In this project I’ve had a lot to say about -toposes for directed . The classical case, of course, is . The case is classical as well, though not generally given that name; a Grothendieck -topos is just a locale, and an elementary -topos is just a Heyting algebra. But what about other values of ?
Recent work, especially by Jacob Lurie, but also by Rezk, Joyal, and others, has shown that there is quite a good notion of -topos, at least Grothendieck ones. In many ways, the -categorical theory parallels the 1-categorical theory more closely than the 2-categorical one does, due to the lack of noninvertible 2-cells. In particular, every map is a fibration and opfibration. An -congruence is just an internal groupoid, and part of the statement of exactness is that internal group objects (groupoids on 1) are equivalent to pointed connected objects (objects equipped with an eso from 1); and interpreted in this becomes the classical characterization of loop spaces.
I also expect that, given a good tractable notion of -category, the theory of -toposes can be combined with the theory of 2-toposes to give a good theory of -toposes. One could probably do this fairly explicitly in the case of (3,2)-toposes if there were seen to be any particular benefit.
However, as soon as there are noninvertible 3-cells, things start to go haywire. This happens already at (2,3)-categories, whose coherence theory is no harder than that of 2-categories (we can consider them as poset-enriched bicategories), but let’s talk about 3-categories since those may be more familiar, intuitively. The problem is that the naive 3-categorical comma object is, for all relevant purposes, wrong.
For instance, let be a 2-category, let , and consider the comma object
in . By this I mean the 3-limit of weighted by the obvious diagram , just as for the corresponding 2-limit. Then an object of is, as expected, a morphism in . But a morphism in from to is an isomorphism in . In other words, the comma object has lost all the information about noninvertible 2-cells in .
This certainly messes up regularity and exactness; there’s no way we can expect to recover from the kernel of an eso if that kernel doesn’t know about the noninvertible 2-cells in (which manifest as noninvertible 3-cells in the 3-category ). It also messes up the theory of fibrations: for any two morphisms and , the projections and are not 2-fibrations (or any of the three possible op-variants). This makes it hard to show that the 3-category of 2-fibrations is monadic, let alone comonadic.
Finally, it messes up logic, since comma objects are used to represent arrow-types, and if is a 2-type, then should be a 1-type, which includes the noninvertible 2-cells in . This puts us in the interesting position of knowing more about the internal logic of a 3-topos than we do about what a 3-topos itself might be.
One possible fix for regularity and exactness would be to define the kernel 3-congruence of a morphism to consist, not only of the comma object, but of the “2-comma object,” meaning the limit weighted by the diagram
which of course contains the information about noninvertible 2-cells. Now, instead of being an internal category, a 3-congruence starts to look more like a -object in the sense of Joyal, Berger, Rezk and others. This might work to solve the problem of regularity and exactness, but it seems unlikely to help much with the problems of fibrations and logic.
Another possible fix is to consider, instead of the universal object equipped with a natural transformation
the universal object equipped with a lax natural transformation
It’s easy to check that for 2-categories, this gives us precisely what we want. It includes data on the noninvertible 2-cells. The projections are fibrations. The fiber over is the hom-category . And so it gives the right thing to represent arrow-types as well. These objects were studied by Gray in the 60s and 70s under the name “2-comma categories.”
The only problem, and it’s a biggie, is that of course lax natural transformations do not live in the 3-category , so how are we to generalize this to 3-toposes other than ? In fact, lax natural transformations do not live in any 3-category: if denotes the 2-category of (strong) functors, lax transformations, and modifications between two 2-categories and , then we do not have a composition functor
We almost do: you can whisker lax transformations by a functor, but the interchange law for horizontal composition of lax transformations only holds up to a noninvertible 2-cell.
This means that we could make the composition functor into a lax functor. But introducing general lax functors is fraught with peril, since you can’t even whisker a (strong) transformation by a lax functor. A better solution is to recall that Gray also defined a tensor product that represents functors of two variables with lax interchange. The “Gray tensor product” usually refers to the one with interchange up to isomorphism, but Gray actually focused on the one with only lax interchange, for the same reason: he cared about lax transformations. The lax Gray tensor product makes the 1-category (of strict 2-categories and strict functors) into a biclosed non-symmetric monoidal category, with the right and left homs containing lax and oplax transformations, respectively:
There should be no problem discarding strictness here; the lax Gray tensor product should make the 3-category of non-strict 2-categories into a biclosed non-symmetric monoidal 3-category in an analogous way. (The pseudo Gray tensor product is only distinguishable from the cartesian product in the fully strict world, but the lax one is different even in the fully weak world.)
