Is there an accepted definition of $(\infty,\infty)$ category?

Question

For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with some weak forms of composition and associativity and .... It was recognized a bit later that it is worth recording two numbers for each category: an $(m,n)$-category for $m,n \in \mathbb N\cup\lbrace \infty\rbrace$ with $n\leq m$ has morphisms up to dimension $m$ (and above that only equalities), but everything above dimension $n$ is invertible (in some weak sense). Thus an $(\infty,0)$-category is a "higher groupoid", which Grothendieck's homotopy hypothesis says should be the same as a homotopy type. (Many people take the homotopy hypothesis to be a definition; others prefer to take it as a check, so that a definition is good if the homotopy hypothesis is a theorem.) Just to confuse the language, an $(\infty,1)$-category is what many people now call by the name "$\infty$-category". In any case, for quite a while these objects were expected to exist but definitions were not available (or perhaps too available, but without enough known to decide which was the correct one).

More than a decade ago, Leinster provided A survey of definitions of n-category (Theory and Applications of Categories, Vol. 10, 2002, No. 1, pp 1-70.). Much more recently, Bergner and Rezk have begun a Comparison of models for (∞,n)-categories (to appear in Geometry and Topology). These works tend to focus on $(\infty,n)$-categories with $n<\infty$ (although some of Leinster's definitions do include the $n=\infty$ case).

My impression, almost certainly very biased by the people that I happen to hang out with, is that by now it is known that "$(\infty,0)$-category" might as well mean "Kan simplicial set" and "$(\infty,1)$-category" might as well mean "simplicial set in which all inner horns are fillable". By "might as well mean", I mean that there are many reasonable definitions, but all are known to be Quillen-equivalent. For $n< \infty$, "$(\infty,n)$-category" can mean "iterated complete Segal space", for example. It is not clear how to take $n=\infty$ in this approach.

A very tempting definition of "$(\infty,n)$-category", which I believe is discussed in Lurie's (to my knowledge not yet fully rigorous) paper On the Classication of Topological Field Theories, is that an $(\infty,n)$-category should be an $(\infty,1)$-category enriched in $(\infty,n-1)$-categories. This is based on the observation that the coherence axioms of being an enriched category only require invertible morphisms above dimension $1$. Thus as soon as you've settled on a notion of "$(\infty,1)$-category" (this seems to have been settled) and a notion of "enriched" (settled?), you get a notion of "$(\infty,n)$-category". This approach is developed in Bergner–Rezk, for example.

In any case, it is less clear to me how to use these inductive definitions to get to $(\infty,\infty)$. Let $\mathrm{Cat}_n$ denote the $(\infty,n+1)$-category of $(\infty,n)$-categories. Then there are clearly inclusion functors $\mathrm{Cat}_n \to \mathrm{Cat}_{n+1}$, but the colimit of this sequence does not deserve to be called $\mathrm{Cat}_\infty$, since any particular object in it has morphisms only up to some dimension. I have seen it proposed that one should instead take the (inverse) limit of the "truncation" functors $\mathrm{Cat}_{n+1} \to \mathrm{Cat}_n$; I am unsure exactly how to define these, and even less sure why this deserves the name $\mathrm{Cat}_\infty$.

Thus my question:

Is there by now an accepted / consensus notion of $(\infty,\infty)$-category? For instance, if I have decided on my favorite meaning of $(\infty,1)$-category, is there some agreed-upon procedure that produces a notion of $(\infty,\infty)$-category? Or are there known comparison results that can push all the way to $n=\infty$?

A closely related MathOverflow question was asked a little less than a year ago (or via Wayback Machine here), but received no answers. Also, I am certainly unaware of many papers in the literature, and likely I have misrepresented even those papers I mentioned above. My apologies to all parties for doing so. It is precisely because of these gaps in my knowledge of the literature that I pose this question.

Answer 1

One thing that might interest you is my result with Clark Barwick which gives an axiomatiation + uniqueness result for the homotopy theory of higher categories: arXiv:1112.0040 (i.e. $(\infty,n)$-categories). This axiomatization includes several variants of $(\infty,n)$-category for finite n, such as what you mention in your question. In particular, when also taken in context with the comparison results of Bergner-Rezk (which you mentioned) and Lurie arXiv:0905.0462, I would say that there is a clear consensus for what the homotopy theory of $(\infty,n)$-categories should be, and that it is fairly rigid (few automorphisms). It is at least fair to say that such a consensus is forming.

Once you pin down the theories of $(\infty,n)$-categories, I would say that there are two distinct reasonable choices for what the (homotopy) theory of $(\infty,\infty)$-categories should be. So in that sense the answer to your question is no, there is not a single theory of $(\infty,\infty)$-categories; there are exactly two. But this "no" is a far cry from saying that there is a vast uncharted landscape of possibilities.

