Definition of an n-Category – Higher Category Theory Reference

ct.category-theoryhigher-category-theoryhomotopy-theoryquantum-field-theoryreference-request

What's the standard definition, if any, of an $n$-category as of 2020? The literature I can tap into is quite limited, but I'll try my best to list what I had so far.

In [Lei2001], Leinster demonstrated 10 different definitions for an $n$-category, and made no comment on whether they are equivalent or not. In [BSP2011], the authors set up axioms and claimed that all (many?) definitions of an $(\infty,n)$-category so far satisfy their axioms, and therefore are equivalent (up to some action). I include those definitions here for completeness:

  • (a) Charles Rezk’s complete Segal Θn-spaces,
  • (b) the n-fold complete Segal spaces,
  • (c) André Hirschowitz and Simpson’s Segal n-categories,
  • (d) the n-relative categories of Clark Barwick and Dan Kan,
  • (e) categories enriched in any internal model category whose
    underlying homotopy theory is a homotopy theory of (∞,
    n)-categories,
  • (f) when n = 1, Boardman and Vogt’s quasicategories,
  • (g) when n = 1, Lurie’s marked simplicial sets, and
  • (h) when n = 2, Lurie’s scaled simplicial sets,

However, all cases in [Lei2001] do not seem to be covered, and there are even more here. What's the crucial difference between defining an $n$-category and an $(\infty,n)$-category?

Question

In short, there are many definitions for higher categories.. so which one should we use? Is there a list of all definitions made, and a discussion on which is equivalent to which under which sense? Are there also discussions on which definition satisfies the three hypotheses

  1. stabilization hypothesis
  2. tangle hypothesis
  3. cobordism hypothesis

postulated in [BD1995]?

Reference

  • [Lei2001]: A Survey of Definitions of n-Category-[Tom Leinster]-[Theory and Applications of Categories, Vol. 10, 2002, No. 1, pp 1-70]
  • [BSP2011]: On the Unicity of the Homotopy Theory of Higher Categories-[Clark Barwick and Christopher Schommer-Pries]-[arXiv:1112.0040]
  • [BD1995]: Higher-dimensional Algebra and Topological Quantum Field Theory-[John C. Baez and James Dolan]-[arXiv:q-alg–9503002]

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Best Answer

First of all, there are important differences between the notions of strict $n$-category, weak $n$-category, and $(\infty,n)$-category. The easiest notion is that of a strict $n$-category, and there's no doubt about the definition there: a strict $0$-category is a set, and by induction a strict $n$-category is a category enriched in the category of $(n-1)$-categories.

It's good that you cited Baez and Dolan's paper, which introduced an early model for the notion of a weak $n$-category. Between 1995 and 2001 there was a huge proliferation of other models. Morally, they should be categories weakly enriched in the category of weak $(n-1)$-categories, but there are many ways to define a weak enrichment, because there are many ways of keeping track of higher cells and how they combine. In 2004 there was a conference to try to get everyone together and figure out the commonalities between the models, and which were equivalent to which others. It did not result in one emerging as the "standard" model, and I don't think you should expect that to happen any time soon. However, we now know that models for weak $n$-categories broadly fall into two camps. Wikipedia says it nicely:

There are basically two classes of theories: those in which the higher cells and higher compositions are realized algebraically (most remarkably Michael Batanin's theory of weak higher categories) and those in which more topological models are used (e.g. a higher category as a simplicial set satisfying some universality properties).

Wikipedia also says "Several definitions have been given, and telling when they are equivalent, and in what sense, has become a new object of study in category theory." This matches my understanding of the field as it currently stands. I think of higher category theory as being interested in questions about the many models for weak $n$-categories. That's different from the study of $(\infty,n)$-categories, which is situated more in homotopy theory.

Now, others might come along and say "$(\infty,n)$-categories are the right thing" because MathOverflow has a larger representation of homotopy theorists than higher category theorists. You might get the same feel from reading the nLab, again based on who writes there. But if you go hang out in Sydney, Australia, where higher category theory is alive and well, you will not hear people saying $(\infty,n)$-categories are the "right" model or that the unicity theorem for $(\infty,n)$-categories solves the problem from 2004 of figuring out which models of weak $n$-categories are equivalent.

There is also plenty of ongoing work related to the stabilization hypothesis, tangle hypothesis, and cobordism hypothesis in various models of weak $n$-categories. For example, Batanin recently proved the stabilization hypothesis for Rezk's model based on $\Theta_n$-spaces. Then Batanin and I gave another proof that holds for a whole class of definitions of weak $n$-categories, including Rezk's model. Way back in 1998, Carlos Simpson proved the stabilization hypothesis for Tamsamani's definition of weak n-categories. Gepner and Haugseng proved the stabilization hypothesis for $(\infty,n)$-categories and the type of weak enrichment you'd get using Haugseng's PhD thesis (on enriched $\infty$-categories). Of course, famously, Lurie wrote thousands of pages towards proving the cobordism hypothesis for $(\infty,n)$-categories, and Ayala and Francis gave a shorter proof using factorization homology.

I'm sure there's lots of literature I missed, and I'm sure some will disagree with me in saying "yes, it is still valuable to study models of weak $n$-categories instead of only studying $(\infty,n)$-categories." But you asked for references so here are a bunch to get you started.

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