Another notion of exactness: how to refine it, and where does it fit?

Question

There are many notions of “exactness” in category theory, algebraic geometry, etc. Here I offer another that generalizes the category of frames, the notion of valuation (from probability theory), and touches on aspects of abelian categories and the Seifert-van Kampen theorem.

My intention is to hear comments from the community on

• how this notion should be refined and improved,
• other examples and fields where this notion arises,
• whether this notion fits into a larger theory or extends an existing theory.

The rough idea is that an exactness structure on a category is a set of commutative squares, such as pushout-pullbacks, and that a functor is exact when it preserves the chosen squares. In this sense, it is something like a “limit sketch”.

One last note: I am aware that the term “exact square” already exists—and I'll give it as an example of what I call exact squares—so although I think the name “exact square” is fitting, I would also be happy to hear alternatives.

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Let $2=\fbox{$\bullet\to\bullet$}$ denote the free arrow category, so $2\times 2$ is the free commutative square.

Definition: Let $\mathcal{C}$ be a category with an initial object $\bot$. An exactness structure on $\mathcal{C}$ is a set $E$ of squares, $e\colon 2\times 2\to \mathcal{C}$, called exact squares

$$\require{AMScd}\tag*{$e$}\begin{CD} A @>f>> B \\ @VgVV @VVhV \\ C @>>i> D \end{CD} $$ satisfying the following conditions:

1. The composite of either projection $2\times 2\to 2$ and any morphism $2\to \mathcal{C}$ (“any degenerate square”) is exact;
2. The composite of the swap map $\sigma\colon 2\times 2\to 2\times 2$ and any exact square $e\colon 2\times 2\to \mathcal{C}$ is exact;
3. The pasting of any two exact squares in $\mathcal{C}$ $$ \begin{CD} \bullet @>>> \bullet @>>> \bullet\\ @VVV @VVV @VVV \\ \bullet @>>> \bullet @>>> \bullet \end{CD} $$ is exact; and
4. if $e\cong e'$ are isomorphic squares then $e$ is exact iff $e'$ is.

We refer to a category with an exactness structure as an exacting category. We say that a functor is exacting if it preserves initial objects and exact squares.

We say that an exacting category $(\mathcal{C}, \bot, E)$ is normalized if it has a final object and continuous if it has filtered colimits, and similarly morphisms are normalized and/or continuous if they preserve these structures. Let $\mathsf{ExCat}$, $\mathsf{CtsExCat}$, $\mathsf{NrmExCat}$, and $\mathsf{NrmCtsExCat}$ denote the various combinations of these adjectives.

Example: If $\mathcal{C}$ is an abelian category, then it can be given the structure of a normalized exacting category. The top element is $0$, and a square $$ \begin{CD} A @>f>> B \\ @VgVV @VVhV \\ C @>>i> D \end{CD} $$ is exact in the present sense iff the sequence $$0\to A\xrightarrow{(f,g)}B\oplus C\xrightarrow{h-i}D\to 0$$ is exact in the sense of chain complexes.

Example: The classical Seifert–van Kampen theorem is the statement that the fundamental group functor $\pi_1\colon\mathsf{Top}\to\mathsf{Grp}$ from topological spaces to groups is exact if we choose the exact squares in $\mathsf{Top}$ to be pushout–pullback squares with simply connected pullback, and those in $\mathsf{Grp}$ to be the pushout squares.

Example: The category $\mathsf{Cat}$ of categories can be given the structure of a (normalized continuous) exacting category, where a square $$ \begin{CD} A @>f>> B \\ @VgVV @VVhV \\ C @>>i> D \end{CD} $$ is called exact iff it is exact in the sense of the nLab, i.e. if $g_!f^*=i^*h_!$ as functors $\mathsf{Psh}(B)\to\mathsf{Psh}(C)$.

Example: A frame (a.k.a. a locale), e.g., the poset of open sets in any topological space, has a continuous, normalized exactness structure. It can be regarded as a category in the usual way, its top / bottom elements serve as initial / final objects, and we say that a square is exact if it is both a pullback and a pushout: $$ \begin{CD} A\cap B @>>> A\\ @VVV @VVV \\ B @>>> A\cup B. \end{CD} $$ This is a full and faithful embedding $\mathsf{Frm}\to\mathsf{NrmCtsExCat}$. Indeed any monotone map between the underlying posets of frames $F$ and $F'$ that preserves top and bottom elements and filtered colimits (directed sups), is a map of frames iff it preserves binary meets and binary joins. But this is the case iff it preserves exact squares. [In fact, the functor $\mathsf{Frm}\to\mathsf{CtsExCat}$ is also fully faithful.]

Example: The poset $\mathbb{R}^+:=\{r\in\mathbb{R}\mid 0\leq r\}\cup\{\infty\}$ of nonnegative real numbers plus infinity under the usual $\leq$ ordering can be given a normalized continuous exactness structure where a square $$ \begin{CD} m @>>> n\\ @VVV @VVV \\ m' @>>> n' \end{CD} $$ is exact iff $m+n'=m'+n$.

Remark: If $(C,\bot,E)$ is a (continuous) exacting category and $c\in C$ is an object, then the slice category $C_{/c}$ inherits a (continuous) exacting structure. Let $U\colon C_{/c}\to C$ be the forgetful functor. Then $C_{/c}$ inherits an initial object and filtered colimits from $C$, and we take a square $e$ to be exacting in $C_{/c}$ iff $U(e)$ is exacting in $C$.