What are these theories? As Charles Rezk mentions, the inclusion of $(\infty,n)$-categories into $(\infty, n+1)$-categories has both a left and right adjoint and this gives rise to two towers of homotopy theories of higher categories. The limits of these towers give the two potential models of $(\infty,\infty)$-categories. They are not equivalent. Let's call the limit using right adjoints $Cat_{(\infty,\infty)}$ and the limit using left adjoints $LCat_{(\infty,\infty)} $.

One consequence of the unicity result is that these towers are essentially uniquely defined and are essentially model independent (provided the models satisfy our axioms). So in that sense there are these two canonical (established?) choices for the homotopy theory of $(\infty,\infty)$-categories. I know Clark and I have discussed this idea with many people, but the idea is certainly not new.

I think one of the important parts of this story is Eugenia Cheng's theorem from her paper "An omega-category with all duals is an omega groupoid". Her result applies in the tower using the left adjoints, $LCat_{(\infty,\infty)} $, the "coinductive" version. There an $(\infty, \infty)$-category can be tought of as a sequences of $(\infty, n)$-categories, where each truncates to the previous theory. Cheng's result implies that in such a higher category if you have all duals you are an $\infty$-groupoid.

I prefer the other limit, the limit of right adjoints. There an $(\infty,\infty)$-category is a sequence of $(\infty,n)$-categories where the previous is the maximal $(\infty,n-1)$-category. For some reason this seems more natural to me, though I know of others who disagree. This version includes one of my favorite examples: the infinite cobordism category. There are cobordisms and cobordisms between cobordisms and cobordisms between these and so on forever. This is an object in the tower of right adjoints which has duals for all objects, but which is not an $\infty$-groupoid. The limit of right adjoints fails Cheng's theorem. It also has a sort of inductive notion of equivalence, rather than coinductive.

Your idea of taking the union of the theories of $(\infty,n)$-categories is actually not as far off as you might think. One problem with the naive union is that it is not cocomplete. There are diagrams which increase category number and have no colimits, or at least not the colimit which should exist. One way out of this is to note that you probably want the homotopy theory of $(\infty, \infty)$-categories to be presentable. So instead of taking the colimit naively you should take the colimit in presentable $(\infty,1)$-categories. Now since the inclusion functors preserve both limits and colimits, you can take this colimit in either $Pr^L$ or $Pr^R$. In either case you can use the equivalence $Pr^R \simeq (Pr^L)^{op}$ and Higher Topos Theory 5.5.7.6 or 5.5.3.13 to compute these colimits. In short you compute the colimit by taking adjoints and computing the limit naively. Thus you find:

Taken in $Pr^L$: $$ \mathrm{colim} \; Cat_{(\infty,n)} = Cat_{(\infty,\infty)} $$

taken in $Pr^R$: $$ \mathrm{colim} \; Cat_{(\infty,n)} = LCat_{(\infty,\infty)} $$

Which is the "right" notion of $(\infty,\infty)$-category probably depends on taste and what you want to do with the notion. Both are useful. The above descriptions gives you a variety of universal properties for these two theories. I would also love to know if there are any theories in between these two. I also strongly believe that $LCat_{(\infty,\infty)} $ is a localization of $Cat_{(\infty,\infty)} $, but I haven't written down a proof. Perhaps it is easy to see this from the above description?

You can be more explicit about these models too. Dominic Verity has a model of higher categories based on "weak complicial sets". There are $(\infty,n)$-versions of this and also $(\infty, \infty)$-versions. One of the conjectures that Clark and I made at the end our paper is that some variant of Dominic's $(\infty,n)$-theory satisfies our axioms (Dominic, Emily Riehl, and I have a partial sketch of this, so hopefully the truth of this conjecture will be known... soon?). If that is true, then it is straighforward to show that the Dominic's $(\infty,\infty)$-version of weak complicial sets is a model of the tower of right adjoints, the one which includes the infinite bordism category. So there are also concrete models of these theories.

There are also others whose work will yield explicit models of this. Rune Haugseng's work was already mentioned. Jeremy Hahn's (upcoming?) work will provide nice model of both limits. I am sure there are many ways to model these two theories.

So to summarize:

• There are now a variety of uniqueness and comparison results which pin down the theory of $(\infty,n)$-categories as well as it is pinned down for $(\infty,1)$-categories.
• It actually is clear how to send $n \to \infty$; there are two ways to do it, giving two such infinite theories.
• We can describe and study these theories explicitly and they have interesting properties and possess interesting examples.
• Does this constitute a "consensus"? Of course not, but I hope it goes a little towards answering the real question.

