Relations et carrés exacts

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translated by Fosco Loregian

[0014] Translator’s Note a.

This is a hypertext English translation of René Guitart’s paper Relations et carrés exacts (Guitart, 1980).

This version of the original text should be understood as a tribute to the author, as a way of making his paper more accessible to those who want to read it and do not know French well enough, as an excuse to understand its content myself, while making it clearer when needed (there are a few more words of explanation around definitions and theorems, and more diagrams/pictures wherever I felt it would have helped the reader).

All typos and mistakes that might result from this light editing come from my poor understanding and should not be attributed to the original author.

-1. Introduction [001B]

[0015] -1·a (Goals).

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).

[0019] Translator’s Note -1·a·a.

In the original paper, H. appears in the form of a reference, but it is not listed in the Références section. I have no idea what Guitart was referring to.

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 $\Cat$.

[001G] -1·b.

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.

[001H] -1·c.

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.

[001I] -1·d.

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)).

[001J] -1·e.

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].

[001K] -1·f.

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).

0. Notation [0016]

[0017] Definition 0·a (Square in a 2-category).

We call square in a 2-category a quadruple of objects $A,B,X,Y$, a quadruple of morphisms $S : A \to X, T : A \to Y, U : X\to B, V : Y \to B$, and a 2-morphism $\vf : U\circ S \Rightarrow V\circ T$. Such a square will be denoted by a diagram

We will also employ the condensed notation

$$ \vf : \esq{S}{U}{T}{V}, \qquad S \xto{(U,T;\vf)} V, \qquad \vf : US \To VT $$

1. Exactness in the 2-category $\Cat$ [001A]

[001L] 1·a.

We denote the bicategory of bimodules (or profunctors, or distributors) as $\Prof$, and its composition [of 1-cells, TN] as $\otimes$.

We know (cf (Bénabou, 1973) (Gouzou & Grunig, 1973) (Harting, 1977) (Thiébaud, 1971)) that the immersion $\Cat \to\Prof : F\mapsto \Cat[\firstblank, F\sndblank]$ is 2-full, and that $F\approx \Cat[\firstblank, F\sndblank] \dashv \Cat[F\firstblank, \sndblank] =: F^o$, in such a way that if $\vf : U\circ S \To V\circ T$ is a square in $\Cat$, it determines a 2-morphism in $\Prof$ $\tilde\vf : S\otimes T^o \To U^o\otimes V$, with components

[001M] Definition 1·b (Exact square, BC criterion).

A square $\vf : US \To VT$ is called exact if the map $\tilde\vf : X[\firstblank,S\firstblank] \times Y[T\firstblank, \firstblank] \to B[U\firstblank, V\firstblank]$ is a natural isomorphism.

[001N] Theorem 1·c (Zigzag criterion).
[001O] 1·c·a (BC' criterion).

Given the definition of $\tilde\vf$ and since the profunctor composition $\otimes$ is defined by a coend (Linton, 1969), $\vf$ exact is equivalent to the fact that for all $x\in X_0, y\in Y_0$, the maps $\tilde\vf_{x,y}(a)$ induce an isomorphism

$$ \int^a X[x,Sa] \times Y[Ta, y] \overset\cong\to B[Ux, Vy]. $$
[001P] 1·c·b (BC'' criterion).

Since coends can be computed as colimits (MacLane, 1971), $\vf$ exact is equivalent to the fact that for all $x\in X_0, y\in Y_0$, the maps $\tilde\vf_{x,y}(a)$ induce an isomorphism

$$ \hat\vf_{x,y} : \colim\left( T\downarrow y \to A \xto{X[x,S\firstblank]} \Set \right) \overset\cong\to B[Ux, Vy]. $$

Symmetrically, we obtain the equivalent condition: for all $x\in X_0, y\in Y_0$, the maps $\tilde\vf_{x,y}(a)$ induce an isomorphism

$$ \check\vf_{x,y} : \colim\left( (x\downarrow S)^\op \to A^\op \xto{X[T^\op\firstblank,y]} \Set \right) \overset\cong\to B[Ux, Vy]. $$
[001Q] 1·c·c (ZZ criterion).