Valuations are a constructive approach to probability theory, which agrees with the usual Kolmogorov definition in nice cases. It does not use $\sigma$-algebras but instead is defined on frames. Here we give the usual definition, except with the present terminology. Note that $\mathbb{R}^+_{/1}$ has as objects the closed interval $[0,1]$.

Definition: Let $F$ be a frame. A valuation on $F$ is an exacting functor $\mu\colon F\to\mathbb{R}^+_{/1}$. It is called normalized and/or continuous if it is normalized and/or continuous as an exacting functor.

In other words, our terminology "normalized" and "continuous" was chosen to match that of valuations. The above definition situates valuations in a much broader context.

Proposition: Any left-exact functor preserves exacting category objects and exacting functors, normalized or not. Moreover, the direct image part of a geometric morphism preserves continuous exacting posets, such as frames and the nonnegative lower reals as described above.

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Again, my question is “how will the community respond”? In other words, I'm looking for insights into this notion, how it fits with other notions I haven't discussed above, other examples of it, whether it already exists, whether there are additional requirements that should be made, etc.

Answer 1

I'd like to argue that the current definition is too minimal to allow for much theory development, since the only substantial axiom is the pasting condition. In particular, it would be possible to take all commutative squares in a given category to be exact squares, which may seem like too large a class. So one might expect to need some condition which limits the size of an exactness structure.

Concerning further potential axioms, I want to propose the 2-out-of-3 property:

2-out-of-3 property: If a composite of two squares is exact and also one of the original squares is, then so is the other.

I will show that it holds in the examples of frames and in the example of abelian categories. I haven't thought about whether it holds in the other examples yet.

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Frames, modular lattices: The frame example doesn't use the infinitary distributivity that is characteristic of frames: David's definition of the exactness structure in frames applies to all lattices. It's not hard to construct lattices in which the 2-out-of-3 property for David's exactness structure fails. But there is a large class of lattices with importance to homological algebra in which it is true:

Proposition: In a modular lattice, the above 2-out-of-3 property holds.

Since every frame is a distributive lattice and therefore modular, this covers the case of frames as well.

Proof. By duality, it is enough to show that if we have a diagram

$\require{AMScd}$

\begin{CD} x \land y @>>> y @>>> z \\ @VVV @VVV @VVV \\ x @>>> x\lor y @>>> x \lor z \end{CD} with $x\land y = x \land z$, then also $y = (x \lor y) \land z$, so that the right square is also a pullback. (It's automatically a pushout by the pushout lemma.)

Indeed by modularity, we have $$ (x\lor y) \land z = y\lor (x\land z) = y\lor (x\land y) = y, $$ as was to be shown.

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Abelian categories: Here, David's definition of exact square is equivalent to saying that a square is exact if and only if it is both a pullback and a pushout.

Proposition: The exactness structure on an abelian category satisfies the 2-out-of-3 property.

Proof. By Mitchell's Embedding Theorem, we can reason in terms of diagram chasing with modules over a ring. Again by duality, it is enough to show one direction: if

\begin{CD} A @>f>> B @>g>> C \\ @VhVV @ViVV @VjVV \\ D @>k>> E @>l>> F \end{CD}

is such that the left square and the composite square are exact, then so is the right square. Again by the pushout lemma, we only need to show that the right square is a pullback. So therefore let $c\in C$ and $e\in E$ be elements such that $j(c) = l(e)$. We need to show that there is a unique $b\in B$ with $e = i(b)$ and $c = g(b)$. For uniqueness, it is enough to consider the case $c = e = 0$. Then since the left square is a pullback, we have $a\in A$ with $b = f(a)$ and $h(a) = 0$. But since $g(f(a)) = g(b) = 0$, the assumption that the composite square is a pullback as well implies $a = 0$, as was to be shown.

For existence, we start with $c$ and $e$ as above. Since the left square is a pushout, we can find $b_1\in B$ and $d\in D$ such that $e = i(b_1) + k(d)$. Then we have $$ l(k(d)) = l(e) - l(i(b_1)) = j(c - g(b_1)). $$ Since the composite square is a pullback, we get $a\in A$ with $d = h(a)$ and $c - g(b_1) = g(f(a))$. So with $b_2 := f(a)$, we take $b := b_1 + b_2$, resulting in $$ g(b) = g(b_1) + g(b_2) = c $$ and $$ i(b) = i(b_1) + i(b_2) = (e - k(d)) + i(f(a)) = e - k(d) + k(h(a)) = e, $$ as was to be shown.

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Addendum: Another source of exactness structures is given by systems of bicoverings, as in Definition 4.2.1 of Compositories and Gleaves. There we also threw in stability under pullbacks, and we were working with cospans only, or equivalently with pullback squares.

Answer 2

If you were willing to allow a generalization where "being exact" is structure on a square rather than a property of it, then there would be an example in the homotopy category of a stable $\infty$-category (or stable derivator), where an "exact structure" on a (homotopy) commutative square is a choice of homotopy filling it and making it into a pushout+pullback square.

Similarly, one could put a "proof-relevant" exactness structure on Ho(Cat) where an exact structure on a commutative-up-to-isomorphism square is a choice of isomorphism filling it and making it into an exact square in the sense of your example above. If you further relaxed the requirement that exact squares have to actually commute, you could allow non-invertible transformations with the same exactness property.