Answer 2

As you point out, there are "inclusion" functors $\mathrm{Cat}_n\to \mathrm{Cat}_{n+1}$. These inclusion functors admit both left and right adjoints (in the sense of functors between $(\infty,1)$-categories).

The right adjoint $r\colon \mathrm{Cat}_{n+1}\to \mathrm{Cat}_n$ is a kind of truncation functor which effectively removes all $(n+1)$-morphisms which are not $(n+1)$-equivalences.

The left adjoint $\ell\colon \mathrm{Cat}_{n+1}\to \mathrm{Cat}_n$ is a different functor, which inverts all $(n+1)$-morphisms.

This gives at least two different possible towers $\{\mathrm{Cat}_n\}$ of $(\infty,1)$-categories. You can form the (homotopy) inverse limit of each in $(\infty,1)$-categories, and perhaps either of these could be considered a definition of $\infty$-category. (Other people have thought about his more than I have, and may be able to tell you if these are really workable.) The main thing to take here is that there is certainly more than one decent choice.

The kind of phenomenon can also be seen in strict $\infty$-categories, where there are at least two different definitions of the notion of a $k$-morphism being a $k$-equivalence: an inductive definition, and a coinductive definition. Which choice you make inevitably feeds into the choice of definiton of weak equivalence between strict $\infty$-cats. For instance, Yves Lafont, Francois Metayer, Krzysztof Worytkiewicz, in "A folk model structure for omega cat", use a coinductive definition of $k$-equivalence.

Answer 3

This is far from a complete answer, but perhaps it will help make rigorous the idea of $(\infty,n)$ as $(\infty,1)$ enriched in $(\infty,n-1)$, and thereby give you another approach to $(\infty,\infty)$. Recently, Rune Haugseng defended his PhD thesis at MIT under Haynes Miller and did the necessary legwork to discuss enriched infinity categories. You can find his website here, and he's posted both his thesis and his research statement there. Rune's work should link up nicely with any model for $(\infty,n)$ category you choose.

Answer 4

This is more of a general comment on the direction of multiple category theory rather than an answer to the question but I hope it will be useful in directing attention to the origins of some of these questions in homotopy theory and for suggesting lines, or a line, for further development. In this I will possibly still continue Out of Line!

I looked at the Berner-Rezk paper referred to in the above, and found the following diagram for compositions in $2$-categories.

^(source)

Compare this with the simplicity of describing the following in a double category.

^(source)

One writes this as an array $(a_{ij})$ with composition $[a_{ij}]$, and this notation is easily extended to $n$ dimensions without using operads, though these might be useful in the cubical context. Note also that an inverse in a specific direction is easy to envisage.

This is why I have found it very difficult to conjecture and prove results in the globular setting, while the cubical setting has allowed a new approach to basic algebraic topology using filtered spaces and homotopy classes of mappings.

My writing to Alexander Grothendieck in 1982 did stimulate his writing of "Pursuing Stacks" in 1983, but he noted that the model of $\infty$-groupoids did not model all homotopy types, and proceeded to look for weak categorical models, which are clearly important. But when in 1985 (or 6) I told him of Loday's work that (strict) $n$-fold groupoids model homotopy $n$-types, he exclaimed: "That is absolutely beautiful!" This work relates to classical parts of homotopy theory, such as $n$-ads, and the Blakers-Massey and Barratt-Whitehead theorems, and gives some new explicit calculations of homotopy $n$-types in a nonabelian context.

Another aspect of Loday's work is his notion of "resolution" of a space: the first stage resolves in one direction, the second in a different direction, and so on, ending up with a resolution of spaces by $n$-cubes of fibrations, which has been clearly explained by Richard Steiner, J. London Math. Soc. (2) {34} (1986) 169--176.

What I am arguing for is an eclectic approach, in which the advantages and difficulties of various approaches are considered.

Update: I'll add a reference to J.C. Morton, "cubical $n$-categories and finite-limits theories", arxiv: 1001.2628, and some further references there. Another relevant paper is arXiv:1204.5101 [pdf, ps, other] n-Fold groupoids, n-types and n-track categories, David Blanc, Simona Paoli

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September 1, 2014 Here is a link to a presentation I gave at the IHP, Paris, June 5, 2014 on Intuitions for cubical methods in nonabelian algebraic topology) (Wayback Machine).

December 26, 2015 I would also like to add this presentation in June 2015 at Aveiro on

A philosophy of modelling and computing homotopy types (Wayback Machine),

which illustrates the aim of doing specific nonabelian colimit computations in homotopy theory.

October 26, 2016 A writeup of part of the above presentation is available on the arXiv as Modelling and Computing Homotopy Types:I.