transactions of the american mathematical society Volume 284, Number I. July 1984 THE FAMILY APPROACH TO TOTAL COCOMPLETENESS ANDTOPOSES BY ROSS STREET Abstract. A category with small homsets is called total when its Yoneda embedding has a left adjoint; when the left adjoint preserves pullbacks, the category is called lex total. Total categories are characterized in this paper in terms of special limits and colimits which exist therein, and lex-total categories are distinguished as those which satisfy further exactness conditions. The limits involved are finite limits and intersec- tions of all families of subobjects. The colimits are quotients of certain relations (called congruences) on families of objects (not just single objects). Just as an arrow leads to an equivalence relation on its source, a family of arrows into a given object leads to a congruence on the family of sources; in the lex-total case all congruences arise in this way and their quotients are stable under pullback. The connection with toposes is examined. In the introduction to [7] and elsewhere, I have maintained that a Grothendieck topos [1] is a Barr exact category [2] for which the exactness axioms are extended to deal with families of arrows and not just single arrows. The present paper makes this explicit without the bicategorical paraphernalia. The concepts involved are far-reach- ing and the main result (Theorem 14) characterizes both total categories [8] and lex-total categories [8, 6] in familial terms. Total categories are precisely the categories, algebraic and topological [12, 10], at which traditional category theory was aimed. Lex-total categories are essentially toposes [6]. Recall that a category is regular when finite limits exist, every arrow factors as a strong epic arrow followed by a monic arrow, and every pullback of each strong epic arrow is strong epic. (Strong epics are defined from the monies by a diagonal property [5].) That this agrees with the definition of Barr [2] when finite limits exist is an observation of Joyal [9]. I am indebted to Max Kelly for providing a proof of Joyal's observation, on which proof the present family version (Theorem 3) is based. The family version of a regular category, a familially regular category, takes the notion of "strong epic family of arrows" as basic, requires that families of arrows should factor through strong epic families via monies (this already implies the existence of many limits including all finite ones; see Proposition 1), and requires strong epic families to be stable under pullback. Received by the editors May 10, 1983 and, in revised form, September 21, 1983. 1980 Mathematics Subject Classification. Primary18A30, 18A32, 18B25; Secondary 18F10, 18F20. Key words and phrases. Total and lex-total category, exact category, factorization of families, Grothendieck topos, finitely presentable, universal extremal epimorphic family. ©1984 American Mathematical Society 0002-9947/84 $1.00 + $.25 per page 355 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use