Why is "everything staying correct" for simplicial spaces?

Question

I recently need a simplicial generalization of some theorem for rigid spaces, namely Theorem A holds for a rigid space $X$ and I want a Theorem $A_\bullet$ for a simplicial rigid space $X_\bullet$. However, I found lots of steps formally true and it's unreasonable and impossible to check everything. I'm wondering what is the key point behind simplicial generalization is "always true" and if there is any interesting examples where simplicial generalization is false.

Answer 1

In general, it is not true that in the context you are interested in, simplicialization of statements preserves their truth. For example, already “every epimorphism of sets is a retraction”, but not “every epimorphism of simplicial sets is a retraction”. But as long as your concepts and theorems are based on common properties of $\mathrm{Set}$ and $\mathrm{sSet}$, you can interpret them directly into $\mathrm{sSet}$ and obtain valid simplicial versions of the theorems. So, for example, by “groups” we can understand not group objects in $\mathrm{Set}$ (ordinary groups), but rather group objects in $\mathrm{sSet}$ (which are the same as simplicial objects in $\mathrm{Group}$ and are classically called simplicial groups). Algebraic objects themselves are a trifle (they can be interpreted in any category with products), it is interesting that almost the entire layer of mathematical ideas, arguments, styles of reasoning (quantifiers, etc.) can be interpreted in a wide class of categories that generalize $\mathrm{Set}$. This is one of the key ideas of topos theory. Sometimes you need to modify the definition a little to make it work the way you intended in a more general context. But really, a lot of math can be done this way (you can find a number of detailed discussions about this on the forum. For example: 1, 2).

So, to check whether the theorem carries over, you need to

• make sure that its proof does not use the axiom of choice, the law of the excluded middle and other set-theoretic facts that are not satisfied in $\mathrm{sSet}$. If the proof is constructive (true in intuitionistic set theory), then this is sufficient (such theorems are true in all topoi).
• make sure that your external definition of the simplicial version of the concept (as the functor $\Delta^\text{op} \to C$) matches the internal one. This is automatically true for all essentially algebraic theories (which correspond to a very wide class of categories called locally presentable). And for all categories of algebras over monads, it seems to me. If this is true, then it is also true for spaces (by which I mean, of course, locales), because they are the dual of the monadic category of frames.

I hope this is useful for you.