Progress in Mathematics

This volume contains five review articles, two in the Algebra part and three in the Geometry part, surveying the fields of cate­ gories and class field theory, in the Algebra part, and of Finsler spaces, structures on differentiable manifolds, and packing, cover­ ing, etc., in the Geometry part. The literature covered is primar­ Hy that published in 1964-1967. Contents ALGEBRA CATEGORIES ............... . 3 M. S. Tsalenko and E. G. Shul''geifer § 1. Introduction........... 3 § 2. Foundations of the Theory of Categories . . . . . 4 § 3. Fundamentals of the Theory of Categories . . . . . 6 § 4. Embeddings of Categories ... . . . . . . . . . . . . 14 § 5. Representations of Categories . . . . . . . . . . . . . 16 § 6. Axiomatic Characteristics of Algebraic Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 18 § 7. Reflective Subcategories; Varieties. . . 20 § 8. Radicals in Categories . . . . . . . 24 § 9. Categories with Involution. . . . . . 29 § 10. Universal Algebras in Categories . 30 § 11. Categories with Multiplication . . . 34 § 12. Duality of Functors. .. ....... 37 § 13. Homotopy Theory . . . . .. ........... 39 § 14. Homological Algebra in Categories. . . . . . 41 § 15. Concrete Categories . . . . .. ......... 44 § 16. Generalizations.. . . . . . . 45 Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 CLASS FIELD THEORY. FIELD EXTENSIONS. . . . . . . . 59 S. P. Demushkin 66 Literature Cited vii CONTENTS viii GEOMETRY 75 FINSLER SPACES AND THEIR GENERALIZATIONS ..