By Li J., Su Y., Zhu L.

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**Example text**

Pairs LS®RS(LR(S,S),LR(M,M)) MOdR ~ in further ~ of ~ , isomorphism are descent LR(S,S)-module an e q u i v a l e n c e ~ data the obtain the retricts and p a i r s structure of we on following to a b i j e c t i o n (M,#) between , where the S - m o d u l e categories result. Des(f)~ M ; thus, and it ModL~(S,S). R PROOF. 7 The in close into the algebra R retract in M°dR ~ M°dL n $ R i=l with MOdR ~ i. with . (S,S) Let result of Then categories the directly that the 2. finite the from are the similar to categories (r I .....

1956. A. Grothendieck, Sur quelques points d'alg~bre homologique. T o h 6 k u M a t h . J o u r n a l 2 (1957), 199-221. M. Johnstone, Topos Theory. C a m b r i d g e U n i v e r s i t y P r e s s , 1977. S. Mac Lane, Duality for groups. Bull. A m e r . M a t h . Soc. 56 (1950), 485-516. This paper is in final form and will not be published elsewhere. DESCENT THEORY FOR BANACH Francis Borceux* I n s t i t u t de M a t h ~ m a t i q u e P u r e et A p p l i q u ~ e Universit~ Catholique de L o u v a i n B-1348 Louvain-la-Neuve, Belgium In the context modules can be of described commutative rings there obvious is an Moreover, given S-module , with of relation mathematics and As w e l l theory case has is of theory in the In commutative modules necessary Banach and and sufficient (in the to y i e l d by we an some locales of [5] for to be a Morita and of the National Sciences of as .

5. Denormalization at level 2. : An internal 2-groupoid [5] in /A is an internal groupoid do C1 4 C2 in Grd /A : do ~ dl m C2 such that its image by the functor ( )0 words such that each structural map of d1 z m2 C2 d2 is a discrete groupoid in &, in other C2 is ( )o-invertible. Clearly it is suffi- cient that any structural map is ( )o-invertible. A 2-functor is a natural transformation of such diagrams in Grd &. Let us denote by 2-Grd nal 2-groupoids in ~ the category of inter- A. Again we have a forgetful functor ( )I : 2-Grd fA + Grd fA , associating CI to C 2 , which has a fully faithful right adjoint groupoid C I , by the kernel ,E, Or C O + - - - Gr , given, for each internal equivalence associated to PO CI ~ C I ÷ Gr C O : ~ C I x0 C I ~ C I x 0 C I x0 C I Pl where (Po' Pl ) is the kernel pair associated to the dotted arrow, or equivalently where C I x0 C I is the product in the fiber above again a fibration.