The basic theme of our monograph is the syntax and semantics of a categorical theory of mathematical structures (as used in algebra, model theory, computer science, etc.). The semantics part is the study of properties of the categories of structures. We concentrate on two kinds of categories: locally presentable categories, which are rather close to quasi varieties of algebras, and the broader (and less “pleasant”) accessible categories, which are close to classes of structures axiomatizable in first-order logic. The syntax part describes categories of structures by means of sketches: a sketch is a small category with specified limits and colimits, and a model of the sketch is a set-valued functor preserving the specified limits and colimits. Locally presentable categories are precisely the categories of models of limit-sketches (i.e., no colimit is specified), and accessible categories are precisely the categories of models of sketches. A different approach to syntax is by means of first-order logic: we characterize theories needed to axiomatize locally presentable and accessible categories.

The fundamentals of the theory of locally presentable categories are exhibited in Chapter 1, and those of accessible categories in Chapter 2. The rest of our monograph is devoted to some related topics: algebraic categories, injectivity, categories of models, and Vopěnka's large-cardinal principle. The book is completely self-contained: we expect the reader to be familiar with the basic category-theoretical concepts (such as adjoint, limit, etc.) which we mention briefly in the Preliminaries, but all the more advanced concepts are carefully explained in the text. Facts about large cardinal numbers, used in the last chapter, are presented in the Appendix.