In the usual theory of categories, with any two objects A, B of a category A there is associated a set A (A B) of morphisms of A into B. Frequently the set A (A B) is endowed with an additional structure such as a privileged element or an abelian group structure. It has become clear that as the ramifications of the theory of categories increase, the structures that A (A B) will carry will be richer and more complex. The need for a general theory has been widely felt for some time, and beginnings have been made in various directions and often under restrictive hypotheses; e.g. by Mac Lane [15], Kelly [10], Bénabou [3], Linton [12]


Natural Transformation Natural Isomorphism Monoidal Category Follow Diagram Commute Monoidal Structure 
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  1. [1]
    Bénabou, J., Catégories avec multiplication. C. R. Acad. Sci. Paris 256 1887–1890, (1963)MathSciNetMATHGoogle Scholar
  2. [2]
    —, Algèbre élémentaire dans les catégories avec multiplication. C. R. Acad. Sci. Paris 258 771–774, (1964)MathSciNetMATHGoogle Scholar
  3. [3]
    —, Catégories relatives. C. R. Acad. Sci. Paris 260 3824–3827, (1965)MathSciNetMATHGoogle Scholar
  4. [4]
    Birkhoff, G., and Mac Lane, S., Algebra. Macmillan (1967) (to appear).Google Scholar
  5. [5]
    Brown, R., Function spaces and product topologies. Quart. J. Math. Oxford Ser. II 15 238–250, (1964)MATHCrossRefGoogle Scholar
  6. [6]
    Ehresmann, C., Catégories structurées. Ann. Sci. École Norm. Sup. 80 349–425, (1963)MathSciNetMATHGoogle Scholar
  7. [7]
    Eilenberg, S., and Kelly, G. M., A generalization of the functorial calculus. J. Algebra 3 366–375, (1966)MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Freyd, P., Algebra-valued functors in general and tensor products in particular. Colloq. Math. 14 89–106, (1966)MathSciNetMATHGoogle Scholar
  9. [9]
    Kelly, G. M., On Mac Lane’s conditions for coherence of natural associativities, commutativities, etc. J. Algebra 1 397–402, (1964)MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    — Tensor products in categories. J. Algebra 2 15–37, (1965)MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Lawvere, F. W., Functorial semantics of algebraic theories. (Dissertation, Columbia Univ., 1963).Google Scholar
  12. [12]
    Linton, F. E. J., Autonomous categories and duality of functors. J. Algebra 2 315–349, (1965)MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    — Autonomous equational categories. J. Math. Mech. (to appear).Google Scholar
  14. [14]
    MacLane, S., Natural associativity and commutativity. Rice Univ. Studies 49 28–46, (1963)MathSciNetMATHGoogle Scholar
  15. [15]
    — Categorical algebra. Bull. Amer. Math. Soc. 71 40–106, (1965)MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Spanier, E., Quasi-topologies. Duke Math. J. 30 1–14, (1963)MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Yoneda, N., On the homology theory of modules. J. Fac. Sci. Univ. Tokyo Sect. I. 7 193–221, (1954)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1966

Authors and Affiliations

  • Samuel Eilenberg
    • 1
    • 2
  • G. Max Kelly
    • 1
    • 2
  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Pure Mathematics DepartmentUniversity of SydneySydneyAustralia

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