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.
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
$$
\begin{array}{ccc}
A&\xrightarrow{f}&B\\
\scriptstyle g\textstyle\downarrow\;&e&\;\downarrow \scriptstyle h\\
C& \underset{i}{\to}&D
\end{array}
$$
satisfying the following conditions:
- The composite of either projection $2\times 2\to 2$ and any morphism $2\to \mathcal{C}$ ("any degenerate square") is exact;
- 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;
- The pasting of any two exact squares in $\mathcal{C}$
$$
\begin{array}{ccccc}
\bullet&\to&\bullet&\to&\bullet\\
\downarrow&&\downarrow&&\downarrow\\
\bullet&\to&\bullet&\to&\bullet
\end{array}
$$
is exact; and
- 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
$$
\begin{array}{ccc}
A&\xrightarrow{f}&B\\
\scriptstyle g\textstyle\downarrow\;&&\;\downarrow \scriptstyle h\\
C& \xrightarrow{i}&D
\end{array}
$$
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
$$
\begin{array}{ccc}
A&\xrightarrow{f}&B\\
\scriptstyle g\textstyle\downarrow\;&&\;\downarrow \scriptstyle h\\
C& \xrightarrow{i}&D
\end{array}
$$
is called exact iff it is exact in the sense of the [nlab][1], 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:
$$
\begin{array}{ccc}
A\cap B&\to&A\\
\downarrow&&\downarrow\\
B& \to&A\cup B
\end{array}
$$
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
$$
\begin{array}{ccc}
m&\to&n\\
\downarrow&&\downarrow\\
m'& \to&n'
\end{array}
$$
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.
----------
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.
Thanks!
[1]: https://ncatlab.org/nlab/show/exact+square