Since there is an isomorphism of functors $\colim \cong \pi_0 \Elts$ (Guitart & den Bril, 1977) [this means that the square is (pseudo)commutative, i.e. that the colimit of a functor $F : C \to \Set$ is the $\pi_0$ of its category of elements, TN] we can make condition 2 above explicit, and $\vf$ exact equals the following criterion ZZ:

  • (ZZ1) for all $x\in X_0, y\in Y_0$, $p : Ux \to Vy$ in $B$, there exist $a\in A, \, m : x \to Sa, \, n : Ta \to y$ with $Vn\circ \vf_a\circ Um =p$;
  • (ZZ2) For every $(a,m,n), (\bar a,\bar m, \bar n)$ such that $Vn\circ \vf_a\circ Um = V\bar n\circ \vf_{\bar a}\circ U\bar m$, there exists a zig-zag of maps [a headless dotted arrow denotes a morphism in either direction connecting $u,v\in A$, TN] in $A$ such that applying $T$ and $S$ as follows there exist ‘lantern’ diagrams as in the following picture: [i.e. with the property that the components of $\vf$ “connect” the upper half and the lower half of the “Chinese lantern” diagram on the right. TN]
[001R] Theorem 1·d (Local criteria).
[001S] 1·d·a (LI/LF criteria).

The fact that $\vf$ is exact is equivalent, through ZZ, to a condition LI of local initiality or to a condition LF of local finality:

  • (local initiality) $x\in X_0$, $x\downarrow S \to Ux\downarrow V$ is initial [this is the functor sending $(x\to Sa)$ into $(Ux \to USa \xto{\vf} VTa)$, TN];
  • (local finality) $y\in Y_0$, $T\downarrow y \to U\downarrow Vy$ is final [this is the functor sending $(Ta\to y)$ into $(USa \xto{\vf} VTa \to Vy)$, TN].
[001T] 1·d·b ($\pi_0\triangledown$ criterion).

For every $P,Q, x : P \to X, y : Q \to Y$ we consider the diagram of comma objects, where $x \tdown_\vf y$ is a strict pullback, and $x\btdown y : x \tdown_\vf y \to Ux\downarrow Vy$ [the dotted arrow, TN] is the functor induced by the universal property of said comma object.

If $\pi_0(x\btdown y)$ is a bijection we say that $\vf$ is exact at $\smat{x \\ y}$ and we write $\smat{x \\ y}\vDash \vf$. So, $\vf$ exact equals the following condition ($\pi_0\tdown$): for all $P,Q, x : P \to X, y : Q \to Y$, we have $\smat{x \\ y}\vDash\vf$.

The proof is by inspection of the strict pullback $x \tdown_\vf y$ and of the functor $x\btdown y$; $x \tdown_\vf y$ is the subcategory of the product on objects of the form $\left\{ p,a,q \mid \nbsmat{ Xp &\overset{f}\to& Sa \\ Ta &\underset{g}\to& Yq } \right\}$, and $x\btdown y$ is defined sending such an object to $\left\{ p,q \mid \nbsmat{UXp &\overset{f}\to& USa \\ &\swarrow&\\ VTa &\underset{g}\to& VYq } \right\}$.

[001U] Theorem 1·e (Comparison with comma and co-comma objects).

We denote the comma square of the pair $U,V$, and $\bar\vf : A \to U\downarrow V$ the only morphism such that $a * \bar\vf = \vf$.

For every $Z, P : X \to Z, Q : Y \to Z$ we have where $\sum_{P,Q}\tilde\vf$ and $\sum_{P,Q}\bar\vf$ denote the composition functors with $\tilde\vf$ and $\bar\vf$. [The correspondences are defined as follows: TN]

[001V] 1·f (H criteria).

Since every bimodule can be written, in $\Cat$, in the form $P^o\otimes Q$, as a consequence of Yoneda lemma, $\tilde\vf$ is invertible if and only if the $\sum_{P,Q}\tilde\vf$ are bijective so that $\vf$ exact equals the following condition: for all $Z$, $P : X\to Z, Q: Y\to Z$, the function

$$ \textstyle\sum_{P,Q} \bar\vf : \Cat(Pd_0, Qd_1) \to \Cat(PS,QT) $$

is a bijection.

