# What are simplicial topological spaces intuitively?

(This is a repost of a question from MSE. I hope there is more to say.)

I tend to imagine simplicial objects in a category as some kind of "topological objects", with a notion of homotopy. Simplicial sets are like topological spaces, simplicial groups are like topological groups, and so on. This rough idea has some precise formulations as mentioned here. However, as it was pointed to me in an answer to MSE question, this heuristic could not be expected to work when weak equivalences are defined not by equivalences of geometric realizations.

Simplicial spaces are useful and quite common: for instance, the nerve of a topological category is a simplicial space. Applying this to the simplicial construction of $\mathrm{B}G$, where $G$ is a Lie group, we see that $\mathrm{B}G$ is a simplicial manifold, hence we can do Chern-Weil theory on it. Segal's $\Gamma$-categories use simplicial spaces essentially. Well, actually bisimplicial sets, since all they need is to form nerves of various categories, but that's more or less the same thing in the context of this question.

Singular simplicial set functor $\mathrm{Top} \to \mathrm{sSet}$ can be readily modified to produce a simplicial space by using a compact-open topology on the space of mappings. Unfortunately here the additional data is redundant, so there are no lessons to extract from the most familiar example.

I can follow simple arguments involving simplicial spaces, since they are more or less the same as arguments about simplicial sets, which I learned to be (somewhat) happy about. However, this understanding is purely formal.

So my question is: how to imagine simplicial spaces? Is there an informal interpretation in terms of topological spaces with additional structure (probably not)?

The answer to the same question on MSE by Zhen Lin seems to indicate there is no interpretation beyond diagrams of spaces. I suppose it's absolutely true for trisimplicial (or even higher) sets, but bisimplicial sets look feasible for a direct intuitive description.

• In principle, simplicial spaces are purely combinatorial objetcs. If you wish to have a geometric interpretation of them, I'd say you should at least put a model structure on them, and there are different options. Jan 11, 2014 at 19:06
• Your question has no answer. Or that depends what you mean by intuition. What is the intuition behind the number $\pi$ for example ? It is the surface of the unit disk. So what ? Is it helpful ? Do you have a better intuition of $\pi$ after that ? Jan 12, 2014 at 10:20
• @PhilippeGaucher: Yes! If I had never seen any definition of $\pi$ other than continued fractions or series that converge to it, I think your comment that $\pi$ can be characterized as the area of the unit disk would have given me a much better feel for what $\pi$ is and why we care about it, as well as giving me a useful new tool for computing with $\pi$. May 14, 2014 at 1:35

If $\mathcal{A}$ is an abelian category, then the Dold-Kan correspondence supplies an equivalence between the category of simplicial objects of $\mathcal{A}$ and the category of nonnegatively graded chain complexes in $\mathcal{A}$. One can therefore think of simplicial objects as a generalization of chain complexes to non-abelian settings.
In homological algebra, chain complexes often arise by choosing resolutions'' of objects $X \in \mathcal{A}$: that is, chain complexes $$\cdots \rightarrow P_2 \rightarrow P_1 \rightarrow P_0$$ with homology $$H_{i}( P_{\ast} ) = \begin{cases} X & \text{ if } i = 0 \\ 0 & \text{ otherwise }. \end{cases}$$ Among these, a special role is played by projective resolutions: that is, resolutions where each $P_n$ is a projective object of $\mathcal{A}$.
If $X_{\ast}$ is a simplicial space, it might be helpful to think of $X_{\ast}$ as a resolution of the geometric realization $| X_{\ast} |$. It plays the role of a projective resolution'' if $X_{\ast}$ is degreewise discrete: that is, if it is a simplicial set.