## Notes on Tetracategories

### 1. History

In 1995, at Ross Street's request, I gave a very explicit description of weak 4-categories, or tetracategories as I called them then, in terms of nuts-and-bolts pasting diagrams, taking advantage of methods I was trying to develop then into a working definition of weak n-category. Over the years various people have expressed interest in seeing what these diagrams look like -- for a while they achieved a certain notoriety among the few people who have actually laid eyes on them (Ross Street and John Power may still have copies of my diagrams, and on occasion have pulled them out for visitors to look at, mostly for entertainment I think).

Despite their notorious complexity, there seems to be some interest in having these diagrams publicly available; John Baez has graciously offered to put them on his website. The theory of (weak) n-categories has come a long way since the time I first drew these diagrams up, and it may be wondered what there is to gain by re-examining the apparently primitive approach I was then taking. I don't have any definitive answer to that, but I do know that there are interesting combinatorics lurking within the methods I was using, which I suspect may hold clues to certain coherence-theoretic aspects of weak n-categories; therefore in these notes I wanted to give some idea of the methods used to construct these diagrams (as well as to provide explanatory notes in case the diagrams are too baffling), in case anyone wants to push them further.

Disclaimer: there is an "operadic" definition of weak n-category, described by me in a talk at Cambridge in 1999, and subsequently written up by Tom Leinster, Eugenia Cheng and Aaron Lauda. The notion of tetracategory given here is not the operadic one in dimension 4. The truth be told, I have never considered the operadic version as anything other than a somewhat strictified case of the "true" notion of n-category; it was invented primarily to give a precise meaning to Grothendieck's "fundamental n-groupoid of a topological space", and is conjectured to be an algebraic characterization of the n-type of a space. (The operadic definition does have the advantage of being short, almost a "two-line definition", and it does embody a great deal of structure one expects for weak n-categories, which may make it appealing for didactic purposes. There are also some conjectures about fundamental n-groupoids which I consider important and which, to my knowledge, have not been completely addressed: one is that they do characterize n-types; another is a van Kampen theorem.)

There is a strong connection with operads, however: as already noticed by a number of people in the early 90's, there is a strong resemblance between the "higher associativity conditions" in the notions of bicategories and tricategories, and the convex polyhedra Kn called "associahedra", introduced 30 years earlier by Stasheff in his work on the homotopy types of loop spaces, and which together form an operad, whose formal definition was given later by May.

My early efforts to define weak n-categories (in 1994-1995) were really just an attempt to take this visual resemblance seriously and build a theory around it. Thus, the higher associativities could all be described by the polytopes Kn, outfitted with orientations or directions on each cellular face t, which divide the boundary of t into negative and positive parts. The negative boundary cells are oriented in such a way that they paste together sensibly (according to one or another formal framework, e.g., Street's parity complexes, Mike Johnson's pasting schemes, etc.) to form the "domain" of t; similarly the positive boundary cells paste together to form the codomain of t.

In addition to the associahedra used for the higher associativities, one needs some allied polyhedra for the higher unit conditions; in short, I needed a suitable collection of "monoidahedra" to capture the combinatorial structure of the n-category data and axioms. This turned out to be easy; a minor tweak on the machine used to define associahedra does the trick. There are also "functoriahedra" for describing the data and equations occurring for (weak) n-functors. All these polyhedra are in fact describable by means of bar constructions, as I've tried to describe informally to a few people, and especially to Baez. (In part this has long been known: the ubiquity of abstract bar constructions probably first hit me on the head while reading May's Geometry of Iterated Loop Spaces [where operads were first introduced]. The specific application to functoriahedra is, I would guess, not generally known.)

The next step is more technical: putting parity complex structures on these polyhedra. Here I received vital input from Dominic Verity: using some ideas from the "surface diagrams" he and Street had begun working on, together with a general position argument, we had a theorem which guarantees that such parity complex structures exist. (I think it doesn't matter which parity structure is chosen, i.e. that one should get the same notion of n-category regardless, or that different choices of parity structure correspond to different presentations of the same theory.)