There is a similar criterion H$^\op$ with co-comma squares. A 2-category $\mathbb{K}$ that is representable and co-representable, where H $\iff$ H$^\op$ is called symmetric. This is the case of $\Cat$.

[001W] Definition 1·g (Multiplicative square).

In a 2-category $\mathbb{K}$, we call multiplicative square the following data: a square $\vf : S \xto[U]{T} V$ and, for each $R,\Omega, P : R \to X$, $Q : R \to Y$, $F : X \to \Omega$, $G:Y\to \Omega$, $\alpha : UP \To VQ$, $\beta : FS \To GT$, a 2-morphism $\beta\otimes_\vf\alpha : FP \To GQ$ satisfying the conditions

  • (un) $\beta\otimes_\vf\vf =\beta$, $\vf\otimes_\vf\alpha=\alpha$ for all $\alpha,\beta$;
  • (bil) $(b*\beta) \otimes_\vf (\alpha * a)=b * (\beta\otimes_\vf\alpha) * a$ for every $a : R\tick \to R$, $b : \Omega \to \Omega\tick$.
[001X] Remark 1·h.

If $\mathbb{K}$ has comma and co-comma objects, the presence of $\firstblank\otimes_\vf \sndblank$ amounts, with $(U\downarrow V)$ a comma object and $(S\diamond T)$ a cocomma object, to a 2-cell $\otimes : i_0 d_0 \To i_1d_1$ such that in such a way that $\beta \otimes_\vf \alpha = \underline\beta * \otimes * \overline\alpha$.

In a category $\mathbb{K}$ (regarded as a 2-category without nontrivial 2-morphisms) we recover the definition of exact square given in Grandis (Grandis, 1977).

[001Y] Remark 1·i (M criterion).

If $\mathbb{K}=\Cat$, $\otimes : i_0\circ d_0\To i_1 \circ d_1$ amounts to the natural transformation

$$ \tilde\otimes : U^oV = d_0d_1^o \To i_0^oi_1 = ST^o $$

that is such that, for given $\alpha,\beta$,

$$ \tilde\alpha * \tilde\otimes* \tilde\vf = \tilde\alpha \qquad \tilde\vf * \tilde\otimes * \tilde\beta = \tilde\beta. $$

However, since $PQ^o$ is a generic profunctor, as well as $F^oG$, we can take $PQ^o=U^oV$, $\tilde\alpha = S$, $F^oG=ST^o$, $\tilde\beta = 1$, hence $\tilde\otimes = \tilde{\vf}^{-1}$, so $\vf$ is exact. We can observe directly, using 1·f [001V], that if $\vf$ is exact, we can define the multiplicative structure $\nu\otimes_\vf \mu := (\sum_{F,G}\bar\vf)^{-1}\nu * \bar\mu$. From this we deduce the following multiplicative criterion: a square $\vf : US \To VT$ in $\Cat$ is exact if and only if it admits a multiplicative structure $\otimes_\vf$ (so, $\otimes_\vf$ is unique, given by the inverse of $\tilde\vf$).

[001Z] Theorem 1·j (PP criterion; pointwise left extension).

A square $\vf : US \To VT$ in $\Cat$ is exact if and only if, for every pointwise left extension, (cf. (MacLane, 1971) (Street, 1974)), the composite triangle remains a pointwise left extension.

[0020] Remark 1·j·a.

In fact, consider $G : Y \to Z$. Using 1·f [001V]) for $\vf$, we have $\cate{Nat}(FS,GT)\cong \cate{Nat}(Fd_0, Gd_1,G)$, and by the assumption made on $\beta$, we have that the latter is isomorphic to $\cate{Nat}(RV,G)$. Hence, $(R*\vf) \c (\beta * S) : FS \To RVT$ is an extensions, and moreover it is pointwise, thanks to Theorem 1.6.

For the converse, 1·c·b [001P]) means that for every $x\in X_0$ the diagram (with $i_x(y) : X[x,y] \to B[Ux, Uy]$ the action of $U$ on morphisms) is a pointwise exension. To conclude, it remains to check that $B[Ux, \firstblank]$ is the pointwise extension of $X[x,\firstblank]$ along $U$.

[0021] Theorem 1·k (Composition rules).

Using Definition 1·b [001M], exact squares can be composed horizontally or vertically

[0022] Theorem 1·l (D criterion).

In $\Cat$, we have that a square $\vf : US \To VT$ in $\Cat$ is exact if and only if the square is exact.

This is evident using criteria 1·c·c [001Q], 1·f [001V] or Remark 1·i [001Y].

[0023] Theorem 1·m (PP* criterion).

Using the definition of Street, we see that $\lambda : GV \To F$ is a pointwise right extension if and only if $\lambda^\op : F^\op \To G^\op V^\op$ is a pointwise left extension, hence applying the dualisation rule Theorem 1·l [0022] and Theorem 1·j [001Z] we obtain that a square $\vf : US \To VT$ in $\Cat$ is exact if and only if, for every pointwise right extension the composite triangle remains a pointwise right extension.

[0024] Theorem 1·n (Preservation conditions).

In a 2-category with comma and co-comma squares, 1·f [001V] implies Theorem 1·j [001Z] and Theorem 1·m [0023]; this is true for $\Cat$ and $\Cat^\op$. If we call co-pointwise left lifting (resp., right lifting) in $\Cat$ a pointwise left extension (resp., right extension) in $\Cat^\op$, we have that if in $\Cat$ $\vf : US \To VT$ is exact, then

  • (PP) $\vf$ preserves the pointwise left extension under $U$;
  • (PP$^*$) $\vf$ preserves the pointwise right extension under $V$;
  • (PP$^\op$) $\vf$ preserves the co-pointwise left liftings under $T$;
  • (PP$^{*\op}$) $\vf$ preserves the co-pointwise right liftings under $S$.

[0025] Definition 1·o (Strong exact square).

In a 2-category $\mathbb K$, we call strong exact square (resp., absolute exact square) a square $\vf : S \xto[U]{T} V$ such that for every $X\in \mathbb K_0$, the square $\mathbb K[X,\vf]$ is exact in $\Cat$ (resp., for every functor $\Phi : \mathbb K \to \mathbb L$, the square $\Phi(\vf)$ is exact in $\mathbb L$).

[0026] Theorem 1·p (Exponentiation rule).

In $\mathbb K=\Cat$, a square is exact if and only if it is strong exact.

[0027] Example 1·q.

Applying the previous criteria for exactness Definition 1·b [001M], 1·c·a [001O], 1·c·b [001P], 1·c·c [001Q], 1·d·a [001S], 1·d·b [001T], 1·f [001V] and its opposite, Remark 1·i [001Y],Theorem 1·j [001Z], Theorem 1·m [0023] (all equivalent to each other in $\Cat$) as well as the composition, dualisation, exponentiation criteria, we immediately obtain the following examples:

  • (EE1) the squares are exact (this is evident using Remark 1·i [001Y]. Following Street (Street, 1974) this is equivalent to the Yoneda lemma; we will call these squares Yoneda squares.
  • (EE2) a comma square is exact.
  • (EE3) a co-comma square is exact.
  • (EE4) the square is exact if and only if $F$ is fully faithful.
  • (EE5) the square is exact if and only if $F$ is co-fully faithful.
[0028] Translator’s Note 1·q·a.

A “co-fully faithful” functor is a functor $F$ with the property that each pre-composition $_\circ F$ is fully faithful. (Thanks to Nathanael Arkor for the suggestion!)

  • (EE6) a square is exact if and only if $\epsilon$ is the counit of an adjunction $L\dashv U$;
  • (EE7) a square is exact if and only if $\eta$ is the unit of an adjunction $L\dashv U$;
  • (EE8) more generally, the square is exact if and only if $\epsilon$ is the counit of a relative adjunction $L\overset{Y}\dashv U$ (i.e., $L$ is a partial left adjoint to $U$, with respect to $Y$);
  • (EE9) a square is exact if and only if $\langle V,\vf\rangle$ is an absolute left extension of $S$ along $T$ (cf. §5.1);
  • (EE10) a square is exact if and only if $\langle U,\vf\rangle$ is an absolute right extension of $S$ along $T$.
[0029] Remark 1·q·b.

Condition EE8 can be dualised to relative right adjoints. From EE10 we deduce the formal characterisation of adjoints: the counit of an adjunction exhibits the left adjoint as the absolute right extension of the identity along the right adjoint, and dually for the unit.

[002A] Remark 1·r (Connection with exactness in $\Ab$).
  • Conditions 1·f [001V]), Theorem 1·j [001Z]), and Remark 1·i [001Y]) apply in every representable 2-category, and they are in general not equivalent to each other. A fortiori this is true in every 1-category.
  • Considering a 1-category $K$ as a 2-category $\mathbb K$ without nontrivial 2-morphisms, conditions 1·f [001V]), Theorem 1·j [001Z]) make sense; so, (cf. Hilton (Hilton, 1966)) in $\Ab$ a square is exact in the sense 1·f [001V]) if and only if the sequence $A\to X\to B$ is exact at the object $X$.
  • Since $\Ab$ embeds in $\Cat$ [an abelian group is a one-object category, TN], the square above can be regarded in $\Cat$; and it is exact in the 2-category $\Cat$ if and only if $A\to X\to B\to 0$ is exact in $X$ and in $B$.
  • The calculus of exactness in $\Cat$ is thus an enlargement of the one in $\Ab$; in this spirit, the absolute exact squares (e.g. adjunctions and Yoneda squares) generalise the split exact sequences, and $\tilde\vf$ represents the homology of the square $\vf$.

2. Exact grids in a symmetric 2-category [001C]

[002I] Definition 2·a (Exact grid).

In a 2-category like $\Prof$, the formulas $\tilde\vf : S T^o \cong U^oV$ with $T\dashv T^o, U\dashv U^o$ (or $\hat\vf : T_o S \cong VU_o$ with $T_o\dashv T, U_o\dashv U$) can be seen as exchange formulas between “2-fractions” or “2-co-fractions”, left and right.

In such a perspective of a 2-dimensional calculus of fractions, exact grids correspond to higher-order exchange formulas as $(\tilde\vf_1 U_2^oV_2)\circ (S_1T_1^o\tilde\vf_2) : S_1T_1^oS_2T_2^o \cong U_1^oV_1U_2^o V_2.$

Let $\mathbb K$ be a symmetric 2-category (cf. §1.4) and $X$ an object; we say that the diagram is an exact grid at $X$ if and only if the completed figure is a exact square (in the sense of condition 1·f [001V] or it opposite).

Similarly, a more complex grid will be called exact if completing it at its empty places with comma or cocomma squares we obtain a big exact square.

[002J] Definition 2·b (Opaque functor).

If $T : A\to X$ we say that a square $\vf_2 : US\To VR$ is $T$-exact if the grid is exact at $X$, and that $T$ is an opaque functor if the square is $T$-exact.

[002K] Remark 2·c.

The characterisation of exact grids in $\Ab$ or $\Cat$ is immediate, thanks to the calculus of additive relations in $\Ab$ and the calculus of bimodules in $\Cat$. In a generic symmetric 2-category, we have to resort to an adequate `calculus of relations’, as described in the following section. In a category with a Yoneda structure, see the calculus outlined in §4.

3. Calculus of relations and types of exactness [001D]

[002L] 3.0.

If we have two 2-functors

$$ J,J^o : \mathbb{K} \rightrightarrows \mathbb{B},\mathbb{B}^\op $$

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.

[002M] Theorem 3·a.
  • 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
$$ \Span(\Ab)/{\equiv_\mathbf{H}} \cong \Rel(\Ab)\qquad \Span(\Cat)/{\equiv_\mathbf{H}} \cong \Prof. $$

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 $\mathbb{X}’=\Prof$.
[002N] Remark 3·b.

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.

4. Pointwise and exact squares in uniform Yoneda structures [001E]

[002O] Definition 4·a (Yoneda strucure).

Let $\mathbb K$ be a 2-category equipped with a Yoneda structure in the sense of Street-Walters (Street & Walters, 1978) (see also (Street, 1974)), which means $\mathbb K$ is equipped with a right ideal $\mathbb A\subseteq\mathbb K_1$ of “admissible” 1-arrows, and for every $A\in \mathbb A$ a “Yoneda morphism” $Y_A : A \to PA$ is given so that for each $f : A \to B$ in $\mathbb A$ there is a diagram with the property that

  • $\chi^f$ exhibits $B(f,1)$ as the left extension of $Y_A$ along $f$;
  • $\chi^f$ exhibits $f$ as the absolute lift of $Y_A$ along $B(f,1)$;
  • for every $k : B\to PA$ and $\sigma : B(f,1)\To k$, $\sigma$ is an isomorphism in $\mathbb K$ if and only if $(\sigma * f)\circ \chi^f$ exhibits $f$ as absolute lift of $Y_A$ along $k$.
[002P] Remark 4·b.

Street and Walters prove that

$$ PB(Y_B\circ f,1) =: Pf $$

determines a pseudo-functor $P : \mathbb A \to \mathbb K$, and moreover if $j : A \to B$ is such that $A,B$ and $B(j,1)$ are admissible, then $Pj\dashv\forall_j$, where

$$ PA(B(j,1),1)=:\forall_j $$

Examples of such structure: $\Cat$, and more generally $\VCat$; $\Cat(\mathbb E)$ for a topos $\mathbb E$.

[002Q] Definition 4·c (Uniform Yoneda structure).

We call uniform Yoneda structure on $\mathbb K$ a Yoneda structure as defined above in the sense of (Street & Walters, 1978), where in addition

  • each extension $\chi^f$ is pointwise;
  • for every $g : B\to PC$ there exists $Z$ and $B\xto{b} Z \xto{c} C$ such that $g = Pc\circ Y_Z\circ b$.

Examples of such structure: $\Cat$, uniform cosmoi of (Street, 1974).

[002R] Definition 4·d (Exactness from admissibility, BC# criterion).

Let $\mathbb K$ be a 2-category with a uniform Yoneda structure on it.

  • A square $\esq{S}{U}{T}{V} : \nbsmat{A &\to&Y\\ \downarrow && \downarrow \\ X & \to&B}$ is called admissible if $S,T,U,X(s,1)$ are admissible, and $Ps$ has a right adjoint $\exists_s$. Then $\vf$ determines $\widetilde{\vf} : \exists_s\circ Y(T,1) \To B(U,1)\circ V$, i.e. a 2-cell filling $\nbsmat{PA &\leftarrow & Y \\ \downarrow && \downarrow \\ PX &\leftarrow & B}$, like in the case of categories and bimodules discussed in §1 (of which this is a generalization).
  • If $\vf$ is admissible in the sense above, the square $\esq{S}{U}{T}{V}$ is called exact (according to “condition BC#”) if $\widetilde{\vf}$ is an isomorphism.
  • If $\vf$ and $\widetilde{\vf}$ are both admissible, $\vf$ is called $Y$-pointwise if $\widetilde{\vf}$ is also exact.
[002S] Theorem 4·e (PP# criterion).
  • In a 2-category $\mathbb K$ with a Yoneda structure, an admissible square $\vf : \esq{S}{U}{T}{V}$ is exact according to BC# if and only if the diagram exhibits $B(U,1)\circ V$ as the left extension of $Y_X\circ S$ along $T$. This is condition PP#
  • If the Yoneda structure is uniform conditions BC# and PP# are equivalent for the squares $\vf$ such that $U$ is admissible.
  • Relative adjunctions and, more in general, absolute extensions are (as in the case of $\Cat$) exact squares in the sense of BC#.

5. Opaque functors, absolute extensions, and applications to pro-localisation [001F]

[002B] 5·a.

