Abstract
The purpose of these notes is to provide as rapid an introduction to category theory and homological algebra as possible without overwhelming the reader entirely unfamiliar with these subjects. We begin with the definition of a category, and end with the basic properties of derived functors, in particular, Tor and Ext. This was the spirit of the four lectures on which thenotes are based, although there is, needless to say, much more material contained herein an was touched on in the lectures. For example, we have included a fairly complete treatment of the basic facts pertaining to adjoint functors, including Freyd's adjoint functor theorems. Application of category theory in the direction of topos theory and logic were treated in the accompanying lectures of Tierney, and Buchsbaum in his lectures indicated some outlets for homological algebra in commutative algebra and local ring theory. We have therefore not felt compelled to emphasize any specific topic. We have, nevertheless, presented module theory as something associated with ringoids (small, additive categories) rather than with the more conventional and restrictive notion of a ring. This point of view has enabled us recently to incorporate several new examples into the traditional setting of homological algebra as found in the book of Cartan-Eilenberg [2]. One can consult [15] in this regard.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Buchsbaum, D. A., A note on homology in categories. Ann. Math. 69 (1959), 66–74.
Cartan, H., and Eilenberg, S. “Homological Algebra”. Princeton University Press, Princeton, New Jersey, 1956,
Freyd, P., “Abelian Categories”. Harper and Row, New York, 1964.
Freyd, P., On the concreteness of certain categories. Symposia Mathematica IV, 431–456. Academic Press, New York, 1970.
Gabriel, P., Des categories abeliennes. Bull. Soc. Math. France, No. 90, (1962), 323–448.
Gabriel, P., and Popescu, N., Caracteérisation des categories abeliennes avec générateurs et limites inductives exactes. Comp. Rend. Acad. Sci. Paris, No. 258, 1964, 4188–4190.
Gabriel, P., and Zismann, M., “Calculus of Fractions and Homotopy Theory.” Springer Verlag, 1967.
Govorov, V. E., Flat modules (in Russian), Sibirs. Mat. Z. 6 (1965), 300–304.
Grothendieck, A., Sur quelques points d'algebre homologique. Tohôku Math. J., No. 9 (1957), 119–221.
Kan, D., Adjoint functors. Trans. Am. Math. Soc. 87 (1958), 294–329.
Kelley, J. L., “General Topology”. Van Nostrand, Princeton, New Jersey, 1955.
Lazard, D., Autour de la platitude, Bull. Soc. Math. France 97 (1969), 81–128.
MacLane, S., “Homology”. Springer, Berlin,1963.
Mitchell, B., “Theory of Categories”. Academic Press, New York, 1965.
Mitchell, B., Rings with several objects. Advances in Math., Vol. 8, No. 1 (1972), 1–161.
Puppe, D., Über die Axiome für abelsche Kategorien. Arch. Math. 18 (1967), 217–222.
Spanier, E., “Algebraic Topology”. McGraw-Hill, New York, 1966.
Eilenberg, S., and Moore. J.C., Foundationfof relative homological algebra. Mem. Amer. Math. Soc. 55 (1965), 1–39.
Lawvere, F. W., Proc. Nat. Ac. Sciences, Vol. 59, No. 5, (1963), 869–872.
Editor information
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Mitchell, B. (2010). Introduction Category Theory and Homological Algebra. In: Salmon, P. (eds) Categories and Commutative Algebra. C.I.M.E. Summer Schools, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10979-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-10979-9_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-10978-2
Online ISBN: 978-3-642-10979-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)