Current Research in Operational Quantum Logic pp 167-194 | Cite as

# Short Introduction to Enriched Categories

## Abstract

This text aims to be a short introduction to some of the basic notions in ordinary and enriched category theory. With reasonable detail but always in a compact fashion, we have brought together in the first part of this paper the definitions and basic properties of such notions as limit and colimit constructions in a category, adjoint functors between categories, equivalences and monads. In the second part we pass on to enriched category theory: it is explained how one can “replace” the category of sets and mappings, which plays a crucial role in ordinary category theory, by a more general symmetric monoidal closed category, and how most results of ordinary category theory can be translated to this more general setting. For a lack of space we had to omit detailed proofs, but instead we have included lots of examples which we hope will be helpful. In any case, the interested reader will find his way to the references, given at the end of the paper.

## Keywords

Natural Transformation Category Theory Full Subcategory Small Category Left Adjoint## Preview

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## References

- [1]AdArnek, J., Herrlich, H. and Strecker, G.E. (1990) Abstract and Concrete Categories, John Wiley & Sons, New York.Google Scholar
- [2]Barr, M. and Beck, J. (1969) Homology and standard constructions,
*Springer Lecture Notes in Mathematics***80**, 245–335.MathSciNetCrossRefGoogle Scholar - [3]Bénabou, J. (1963) Categories avec multiplication,
*Comptes rendus de l’Académie des Sciences de Paris***256**, 1888–1890Google Scholar - [4]Borceux, F. (1994) Handbook of Categorical Algebra I, I I, III, Cambridge UP.CrossRefGoogle Scholar
- [5]Dubuc, E. (1970) Variations on Beck’s tripleability criterion,
*Springer Lecture Notes in Mathematics***106**, 74–129.Google Scholar - [6]Eilenberg, S. and Kelly, G.M. (1966) Closed categories, in Proceedings of the Conference on Categorical Algebra, La Jolla 1965, pp. 421–562, Springer.Google Scholar
- [7]Eilenberg, S. and Mac Lane, S (1945) General theory of natural equivalences,
*Transactions of the American Mathematical Society***58**, 231–294.MathSciNetzbMATHGoogle Scholar - [8]Eilenberg, S. and Moore, J.C. (1965) Adjoint functors and triples,
*Illinois Journal of Mathematics***9**, 381–398.MathSciNetzbMATHGoogle Scholar - [9]Johnstone, P.T. (1975) Adjoint lifting theorems for categories of algebras,
*Bulletin of the London Mathematical Society***7**, 294–297.MathSciNetzbMATHCrossRefGoogle Scholar - [10]Kan, D. (1958) Adjoints functors,
*Transactions of the American Mathematical Society***87**, 294–329.MathSciNetzbMATHCrossRefGoogle Scholar - [11]Kelly, G.M. (1982) Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Notes 64, Cambridge UP.Google Scholar
- [12]Kelly, G.M. and Mac Lane, S. (1971) Coherence in closed categories,
*Journal of Pure and Applied Algeb*ra**1**, 97–140.MathSciNetzbMATHCrossRefGoogle Scholar - [13]Mac Lane S. (1971) Categories for the Working Mathematician, Springer.Google Scholar
- [14]Yoneda, N. (1954) On the homology theory of modules,
*Journal of the Faculty of Sciences of Tokyo***7**, 193–227.MathSciNetzbMATHGoogle Scholar