Graduate Texts in Mathematics. 5. New York, NY: Springer. xii, 314 p. DM 119.00; öS 869.00; sFr. 108.50; £46.00; $ 54.95 (1998).
For the reviewer this is one of the most intelligible books on categories. For myself I worked with its German translation [“Kategorien. Begriffssprache und mathematische Theorie” (1972; Zbl 0243.18001)] since its appearence.
Now the new edition (first ed. 1971; Zbl 0232.18001) is much more useful. There is a new chapter on “symmetry and braiding in monoidal categories” [including the famous coherence theorem of the author, cf. S. Mac Lane, CMS Conf. Proc. 13, 321-328 (1992; Zbl 0789.18006)]. But here the now also famous coherence theorem of R. Gordon, A. J. Power, and R. Street [see “Coherence in tricategories”, Mem. Am. Math. Soc. 558 (1995; Zbl 0836.18001)] is missing in the bibliography.
A further new chapter is on “structures in categories”. This is on internal categories, 2-categories, and bicategories. The philosophical account of “Structure in mathematics” by the author [S. Mac Lane, Philos. Math., III. Ser. 4, No. 2, 174-183 (1996; Zbl 0905.18001)] is not addressed.
It’s a pity that in the meanwhile the interesting notes have vanished for these new chapters. Therefore, a much too short paragraph on “perspectives” is added, giving some bibliographical hints to more advanced work, esp. coming up from physics. I think this is poor. A second book by the author on these topics with the old standard is desirable, and perhaps lower in price as this reprint brushed up with some additions.
There is now one page on topoi and also an appendix of three sides upon foundations – on these, only the author’s book is mentioned [S. Mac Lane and I. Moerdijk, “Sheaves in geometry and logic” (1992; Zbl 0822.18001)], not P. T. Johnstone’s classic [“Topos theory” (1977; Zbl 0368.18001)]. To whom it may concern?