Are strict higher categories more general than weak higher categories?

Question

I've been exploring the differences between strict and weak higher categories, and I'm curious about their expressiveness and generality. In strict higher categories, morphisms and coherence are explicitly defined at each level, which seems to offer a more comprehensive framework.

My questions:

• Do strict higher categories provide a more general framework than weak categories because they allow explicit definitions that can encompass weak behaviors?

• How does the complexity of strict categories compare to the flexibility of weak categories in terms of modeling mathematical structures?

• Can strict categories inherently express more due to their explicit nature, or does the homotopy equivalence in weak categories offer a fundamentally different form of expressiveness?

• If strict higher categories are able to capture more fine grained detail that is lost in homotopical approach, why is it less used? Is it because working with strict higher category theory would uncover more details, but it would be too hard?

Answer 1

As Qiaochu has explained in the comments, strict higher categories are less general: You can pass from strict to weak higher categories, but many important $\infty$-categories (for example the fundamental $\infty$-groupoid of a space) only exist in the weak world and do not lift to the strict world.

While it should in principle be possible to give combinatorial descriptions with explicit coherences of weak $n$-categories (this seems to be what you asked for in your follow-up comment), note that this can't be easy: The free weak $n$-groupoid on one object with a nontrivial $m$-automorphism looks like (the fundamental $\infty$-groupoid of) $\tau_{\leq n} S^m$, in particular it has $\pi_k S^m$ for $k\leq m$ as the $k$-endomorphisms of the object. So for example, any combinatorial description of, say, weak $7$-categories, must somehow reproduce things like $\pi_7(S^4)=\mathbb{Z}\times \mathbb{Z}/12$. In fact, the fundamental $\infty$-groupoid of $\tau_{\leq n} X$ for any finite CW complex $X$ can be interpreted as $n$-groupoid given by an explicit generators-and-relations description, so a suitably effective combinatorial description of $n$-categories more or less completely subsumes homotopy theory "up to degree $n$".