This is closely related to the delooping hypothesis, and at least in low dimensions the latter is a special case of it. To get some intuition for how to get from delooping to exactness (and understand the meaning of “exact”), let be a category with one object and consider how we might construct the corresponding monoid by 2-categorical methods in . A little bit of thought shows that this monoid is given by the comma object
This gives as a discrete object in , which moreover has the structure of a monoid. Note that this comma object can also be described as the based loop object of when is equipped with the “walking arrow” as its interval object.
Conversely, given a monoid , we can construct a category with one object and as the lax codescent object
In less fancy words, this means that is the universal category equipped with a functor and a 2-cell
which is compatible with the multiplication and unit of . Thus, the statement that “monoids can be identified with one-object categories” can be interpreted as a statement about an idempotent adjunction.
This is strikingly reminiscent of the definition of an exact category. Now the monoid , which we can regard as an internal category in via , plays the role of an equivalence relation, and the comma square above plays the role of the kernel pair of the map . Clearly, then, a natural generalization of the delooping hypothesis for would involve replacing by some more general category. We can define the kernel of an arbitrary functor to be the comma category
an appropriate statement of exactness for a 2-category should then say that certain structures that “behave like ”, in the same way that equivalence relations behave like kernel pairs, all arise as the kernel of some functor.
Possibly some more general notion of exactness could be formulated in any category equipped with a (possibly directed) interval object.
One formal definition of “behave like a kernel” in the case of 2-categories (due essentially to Ross Street) can be found here, and a corresponding definition of “exact 2-category” here. In the case of , one way to state exactness is that
In the case , such a double category is precisely a discrete monoid in (the thin structure is automatic in this case), while an essentially surjective functor just makes a pointed connected category.
In the case of , one way to state exactness is that
Again, in the case such a double category is precisely a discrete commutative monoid, and an essentially surjective monoidal functor makes a pointed connected monoidal category.
In this case an exactness statement has been proven by Lurie:
Taking , this includes delooping in classical homotopy theory. When all 1-cells are invertible, the comma object in the definition of a kernel reduces to a (homotopy) pullback. Thus we recover the observation that the based loop space of a pointed space is the homotopy pullback of the basepoint along itself.
When -categories contain noninvertible -morphisms for , there is an extra subtlety. If is a pointed 2-category, then the comma object
(in the 3-category ) is the monoidal category consists of endomorphisms of and isomorphisms between them.
To recover information about the noninvertible 2-cells in , we can consider, in addition to the comma object, the “2-comma object,” which is the 3-limit weighted by where is the “walking 2-cell.” With this approach, the appropriate notion of “kernel” starts to look more like the “-categories” considered by Joyal, Berger, Rezk, and others. The connection with monoidal objects also becomes less direct.
A different approach is to consider a version of the comma object that does include information about the noninvertible 2-cells. This is not a 3-limit in the sense of a 2Cat-enriched limit, but it is a -enriched limit, where denotes 2Cat with the lax version of the Gray tensor product. This approach might maintain the strong connection with monoidal objects and keep the notion of “kernel” looking simplicial, but it means that instead of the 3-category we have to be working in , which is enriched over itself but is not a 3-category (its interchange law holds only laxly).
What I wanted to say is that in the square above, the 2-cell should be a lax natural transformation (i.e. we take lax natural transformations as 2-cells in ). I’m not sure that it’s the same as working with -enriched limits. —Mathieu
I’m pretty sure that it is the same. Saying that you take the 2-cells in 2Cat to be lax transformations really means the same thing as saying that you’re working in rather than . Note that is not a 3-category. Although it is 3-category-like since it is closed monoidal and thereby enriched over itself, the interchange law only holds laxly. -Mike
For 1-categories, exactness is one of the conditions in Giraud’s theorem characterizing Grothendieck toposes. This is likewise true for Street’s theorem characterizing Grothendieck 2-toposes and Lurie’s theorem characterizing Grothendieck -toposes. In all three cases the other conditions are infinitary extensivity (see here for a 2-categorical version) and the existence of a small generating set. Since should certainly satisfy these two hypotheses as well, a “corollary” of the extensivity hypothesis is the following “topos hypothesis:”
(On the other hand, will not, in general, be extensive.)