The n-Category Café

The paper is about “exact completions”, but to read it you don’t need to know already what an exact completion is, or even what an exact category is. However, let me tell you something about them anyway.

Roughly, an exact category is one with well-behaved quotients of internal equivalence relations. Every exact category is a regular category, i.e. it has well-behaved image factorizations. Specifically, given $f\colon A\to B$, we can define its kernel to be the equivalence relation on $A$ where $a\sim a'$ means $f(a)=f(a')$; then the quotient of this equivalence relation gives the image factorization of $f$.

The easiest sort of “exact completion” to understand is that if you have a regular category $C$, then there is a universal way to make it exact. In fact, we know what the objects of this “exact completion” should be: they should all be quotients of equivalence relations in $C$, so we can just use the actual equivalence relations in $C$ to stand in for them. The morphisms between the quotients of two equivalence relations can also be characterized purely in terms of the equivalence relations — they are binary relations that are “entire” and “functional” with respect to the given equivalence relations — so we can use this to define the morphisms in the exact completion.

People often denote the exact completion of a regular category $C$ by $C_{ex/reg}$ (although I didn’t use that notation in the paper). This construction has a pleasing universal property: it exhibits exact categories as a reflective subcategory of regular ones. In particular, it is idempotent: the exact completion of an exact category is itself.

On the other hand, there is a slightly odd-looking construction, also called “exact completion”, where we start with a merely finitely complete category $C$ and produce an exact category. We can still look at internal equivalence relations in a finitely complete category, but in this case those aren’t enough to give us an exact category; instead we have to look at “pseudo equivalence relations”.

Sadly, the use of “pseudo” here has nothing to do with its usual meaning of “up to isomorphism” in higher category theory. A “pseudo equivalence relation” in this sense consists of an object $X_0$, an object $X_1$, two maps $X_1 \;\rightrightarrows\; X_0$ with a common “reflexivity” section $X_0\to X_1$, a “transitivity” map $X_1 \times_{X_0} X_1 \to X_1$ over $X_0\times X_0$, and a “symmetry” map $X_1\to X_1$ over the twist $X_0\times X_0 \cong X_0\times X_0$. In other words, it’s just like an internal equivalence relation, but we don’t demand that $X_1 \to X_0\times X_0$ be monic. The idea is that if $C$ were regular, then we could take the image of $X_1 \to X_0\times X_0$ and it would give us an equivalence relation — but since it isn’t, we can’t.

A pseudo equivalence relation could also be described as an “internal groupoid” but with no axioms imposed — or, better, as an internal 2-groupoid whose hom-categories are essentially discrete. With that in mind, you can define “functors” between such things, and these (modulo “natural transformations”) give the morphisms in the “exact completion” of our finitely complete $C$. This is denoted $C_{ex/lex}$, to distinguish the fact that we are regarding the input $C$ as only finitely complete (i.e. “left exact” or “lex” for short).

This construction produces a left adjoint to the forgetful functor from exact categories to finitely complete ones. Since this forgetful functor is not full (a functor of exact categories must also preserve quotients, or equivalently regular epimorphisms), the exact completion of finitely complete categories is not idempotent. But as odd as this construction may seem (it certainly did to me at first), it is useful. For instance, the effective topos can be defined as the exact completion of a certain finitely complete category.

One pleasing property of exact completion (of either sort) is that it tends to make “weak” universal properties into “actual” universal properties. (Here “weak” refers to existence-but-not-uniqueness — again an unfortunate clash with the common higher-category-theory meaning of “up to isomorphism”!) For instance, suppose that $C$ has weak exponentials, in the sense that for any $A$ and $B$ there is an object $F$ and a map $F\times A\to B$ such that for any map $X\times A\to B$, there exists a (not necessarily unique) map $X\to F$ making the evident triangle commute. Then the exact completion of $C$ is cartesian closed: we can define exponentials by imposing a suitable (pseudo) equivalence relation on the weak exponentials in $C$.

But this leads us into really weird territory, when we observe that the construction of the exact completion doesn’t actually require $C$ to have finite limits, but only weak finite limits. In particular, the notion of pseudo equivalence relation involves a pullback $X_1\times_{X_0} X_1$, but we can replace that by a weak pullback — roughly, because we only care about the “image” of $X_1 \to X_0\times X_0$, it doesn’t matter which weak pullback we use.

