Michael Shulman
exactness hypothesis

Contents

Idea

I propose the exactness hypothesis as one of the “guiding hypotheses of higher category theory,” alongside the homotopy, delooping, and stabilization hypotheses. The exactness hypothesis states that

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 CC be a category with one object ** and consider how we might construct the corresponding monoid C(*,*)C(*,*) by 2-categorical methods in CatCat. A little bit of thought shows that this monoid is given by the comma object

C(*,*) 1 * 1 * C.\array{C(*,*) & \to & 1\\ \downarrow & \Downarrow & \downarrow * \\ 1 & \overset{*}{\to} & C.}

This gives C(*,*)C(*,*) as a discrete object in CatCat, which moreover has the structure of a monoid. Note that this comma object can also be described as the based loop object of CC when CatCat is equipped with the “walking arrow” 2\mathbf{2} as its interval object.

Conversely, given a monoid MM, we can construct a category BMB M with one object and BM(*,*)=MB M (*,*)=M as the lax codescent object

M×M M 1 \array{ & \to && \to \\ M\times M & \to & M & \leftarrow & 1\\ & \to && \to }

In less fancy words, this means that BMB M is the universal category equipped with a functor 1M1\to M and a 2-cell

M 1 1 BM.\array{M & \to & 1\\ \downarrow & \Downarrow & \downarrow \\ 1 & \to & B M.}

which is compatible with the multiplication and unit of MM. 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 MM, which we can regard as an internal category in CatCat via M1M \;\rightrightarrows\; 1, plays the role of an equivalence relation, and the comma square above plays the role of the kernel pair of the map 1C1\to C. Clearly, then, a natural generalization of the delooping hypothesis for CatCat would involve replacing 11 by some more general category. We can define the kernel of an arbitrary functor f:ABf:A\to B to be the comma category

(f/f) A f A f B;\array{(f/f) & \to & A\\ \downarrow & \Downarrow & \downarrow f \\ A & \overset{f}{\to} & B;}

an appropriate statement of exactness for a 2-category should then say that certain structures that “behave like (f/f)A(f/f) \;\rightrightarrows\; A”, 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.

Particular cases

k=0,n=1k=0,n=1

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 k=0,n=1k=0,n=1 case of CatCat, one way to state exactness is that

In the case D 0=1D_0=1, such a double category is precisely a discrete monoid in CatCat (the thin structure is automatic in this case), while an essentially surjective functor 1C1\to C just makes CC a pointed connected category.

k=1,n=1k=1,n=1

In the k=1,n=1k=1,n=1 case of MonCatMonCat, one way to state exactness is that

  • monoidal double categories D 1D 0D_1 \;\rightrightarrows\; D_0 whose horizontal 2-categories are homwise discrete and which have a thin structure can be identified with essentially surjective (strong) monoidal functors D 0CD_0\to C.

Again, in the case D 0=1D_0=1 such a double category is precisely a discrete commutative monoid, and an essentially surjective monoidal functor 1C1\to C makes CC a pointed connected monoidal category.

k=0,n=(,0)k=0,n=(\infty,0)

In this case an exactness statement has been proven by Lurie:

  • internal groupoids G 1G 0G_1 \;\rightrightarrows\; G_0 in the (,1)(\infty,1)-category of spaces can be identified with maps G 0XG_0\to X which are surjective on π 0\pi_0.

Taking G 0=1G_0=1, 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 ΩX\Omega X of a pointed space XX is the homotopy pullback of the basepoint 1X1\to X along itself.

k=0,n=2k=0,n=2

When nn-categories contain noninvertible jj-morphisms for j>1j\gt 1, there is an extra subtlety. If CC is a pointed 2-category, then the comma object

(*/*) 1 * 1 * C.\array{(*/*) & \to & 1\\ \downarrow & \Downarrow & \downarrow * \\ 1 & \overset{*}{\to} & C.}

(in the 3-category 2Cat2Cat) is the monoidal category consists of endomorphisms of *C*\in C and isomorphisms between them.

To recover information about the noninvertible 2-cells in C(*,*)C(*,*), we can consider, in addition to the comma object, the “2-comma object,” which is the 3-limit weighted by 1T11\to T \leftarrow 1 where TT is the “walking 2-cell.” With this approach, the appropriate notion of “kernel” starts to look more like the “Θ\Theta-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 2Cat l2Cat_l-enriched limit, where 2Cat l2Cat_l 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 2Cat2Cat we have to be working in 2Cat l2Cat_l, 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 2Cat2Cat). I’m not sure that it’s the same as working with 2Cat l2Cat_l-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 2Cat l2Cat_l rather than 2Cat2Cat. Note that 2Cat l2Cat_l 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

Grothendieck nn-toposes

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 (,1)(\infty,1)-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 nCatn Cat should certainly satisfy these two hypotheses as well, a “corollary” of the extensivity hypothesis is the following “topos hypothesis:”

  • nCatn Cat is a Grothendieck (n+1)(n+1)-topos.

(On the other hand, kMonnCatk Mon n Cat will not, in general, be extensive.)

Revised on June 12, 2012 11:10:00 by Andrew Stacey? (129.241.15.200)