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?
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
- stabilization hypothesis
- tangle hypothesis
- cobordism hypothesis
postulated in [BD1995]?
- [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]