(NOTE: I reposted the question to MO. Please answer there.)

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, it doesn't help thinking about simplicial topological spaces.

They are useful: 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, if I'm not mistaken, in this example the additional data is redundant.

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. The heuristic above is to think about "topological topological spaces", which doesn't make any sense to me. (Out of pure curiosity, are simplicial simplicial spaces useful? Or even quadrosimplicial sets?)

So my question is: how to think about simplicial spaces? In particular, does the phrase "topological topological spaces" mean something concrete and simple?

  • $\begingroup$ Concerning the second paragraph: "Topological category" has at least three meanings. Which one do you mean? Probably a category-object in $\mathsf{Top}$? Concerning the fourth paragraph: simplicial simplicial $\cong$ bisimplicial, is useful, as you have already remarked in the second paragraph. $\endgroup$ – Martin Brandenburg Jan 10 '14 at 13:51
  • $\begingroup$ By topological category I mean a category where set of objects and set of morphisms both have a topology, and "source" and "target" mappings are continuous. Probably that's exactly category-object in Top. As for bisimplicial spaces, the question still holds: they are more unusual than bisimplicial sets, and I'm interested whether trisimplicial sets are common and useful. $\endgroup$ – Dmitry Jan 10 '14 at 14:02

The analogy between simplicial $X$-es and topological $X$-es works so long as the notion of weak equivalence between simplicial $X$-es are induced by some kind of geometric realisation. But this is definitely not the case when working with simplicial spaces: weak equivalences of simplicial spaces are usually defined to be componentwise (= levelwise, objectwise) weak equivalences.

It is probably better to think of simplicial spaces as being models for "homotopy-coherent" diagrams $\mathbf{\Delta}^\mathrm{op} \to \mathbf{H}$, where $\mathbf{H}$ is the $(\infty, 1)$-category of homotopy types. This probably sounds a bit silly because a simplicial space is in fact a commutative diagram $\mathbf{\Delta}^\mathrm{op} \to \mathbf{Top}$, but the interesting part is the converse: it is a theorem of Vogt that every "homotopy-coherent" diagram of topological spaces can be rectified to a weakly equivalent strictly commutative diagram. For example, if you wanted to generalise the notion of "category" to the homotopy-coherent setting, it would make sense to look at simplicial spaces (à la Rezk).

That said, there is also the Moerdijk model structure on bisimplicial sets, in which a weak equivalence is a morphism that induces a weak equivalence between the diagonal simplicial sets. The Moerdijk model structure is Quillen equivalent to the standard Kan–Quillen model structure on simplicial sets.

  • $\begingroup$ Honestly, homotopy-coherent diagrams in the category of homotopy types are not things I would call intuitive (yet). Is there any handwaving explanation, maybe with some simplifying lies, of this view of simplicial spaces? I am looking for something really down-to-earth, like motivating simplicial sets by saying that all ways of mapping simplices to a CW-complex capture enough information to do homotopy theory. I am not closely familiar with modern homotopy theory, and (for now) I am more interested in informal explanations than in precise facts. $\endgroup$ – Dmitry Jan 10 '14 at 19:16
  • $\begingroup$ This is quite down-to-earth, really: I'm saying a simplicial space is not much more than a diagram of spaces. There's nothing special about $\mathbf{\Delta}$ either – the same goes for the category of diagrams $\mathcal{C} \to \mathbf{Top}$ for any small category $\mathcal{C}$. $\endgroup$ – Zhen Lin Jan 11 '14 at 0:16
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    $\begingroup$ That's sort of frustrating. Simplicial sets are just diagrams of sets, but they have a powerful interpretation as convenient analogs of topological spaces. I was hoping that simplicial spaces admit, at least informally and partially, some space-like intuition as well, though I have no idea how to formulate what I'd like more precisely. $\endgroup$ – Dmitry Jan 11 '14 at 0:39
  • $\begingroup$ That's not true. Simplicial sets admit (by design) a geometric realisation functor that give them their geometric meaning. I know of no such thing for simplicial spaces. $\endgroup$ – Zhen Lin Jan 11 '14 at 22:13

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