If we have two 2-functors

available, with the property that $J=J^o$ on objects, i.e. for all $X\in \mathbb{K}_o$ we have $JX=J^oX$, we can define a $J$-exact square as a square $\esq STUV$ such that $JS\circ J^oT\cong J^oU\circ JV$.

Conversely, let $\mathcal{E}$ be a class of squares in $\mathbb{K}$. We denote $\Span(\mathbb{K})$ the multiplicative system having as terms the spans $X \xot{a} R \xto{b} Y$ in $\mathbb{K}$, and where composition is defined up to isomorphism by a comma construction We denote $\equiv_\mathcal{E}$ the smallest equivalence relation on $\Span(\mathbb{K})$ such that $\Span(\mathbb{K})/{\equiv_\mathcal{E}}$ is a bicategory, and such that a span $(A,p,q)\equiv_\mathcal{E} (B, r,s)$ when there are 1-cells $u,v$ and 2-cells $\alpha : up\To vq$ and $\beta : ur \To vs$, both in $\mathcal{E}$. If we assume that $\mathcal{E}$ contains all the Yoneda squares, we recover the situation of the above construction.

• Following the above remarks, to each “calculus of relations” on $\mathbb{K}$, we can associate a “notion of exactness” in $\mathbb{K}$, and conversely, if $\mathbf{E}$ is a class of distinguished squares n $\mathbb{K}$ that we deem “exact” we can build a calculus of relations on the bicategory $\Span(\mathbb{K})/{\equiv_\mathbf{E}}$.
• If $\mathbf{H}=\mathbf{H}(\mathbb{K})$ is the class of exact squares in the sense of condition 1·f [001V], we can show that

We know that conversely, $\mathbf{H}(\Ab)$ and $\mathbf{H}(\Cat)$ can be described in terms of $\Rel(\Ab)$ and $\Prof$.

• In theorem 3.4 of Meisen’s paper (Meisen, 1974), we can consider $\hom_{\mathbb{X}’}(X,Y)=\cate{Fix}(C_{X,Y})$, with $C_{X,Y}$ the triple induced on $\Span(X,Y)$ by the adjunction If $\mathbb{K}=\Ab$, we get $\mathbb{X}’=\Rel(\Ab)$. If $\mathbb{K}=\Cat$, we get $\mathbb{X}’=\Prof$.

Many calculi of relations in the literature are of the form $\Span(\mathbb{K})/R$ for some $\mathbb{K},R$ (cf. Monades involutives complémentées, pp. 35, 41 cited in (Guitart, 1977)). Similarly for the relations associated to the decompositions of Coppey (Coppey, 1971) on $\mathbb{K}$, the construction by Meisen (Meisen, 1974) of $\cate{Pull}(\mathbb{K})$ and $\cate{Rel}_M$ where $(E,M)$ is a Kelly decomposition, and the relations in a uniform Yoneda structure (§4).

Given a category $\mathbb{K}$, let $\cate{Rel}(\mathbb{K})$ denote the quotient of $\Span(\mathbb{K})$ by the relation $(R,a,b)\sim (R’, a’, b’)$ if and only if there are $u : R’ \leftrightarrows R: v$ such that $au=a’$, $bu=b’$, $a’v=a$ and $b’v=b$. The corresponding notion of exact square is that of a semi-cartesian square.

We can also obtain the semi-cartesian squares as the exact squares of a topos in terms of its usual calculus of relations. This is of course true for $\Set$, where condition 1·f [001V] has a different meaning.