Skip to main content
SpringerLink
Log in

Categories and Functors

  • Chapter
An Invitation to General Algebra and Universal Constructions

Part of the book series: Universitext ((UTX))

  • 2948 Accesses

Abstract

We define the concepts of category, functor, and morphism of functors (‘natural transformation’). The set-theoretic difficulty in treating cases like the category of all sets is handled using Grothendieck’s Axiom of Universes. Epimorphisms, monomorphisms, and similar concepts are investigated. The concept of “enriched categories” (for example, additive categories) is briefly sketched.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Numbers in angle brackets at the end of each listing show pages of these notes on which the work is referred to. “MR” refers to the review of the work in Mathematical Reviews, readable online at http://www.ams.org/mathscinet/ .

References

Numbers in angle brackets at the end of each listing show pages of these notes on which the work is referred to. “MR” refers to the review of the work in Mathematical Reviews, readable online at http://www.ams.org/mathscinet/ .

  1. Andreas Blass, The interaction between category theory and set theory, pp. 5–29 in Mathematical applications of category theory, 1983. MR 85m :03045.

    Google Scholar 

  2. Samuel Eilenberg and Saunders Mac Lane, General theory of natural equivalences, Trans. AMS 58 (1945) 231–294. MR 7 109d.

    Google Scholar 

  3. Peter Freyd, Abelian Categories, Harper and Row, 1964. Out of print, but accessible online, with a new Foreword, at http://www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf . MR 29 #3517.

  4. F. William Lawvere, The category of categories as a foundation for mathematics, pp. 1–20 in Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) ed. S. Eilenberg et al. Springer-Verlag, 1966. MR 34 #7332. (Note corrections to this paper in the MR review.)

    Google Scholar 

  5. Saunders Mac Lane, Categories for the Working Mathematician, Springer GTM, v.5, 1971. MR 50 #7275.

    Google Scholar 

  6. P. M. Cohn, Algebra, second edition, v. 1, Wiley & Sons, 1982. (first edition, MR 50 #12496) MR 83e :00002.

    Google Scholar 

  7. David S. Dummit and Richard M. Foote, Abstract Algebra, Prentice-Hall, 1991. MR 92k :00007.

    Google Scholar 

  8. Marshall Hall, Jr., The Theory of Groups, MacMillan, 1959; Chelsea, 1976. MR 21 #1996, MR 54 #2765.

    Google Scholar 

  9. Thomas W. Hungerford, Algebra, Springer GTM, v. 73, 1974. MR 50 #6693.

    Google Scholar 

  10. Serge Lang, Algebra, Addison-Wesley, third edition, 1993. Reprinted as Springer GTM v. 211, 2002. MR 2003e :00003.

    Google Scholar 

  11. Joseph J. Rotman, An Introduction to the Theory of Groups, 4th edition, Springer GTM, v. 148, 1995. MR 95m :20001.

    Google Scholar 

  12. George M. Bergman, Supports of derivations, free factorizations, and ranks of fixed subgroups in free groups, Trans. AMS 351 (1999) 1531–1550. MR 99f :20036.

    Google Scholar 

  13. George M. Bergman, Notes on composition of maps, supplementary course notes, 6 pp., at http://math.berkeley.edu/~gbergman/grad.hndts/left+right.ps .

  14. Solomon Feferman, Set-theoretical foundations of category theory, pp. 201–247 in Reports of the Midwest Category Seminar, Springer LNM, v.106, 1969. MR 40 #2727.

    Google Scholar 

  15. Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962) 323–448. MR 38 #1144.

    Google Scholar 

  16. Gregory Maxwell Kelly, Basic concepts of enriched category theory, London Math. Soc. Lecture Note Series, 64, 1982, 245 pp. Readable at http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html .

  17. E. W. Kiss, L. Márki, P. Pröhle and W. Tholen, Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness, and injectivity, Studia Scientiarum Mathematicarum Hungarica, 18 (1983) 79–141. MR 85k : 18003.

    Google Scholar 

  18. A. H. Kruse, Grothendieck universes and the super-complete models of Shepherdson, Compositio Math., 17 (1965) 96–101. MR 31 #4716.

    Google Scholar 

  19. T. Y. Lam, Lectures on modules and rings, Springer GTM, v.189, 1999. MR 99i:16001.

    Google Scholar 

  20. Saunders Mac Lane, One universe as a foundation for category theory, pp. 192–200 in Reports of the Midwest Category Seminar, Springer LNM, v.106, 1969. MR 40 #2731.

    Google Scholar 

  21. Walter Rudin, Fourier Analysis on Groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, 1962. MR 27 #2808.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, Berkeley, CA, USA

    George M. Bergman

Authors
  1. George M. Bergman
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bergman, G.M. (2015). Categories and Functors. In: An Invitation to General Algebra and Universal Constructions. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-11478-1_7

Download citation

Publish with us

Policies and ethics

Search

Navigation