Appl.Alg. 192 (2004)
� � � � � � � � Quotient completion � Mono ( Map ( Rel (c( p ))) ex / reg ) � Mono ( Map ( Rel (c( p )))) � Vert ( p ) E p c( p ) cod cod � Map ( Rel (c( p ))) ex / reg � Map ( Rel (c( p ))) A � E ||| Map ( Split equiv ( Rel (c( p )))) For A with finite products and weak equalizers ( A → ) po � Mono ( A reg ) � M ( Fr ( A )) � Mono ( A ex ) cod � Fr ( A ) � A reg � A ex A
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