The scope of the present text is to build a bridge between logic and homology by exposing the calculus of exactness (and its connections with the calculus of relations) in a symmetric 2-category (Section 3 [001D]) and in a uniform Yoneda structure (Section 4 [001E]) in a general fashion, but first of all in the particular case of the 2-category $\Cat$.

Section 1 [001A] introduces the concept of exact square in a representable 2-category (cf. H. §1.4).

We give, in the case of the 2-category $\Cat$, several equivalent conditions (zig-zag criterion, local criterion, criterion of comparison of comma squares or co-comma squares, multiplicative criterion, criterion of preservation of pointwise left extensions, criterion of preservation of pointwise right extensions, composition, duality and exponentiation rules). We will see that these exact squares constitute a common generalisation of all the following situations: the usual exact sequences, Jaffard-Poitou exact sequences; Hilton’s exact squares in an abelian category; absolute extensions, absolute limits, opaque functors, co-fully faithful or rich; partial adjunctions (i.e., absolute liftings), adjunctions, fully faithful functors; comma squares and co-comma squares. The end of Section 2 [001C] shows why it seems now preferable when studying exactness in $\Ab$ to consider $\Ab$ as a 2-category (without non-identity 2-morphisms) of $\Cat$, and to compute [exactness, TN] in $\Cat$.

In Section 2 [001C], we analyse the problem of exact grids, and in Section 3 [001D] we indicate a possible solution to this problem through “relational calculi”; we observe that all the different ways to approach the calculus of relations (Bénabou, Guitart, Hilton, Meisen, Street-Walters, Coppey) each induce a notion of exactness, and reciprocally, every notion of exactness determines a congruence on a bicategory of spans, in a certain way representing it.

In Section 4 [001E], we define and characterise exact squares in a Yoneda structure in the sense of Street-Walters (Street & Walters, 1978), and we underline the possibility of defining pointwise squares in this framework. Maybe when we have sufficiently understood the link between exact and pointwise squares, we can construct a solid axiomatic approach to exactness in a 2-category.

In Section 5 [001F] we define opaque functors, exposing their relation with absolute extensions and exact squares, and we show their role for Deleanu-Frei-Hilton’s Categorical Shape Theory (cf. A. Frei (Frei, 1976)) and for Kan extensions of cohomological theories (cf. J.L. Mac Donald (MacDonald, 1976)).

These are, in fact, the results obtained first in (Guitart, 1977), because to reach the notion of exact squares, we started from R. Paré (Paré, 1969) absolute limits, we went through the Beck-Chevalley condition (Chevalley, 1964-1965) and the zig-zag theorem of Isbell (Guitart, 1982), and we only belatedly observed the connection of these questions with the exact squares in abelian categories (in the sense of Lambek (Lambek, 1958), Mac Lane, Brinkman-Puppe, Hilton (Hilton, 1966)), which suggested 1·f [001V] and Remark 1·i [001Y] definitions of Section 1 [001A].

A preliminary version of this text circulated in May 1978, and it was presented in Oberwolfach in August 1979 (Guitart, 1979), (Guitart, 1980).

The reader will find extensions (exact squares in $\Set$, in $\Grp$, extensions of these methods to non-representable 2-categories) in van den Bril (Van den Bril, 1980), applications to deduction in (Guitart, 1982), and to categories with models in (Van den Bril, 1982).