By J. Adámek, J. Rosický, V. Trnková (auth.), Francis Borceux (eds.)

Categorical algebra and its purposes comprise a number of basic papers on normal class concept, via the pinnacle experts within the box, and lots of fascinating papers at the purposes of type concept in practical research, algebraic topology, algebraic geometry, basic topology, ring concept, cohomology, differential geometry, workforce idea, mathematical good judgment and desktop sciences. the amount includes 28 rigorously chosen and refereed papers, out of ninety six talks brought, and illustrates the usefulness of class idea this day as a strong device of research in lots of different areas.

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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 This paper is in final form and will not be published elsewhere.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.