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 $\Cat$.