Locally presentable and accessible categories.

*(English)*Zbl 0795.18007
London Mathematical Society Lecture Note Series. 189. Cambridge: Cambridge University Press. xiv, 316 p. (1994).

The accessible categories of M. Makkai and R. Paré [“Accessible categories: The foundations of categorical model theory” (1989; Zbl 0703.03042)] form a natural extreme generalization of locally presentable categories, so a combined treatment of the two topics suggests itself. In a technical sense, Makkai and Paré already gave a combined treatment, re-doing all the main points of P. Gabriel and F. Ulmer [“Lokal präsentierbare Kategorien”, Lect. Notes Math. 221 (1971; Zbl 0225.18004)]; but a reader not already familiar with locally presentable categories would not want to begin with Makkai- Paré. The present book is the only really accessible exposition.

There are six chapters. Predictably, we find (1) Locally Presentable Categories, (2) Accessible Categories, (3) Algebraic Categories, (4) Injectivity Classes, (5) Categories of Models. More distinctive is the last chapter, (6) Vopĕnka’s Principle. The motivation for that is very strong, resting on the remarkable theorem of the authors with V. Trnková, that every limit-closed full subcategory of a locally presentable category is reflective if and only if the weak Vopĕnka principle holds [J. Adámek, J. Rosický and V. Trnková, “Are all limit-closed subcategories of locally presentable categories reflective?”, Lect. Notes Math. 1348, 1-18 (1988; Zbl 0668.18004)]. If the full Vopĕnka principle holds, there are companion results about coreflective, and about weakly reflective, subcategories. The Vopĕnka principle has several equivalent statements; perhaps the simplest is that there exists no large rigid class of graphs. It implies the existence of measurable cardinals; consistency of the principle is implied by the existence of huge cardinals.

Basically, the authors have taken the indicated material, organized it efficiently, written a very lucid, readable development of it in 280 pages, and added helpful historical remarks to each chapter and a brief appendix on large cardinals. There are some novel results in Chapters 1- 5, most notably a significant improvement of the Gabriel-Ulmer theorem on “local generation” of locally presentable categories. As the authors observe, the theorem that locally presentable categories are precisely the categories of models of (small) essentially algebraic theories, though long present in the folklore, seems not to have been publicly proved until now. Chapter 4 covers Diers’ locally multipresentable categories [Y. Diers, Arch. Math. 34, 344-356 (1980; Zbl 0432.18006)].

There are six chapters. Predictably, we find (1) Locally Presentable Categories, (2) Accessible Categories, (3) Algebraic Categories, (4) Injectivity Classes, (5) Categories of Models. More distinctive is the last chapter, (6) Vopĕnka’s Principle. The motivation for that is very strong, resting on the remarkable theorem of the authors with V. Trnková, that every limit-closed full subcategory of a locally presentable category is reflective if and only if the weak Vopĕnka principle holds [J. Adámek, J. Rosický and V. Trnková, “Are all limit-closed subcategories of locally presentable categories reflective?”, Lect. Notes Math. 1348, 1-18 (1988; Zbl 0668.18004)]. If the full Vopĕnka principle holds, there are companion results about coreflective, and about weakly reflective, subcategories. The Vopĕnka principle has several equivalent statements; perhaps the simplest is that there exists no large rigid class of graphs. It implies the existence of measurable cardinals; consistency of the principle is implied by the existence of huge cardinals.

Basically, the authors have taken the indicated material, organized it efficiently, written a very lucid, readable development of it in 280 pages, and added helpful historical remarks to each chapter and a brief appendix on large cardinals. There are some novel results in Chapters 1- 5, most notably a significant improvement of the Gabriel-Ulmer theorem on “local generation” of locally presentable categories. As the authors observe, the theorem that locally presentable categories are precisely the categories of models of (small) essentially algebraic theories, though long present in the folklore, seems not to have been publicly proved until now. Chapter 4 covers Diers’ locally multipresentable categories [Y. Diers, Arch. Math. 34, 344-356 (1980; Zbl 0432.18006)].

Reviewer: J.R.Isbell (Buffalo)

##### MSC:

18C10 | Theories (e.g., algebraic theories), structure, and semantics |

03E55 | Large cardinals |

03G30 | Categorical logic, topoi |

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |