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 $\vf$.