In particular, with this structure (of whichever variety) will become enriched over itself (using the right homs , as always); we call it when so regarded. Now we can identify the above “improved comma object” as a -weighted limit in the -category .
So, should a 3-topos actually be defined to be a certain sort of -category, rather than a 3-category? I’ve convinced myself that a functor between 2-categories is a 2-fibration in if and only if it is a 2-fibration in , so we would get the right notion there. And it seems likely that regularity, exactness, and logic would work out.
However, only having lax transformations around is also problematic. For instance, is not even cartesian closed, so it’s not clear how to aim at any higher-order sort of structure.
As another example, in a 2-category we can construct the object of algebras for a monad as a certain finite 2-limit. This is an important part of category theory, so it’s good that we can do it in any 2-topos. Now in 2-category theory, there are at least three different interesting categories of algebras that we can construct from a 2-monad, depending on whether we take the morphisms to be strong, lax, or oplax. Now in a 3-category, we can construct the object of algebras for a 2-monad and strong morphisms, while in a -category, we can construct the object of algebras and oplax morphisms, and it takes a -category to construct an object of algebras and lax morphisms. So if we really want to be able to do 2-category theory in a 3-topos, it seems as if we need a structure including both pseudo, lax, and oplax transformations.
I have some ideas about what such a structure would look like, building on the notion of F-category, but things are starting to get messy. And then you ask “what about a 4-topos?” and my brain starts to hurt. Should the prototypical 4-topos be ? Or ? Or some more complicated thing? Should the 3-cells be lax as well, or only the 2-cells? Is there a lax Gray tensor product of -categories?
And can any of the proposed definitions of -category deal with this morass? Can they even deal with -categories? The only one that seems to me to have much of a chance is Batanin’s theory of higher operads.
On the other hand, maybe I’m just missing something fundamental, and some helpful reader of this page will point out why a 3-topos doesn’t need to be anything more exotic than a 3-category.
Even before I got to this last paragraph, I was planning to suggest something along those lines, but not in a way that you're going to like.
I've been thinking lately (that is, for less than a month) that maybe the entire notion of -category (beginning with -category) needs to be revised to include lax transformations (and higher). That is, we started with strict -category, now we've got definitions that (while sometimes hard to work with) have moved us to weak -category, but we're not done and still must define lax -category.
Mike: Actually, that doesn’t really sound all that different from what I’m saying (prior to the last paragraph). Weak -categories don’t seem to be enough; we need something that can deal with lax transformations, at least. I’m not sure whether you are saying this or not, but it seems unlikely to me that there is just one notion of “lax -category;” it’s going to depend at least on which dimensions we make pseudo, lax, oplax, or both.
I think evidence has been accumulating for a while that focusing only on -dimensional categorical structures that have only one type of -cell for all is too myopic. Double and -fold categories are one example. My recent work with Steve Lack is another. These observations (which owe a lot to some discussions with Mathieu) are a third.
What has led you to your thoughts about including laxness?
Toby: As far as the laxness is concerned, there may be one maximally general notion of lax -category, which allows both lax and oplax everywhere. Then just as -categories are (non-lax) -categories where some things must be equivalences or trivial, we'll have a multidimensional classification where some things must be strong (or at least lax in only one way) at various places —which will be pretty messy in general. But even this doesn't include double categories or -categories.
I forget why I was thinking of lax -categories. Probably it was just reading the other material on your web, but I'll let you know if I see reasons for it around in other places. Anyway, haven't really thought about what they would be like, only that we might want them.
I agree that -categories as we know them are just the start of a general theory of higher-dimensional transformations. I think that we (or at least some of us) do understand what -groupoids are all about, and that this does give us a complete general theory of higher-dimensional equivalences. But we're only getting started on non-invertible transformations.
Mike: I’m having trouble visualizing a notion that allows “both lax and oplax” while only having one type of -cell for each , since it seems you can’t compose a lax transformation with an oplax one and get much of anything.
I agree that we have a fairly good understanding of -groupoids and higher-dimensional equivalences, due to all the work of homotopy theorists over the past century. But higher non-invertible transformations may be more subtle than many of us realized at first.