The exact completion of weakly finitely complete categories still defines a functor landing in exact categories, but this functor is not adjoint to the forgetful functor, no matter what class of functors we pick for the morphisms of weakly finitely complete categories. (The obvious choice is weak-finite-limit–preserving functors, which turn out to be the same as (representably) flat functors.) Instead, Carboni and Vitale proved that for weakly finitely complete $C$ and exact $D$, the category of exact functors $C_{ex/lex}\to D$ is equivalent to the category of left covering functors $C\to D$. These are functors $F\colon C\to D$ such that for any finite diagram $G$ in $C$, the induced map $F(\lim G) \to \lim F(G)$ is a regular epi in $D$. What the heck?

You could say that the paper I’ve just posted today grew out of my deep dissatisfaction with this universal property.

The answer which makes me much happier is the following. There is a 2-category which contains weakly finitely complete categories and regular categories as two nearly disjoint full sub-2-categories. The sub-2-category of exact categories (sitting inside the sub-2-category of regular categories) is reflective, and when restricted to regular categories or to weakly finitely complete categories, the reflection becomes the two exact completions described above. Moreover, the morphisms in this larger 2-category from a weakly finitely complete category to an exact category are precisely the left covering functors.

In general, the objects of this bigger 2-category are a particular class of sites. The two sub-2-categories I mentioned come from equipping weakly finitely complete categories with trivial topologies (only split epis cover), and equipping regular categories with their regular topologies (all regular epis cover). In general, the sites that we allow have to satisfy two properties: first, that all their covering families are determined by single morphisms (such as “split epis” or “regular epis”), and second, that they have a sort of “weak finite limit relative to the topology”. For a trivial topology, this second condition is equivalent to ordinary weak finite limits; for nontrivial topologies it is a strictly weaker condition (and in particular is always satisfied by a site with finite limits).

I call sites satisfying these two conditions unary sites. The morphisms of unary sites are a slightly generalized version of the usual notion of “morphism of sites”, where we use a notion of flatness relative to the topology. I blogged about this kind of flatness last year.

This is nice enough, but it becomes even nicer when we realize that there is nothing particularly special about the number one. In fact, we can let $\kappa$ be any regular cardinal, and define a $\kappa$-ary site to be a site whose covering families are determined by families of size $\lt\kappa$ and which has “weak finite $\kappa$-ary multi-limits relative to the topology”. Don’t worry about that latter condition at the moment; whatever it means, it gets weaker as $\kappa$ gets bigger, until when $\kappa$ reaches the size of the site itself it becomes automatic.

Similarly, we have a notion of $\kappa$-ary exact category, which has well-behaved quotients of “many-object equivalence relations” of size $\lt\kappa$. This is equivalent to being exact in the unary sense plus having disjoint and universal coproducts of size $\lt\kappa$. For instance, when $\kappa=\omega$, an $\omega$-ary exact category is precisely a pretopos. And the same theorem is true: $\kappa$-ary exact categories are a reflective sub-2-category of $\kappa$-ary sites.

(The astute reader may have noticed that there is no regular cardinal $\kappa$ such that $\kappa$-ary sites and exact categories reduce to unary ones. The closest we can get is $\kappa = 2 = \{0,1\}$, which according to some definitions is a regular cardinal — but $2$-ary sites can have empty covering families, and $2$-ary exact categories must have initial objects. I dealt with this in the paper by letting $\kappa$ be something a bit more general than a regular cardinal, which can also take the value “$\{1\}$”.)

Finally, the very nice conclusion is that when $\kappa$ is the “size of the universe” $\mathbf{K}$ (a “large regular cardinal”), then $\mathbf{K}$-ary exactness becomes precisely the exactness conditions of Giraud’s theorem that characterizes sheaf toposes. Moreover, since $\mathbf{K}$ is large, any small site is $\mathbf{K}$-ary, and therefore has a $\mathbf{K}$-ary exact completion — which will no longer be small. In fact, the $\mathbf{K}$-ary exact completion of a small site is precisely the category of sheaves on that site. And the universal property of this exact completion, as a reflection into a sub-2-category, reproduces precisely the classical theorem that the category of sheaves on a site is the classifying topos for flat cover-preserving functors.

One interesting possibility this raises is that if we start with a large $\mathbf{K}$-ary site, then its $\mathbf{K}$-ary exact completion could be regarded as a “category of small sheaves” on that site: it satisfies all the exactness properties of a sheaf topos, and has a similar universal property. (But in general it lacks a small generating set, is not cartesian closed or an elementary topos, and does not satisfy SAFT.)

I could go on and talk about how to construct the exact completion of a $\kappa$-ary site (it has three equivalent definitions, using either “profunctors” or “anafunctors” or “$\kappa$-sized sheaves”), but maybe I’ll stop here and let you read the paper. It has proarrow equipments, Cauchy completion, and even a coinductive definition!