This is to announce two papers, "Avoiding the axiom of choice in general category theory" and "Generalized sketches as a framework for completeness theorems" both by M. Makkai, McGill University. Both papers are revised versions of ones with identical titles; the original versions were produced about a year ago. Both papers will appear in the Journal of Pure and Applied Algebra. Unfortunately, they are not available electronically. If you are interested in obtaining copies, please send your request to Makkai@triples.math.mcgill.ca . The abstracts of the papers follow. ABSTRACT of "Avoiding the axiom of choice in general category theory", by M. Makkai, McGill University: "The notion of anafunctor is introduced. An anafunctor is, roughly, a "functor defined up to isomorphism". Anafunctors have a general theory paralleling that of ordinary functors; they have natural transformations, they form categories, they can be composed, etc. Anafunctors can be saturated, to ensure that any object isomorphic to a possible value of the anafunctor is also a possible value at the same argument object. The existence of anafunctors in situations when ordinarily one would use choice is ensured without choice; e.g., for a category which has binary products, but not specified binary products, the anaversion of the product functor is canonically definable, unlike the ordinary product functor that needs the axiom of choice. When the composition functors in a bicategory are changed into anafunctors, one obtains anabicategories. In the standard definitions of bicategories such as the monoidal category of modules over a ring, or the bicategory of spans in a category with pullbacks, and many others, one uses choice; the anaversions of these bicategories have canonical definitions. The overall effect is an elimination of the axiom of choice, and of non-canonical choices, in large parts of general category theory. To ensure the Cartesian closed character of the bicategory of small categories, with anafunctors as 1-cells, one uses a weak version of the axiom of choice, which is related to A. Blass' axiom of Small Violations of Choice ("Injectivity, projectivity, and the axiom of choice", Trans. Amer. Math. Soc. 255(1979), 31-59)." ABSTRACT of "Generalized sketches as a framework for completeness theorems", by M. Makkai, McGill University: "A generalized concept of sketch is introduced. Because of their role, morphisms of (generalized) sketches are called sketch-entailments. A sketch is said to satisfy a sketch-entailment if the former is injective relative to the latter in the standard sense; the models of a set R of sketch-entailments are the sketches satisfying all members of R . R logically implies a sketch-entailment s if every model of R is also a model of {s} . A deductive calculus is introduced in which s is deducible from R iff R logically implies s (General Completeness Theorem, GCT). A large number of examples of kinds of structured category is presented showing that the structured categories are selected from among the corresponding generalized sketches as the models of a set of sketch-entailments. As a consequence, the sketch-entailments satisfied by all structured categories of a given kind are exactly the ones that are deducible from a certain, usually finite, set of axioms. In the finitary case, which is the only one considered in detail in the paper, the notion of deduction is effective, and straightforwardly implementable on a computer. One obtains Specific Completeness Theorems (SCT's), each of which asserts that the exactness properties (certain kinds of sketch-entailments) that hold in a specific class of structured categories coincide with the ones that are deducible from a given set of axioms. Each of these specific completeness theorems is deduced from the GCT, and a particular Representation Theorem (RT); RT's are a well-known class of results in categorical logic. The concepts of Compactness and of Abstract Completeness are introduced, and shown to correspond to the same-named concepts in logics in the usual symbolic form, via a translation between the sketch-based syntax and semantics on the one hand, and the Tarskian syntax and semantics on the other. The sketch-based concepts are available for several logics defined categorically for which there are no available symbolic presentations."