The main motivation for our definition and our study of exact squares, as exposed through sections 1 to 4 was the study of Paré’s thesis about absolute limits (Paré, 1969) and the characterizations of absolute extensions given in (Guitart, 1977) (partially continuing on the line of some results in Thiébaud (Thiébaud, 1971) and Harting (Harting, 1977)).

We call absolute left extension a diagram in $\Cat$ of the form such that for each $H: Z\to K$ the composite $H\vf$ is a left extension (i.e. $HR\cong \Lan_J HF$).

[002C] Definition 5·b (Opaque functor).

A functor $F : A \to B$ is said to be opaque if for each $a,a\tick\in A_o$ and $b : Fa\to Fa\tick$ in $ B$ there is a zig-zag in $A$, the image of which under $F$ fits into a lantern diagram

[002D] Theorem 5·c.
  • $\vf$ is an absolute extension if and only if $\vf$ is a pointwise extension and can be computed as an absolute colimit if and only if $Y_Z\circ\vf$ is an extension.
  • Following §1.14 (with 1.1), $\vf$ is an absolute extension if and only if $\vf : \esq FJIR$ is exact, i.e. $\widetilde{\vf}$ is an isomorphism (in such a case, the zig-zag criterion and all local criteria take special forms, see (Guitart, 1977)).
  • $\vf$ is an absolute extension if and only if, taking the comma square $\esq{d_0}{d_1}UF$, the induced functor $J\circ d_1 \downarrow Y \to Z \downarrow R$ is opaque and surjective on objects.
  • As in §2.1, $F : A \to B$ is opaque if and only if the square is exact, i.e. (cf. §1.1) if and only if $F\otimes \widetilde{1_F} : F^\text{o}\otimes F \otimes F^\text{o} \to F^\text{o}$ is an isomorphism, or equivalently if and only if the pro-comonad on $ B$ associated to $F$ (see (Harting, 1977) (Thiébaud, 1971)) is idempotent.
  • If $ B_F$ is the full subcategory of $ B$ on the image of $F$, i.e. on objects of the form $FA$ for $A\in A_o$, $F$ is opaque if and only if the triangle is an absolute left extension.
  • $F$ is opaque if and only if it can be factored as a composition $T\circ Q$ with $T$ fully faithful and $Q$ co-fully faithful.
  • Thanks to Isbell’s zig-zag theorem, if $F$ is bijective on objects, then $F$ is opaque if and only if $F$ is an epimorphism of $\Cat$.
[002E] Definition 5·d (Consistent functor, C' condition).

A functor $F : A \to B$ is called consistent if it satisfies Frei’s “C condition” (Frei, 1976), which following (Hilton, 1966) p. 241 is equivalent to the following condition: for all $A\in A_o$, $FA \cong \varprojlim \big( FA\downarrow F \to A \xto{F} B \big)$, and which, following Linton (Linton, 1969) is equivalent to the fact that $\exp_F^\op$ is fully faithful on the pairs $(B,FA)$.

[002F] Definition 5·e (Very rich functor).

A functor $F : A \to B$ is called very rich if for all $A,A\tick\in A_o$, and $b : FA\to FA\tick$, there exists a span $A\xleftarrow{r} V \xto{r\tick} A\tick$ in $ A$ such that $Fr$ is invertible and $b = Fr\tick\circ (Fr)^{-1}$.

[002G] Theorem 5·f (Pro-localization conditions).
  • For every functor we have the implications

$F$ is very rich $\To$ $F$ is rich (Hilton) $\To$ $F$ is opaque $\To$ $F$ is consistent (Frei)

and each converse implication is false in general.

  • An opaque functor is consistent and co-consistent (i.e. $F^\op$ is consistent).
  • If $L\dashv U$ is an adjunction, $U$ is opaque if and only if $L$ is opaque.
  • If $L\dashv U$ is an adjunction, $U$ is very rich if and only if it is rich, if and only if it is opaque, if and only if it is consistent.
[002H] Remark 5·g.

Another important use of opaque functors is the following: Mac Donald (MacDonald, 1976) shows that Kan extensions of cohomological theories can be taken along rich functors; in fact, it can also be taken along opaque functors (and more in general with exact squares).