In practice, it would be nice to have a systematic way of choosing parity structure to make the pictures as pretty as possible, but for that I have just a handful of tricks and ideas, without any general theory. I believe Jack Duskin told me he had a nice way of getting parity structures on general associahedra, but I don't have details on that.

The final step in my 1995 attempt to define n-categories was to interpret these pasting schemes as cells in local hom (n-1)-categories. Here is where the project got bogged down. With the wealth of n-categorical theory which has appeared in the meantime, it should be possible, I think, to fix this part of the program (perhaps by adapting Penon's definition?). At the time, it seemed as if I needed an (n-1)-categorical coherence theorem in order to interpret these pastings, which I didn't have of course. We did have coherence of tricategories, due to Gordan-Power-Street (GPS), and this enabled me to give a rigorous definition of tetracategory.

I still feel that despite the technical difficulties which blocked my progress (some of which may be spurious), the program may be worth resurrecting, because in fact in dimensions 2 and 3 this approach does yield exactly the classical notions of bicategory and tricategory, without further tweakings or encryptions, and also because the underlying ideas and methods are really quite simple -- a fact which is obscured by the dazzling complexity of the diagrams which were produced at Street's request. The fact is that these diagrams were produced merely by patiently turning the crank of a simple machine -- anyone with time to kill can produce these things automatically. The general machine, and what one might be able to say with its help, was the real point behind my failed attempts back in 1995.

### 2. The Definition of Tetracategory

The actual definition is contained in these PDF files:
1. Tetracategorical data (5 pages)
2. K6 associativity (16 pages)
3. U5,2 unit condition (10 pages)
4. U5,3 unit condition (10 pages)
5. U5,4 unit condition (10 pages)
For further explanation of associahedra, monoidahedra, and functoriahedra, see:

### 3. Explanatory Notes

These notes amplify on the definition given in section 2. They are meant to give some intuition for how the definition works, in some cases more in the way of suggestive hints than precise formal statements. Here and there I mention literature which may provide helpful background.

To read the diagrams for each coherence condition, it is recommended that the reader print them out (large) and spread them out on a wide surface, and arrange them in a circular pattern. Let me illustrate by considering the case U5,2

The sheets numbered 1-, 2-, ..., 6- form a semicircle (on the left), with 1- at the top and 6- at the bottom. They represent stills from a movie where the action is in the 3-dimensional cells (i.e. 3-morphisms in a local hom tricategory; see "tetracategorical data") which mediate from one page to the next. One can visualize the shapes of these 3-dimensional cells by laying two successive pages, ideally transparencies, on top of one another, and noting that nothing has changed except in isolated planar regions where the 3-dim cell is doing the deforming. Thus the 3-morphism has the shape of a bubble in a sheet of plastic (the surrounding sheet is the "whiskering"), and the deformations or 3-morphisms are pasted together to form a polyhedral shape homeomorphic to a 3-disk. The boundary of the 3-disk consists of the hemispheres 1- and 6-, and the pasting represents a complete transformation from 1- to 6-. This pasting is one side of the unit coherence equation U5,2.

The other side of the equation is the pasting represented by the movie 1+ → 6+. Of course, 1- = 1+ and 6- = 6+, so this is just a different movie with the same initial and final scenes. Or rather, the movies just look different; the coherence equation asserts that they are different pasting presentations of the same 3-morphism. The equation itself can be pictured as a 4-disk whose boundary is the union of the negative and positive 3-disks.

The associativity coherence condition is read in similar fashion, with 1- = 1+ and 7- = 9+. It is a photocopy of the original 1995 diagram given to Ross Street, and is rather more heavily annotated than the unit conditions (which were redrawn in 2006, having been lost in the intervening years), to indicate more precisely the categorical semantics of the cells in "old-fashioned" tricategorical language. It's possible that such annotations will be a help rather than a hindrance.

Note that K6 refers to Stasheff's K6, the operadic component of arity 6 in the associahedral operad, which is a 4-dimensional convex polyhedron. Readers not already familiar with Stasheff's classic work Homotopy Associativity of H-Spaces I, II are strongly encouraged to read it (especially part I) before tackling this work.