By A. Carboni, M.C. Pedicchio, G. Rosolini

With one exception, those papers are unique and completely refereed study articles on numerous functions of classification concept to Algebraic Topology, good judgment and computing device technology. The exception is an exceptional and long survey paper through Joyal/Street (80 pp) on a growing to be topic: it offers an account of classical Tannaka duality in the sort of approach as to be available to the overall mathematical reader, and to supply a key for access to extra contemporary advancements and quantum teams. No services in both illustration thought or class concept is thought. subject matters reminiscent of the Fourier cotransform, Tannaka duality for homogeneous areas, braided tensor different types, Yang-Baxter operators, Knot invariants and quantum teams are brought and reviews. From the Contents: P.J. Freyd: Algebraically whole categories.- J.M.E. Hyland: First steps in man made area theory.- G. Janelidze, W. Tholen: How algebraic is the change-of-base functor?.- A. Joyal, R. highway: An advent to Tannaka duality and quantum groups.- A. Joyal, M. Tierney: robust stacks andclassifying spaces.- A. Kock: Algebras for the partial map classifier monad.- F.W. Lawvere: Intrinsic co-Heyting obstacles and the Leibniz rule in definite toposes.- S.H. Schanuel: damaging units have Euler attribute and dimension.-

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**Extra info for Category Theory: Proceedings of the International Conference Held in Como, Italy, July 22-28, 1990**

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MULVEY, Representations of rings and modules, Springer LNM 753, 1980, 542587. This paper is in final form and will not be published elsewhere. NORMALIZATION EQUIVALENCE, KERNEL EQUIVALENCE AND ~ CATEGORIES Dominique Bourn Fac. de MathEmatiques, Universit6 de Picardie 33 rue St Leu, 80039 Amiens France. In a recent paper [4], A. Carboni gave an interesting characterization of the categories of affine spaces, i. e. slices of additive categories, by means of a "modularity" condition, relating coproducts and puUbacks, which is a categorical version of the modularity condition for lattices, in the same way as the distributive categories are the categorical version of the distributive lattices.

S ' = l 2 . s ' ( = o ) . The following diagram of split epimorphisms : k Y (f', 11) ~ Y' ~ ~ iT (]'~) It *| x ......... h x' ~ X~x~ is such that the two upper composites are equal. ) : Y----~X x Y). Consequently (ft, ll) and (re, 12) are equal and so 11 and l 2. II 4] The essentially affine categories. Now to assume that the previous change of base functor h* is an equivalence of categories is to assume the following essentially affine condition : In any commutative square of split epimorphisms : k y .......

I) = id, hence we have a reflexive graph. But all the maps are homomorphisms of monoids, hence it is a monoidal graph. 6) and the map M#s defines a multiplication on the graph. The properties of triples imply that this multiplication is associative and admits M(r/1) as identity, hence M(N) is the object of arrows of art internal category C, having N as object of objects. Moreover C is strictly monoidal since all the maps axe homomorphisms of monoids. Since N is decidable, it is clear that all the objects in the previous diagrams are decidable, and all the monos are complemented.