The question asks whether a simplicial object in finite CW complexes can be approximated by a simplicial object in finite simplicial sets, with levelwise weak equivalences in geometric realization.

Key Analysis:

1. Single Finite CW Complexes: Every finite CW complex ( K ) can be converted into a finite simplicial set via barycentric subdivision. The barycentric subdivision ( \text{Sub...

Short answer for the simplicial case: yes.

Let X• be a simplicial object in finite CW complexes. Then there is a simplicial object K• in finite simplicial sets and a zig–zag of levelwise weak equivalences of simplicial objects |K•| ~→ |Sing X•| ~→ X• so that, in particular, each |Kn| has the homotopy type of Xn and the maps respect the simplicial structure (strictly) in the middle object.

He...

Here is what one really gains by abstracting the usual “triangulation of finite CW‐complexes” argument into an ∞–categorical framework:

1. Let C be a presentable ∞–category which is compactly generated by a set A of compact objects. By definition CW⁽ᶠ⁾(A) is the full sub–∞–category of C spanned by those objects which can be built from A by finitely many homotopy–colimit steps (f...

This is an interesting question about the relationship between finite CW complexes and finite simplicial sets, particularly in the context of simplicial diagrams.

Let me address the main question: Yes, for simplicial objects in finite CW complexes, one can construct simplicial objects in finite simplicial sets whose geometric realizations are levelwise weakly equivalent.

Here's why this work...

This is an excellent question that touches upon the foundational relationship between the combinatorial world of simplicial sets and the geometric world of CW complexes. The difficulty, as you've correctly identified, lies entirely in functoriality.

The answer to your main question is yes.

Given a simplicial object in the category of finite CW complexes, one can indeed construct ...

Yes – every simplicial diagram of finite CW–complexes can be replaced, up to level–wise weak equivalence, by a simplicial diagram whose entries are finite simplicial sets. More precisely

Statement.
Let X• : Δop → Top be a simplicial object such that every space Xn is a finite CW–complex.
Then there is a simplicial object K• : Δop → sSet such that

1. every Kn is a finite simplicial set (i.e....

<think> Okay, the user is asking a pretty advanced question in algebraic topology, specifically about the equivalence between simplicial finite CW complexes and simplicial finite simplicial sets. Let me unpack this carefully.

First, I recall that for individual spaces, we know every finite CW complex can be approximated by a finite simplicial set up to weak equivalence. But the user is asking abo...

Yes for topological spaces; in general only under additional hypotheses.

PART 1. Topological spaces

Let X• be a simplicial object in the category CWfin of finite CW–complexes.
We claim that there is a simplicial object K• of finite simplicial sets and a level-wise homeomorphism

  |K n |  ≅  X n ,		n ≥ 0,

so that the two simplicial objects are connected by a zig–zag of level-wise weak e...

Simplicial Finite CW Complexes vs. Simplicial Finite Simplicial Sets

This is a question about representing simplicial objects in finite CW complexes by simplicial objects in finite simplicial sets, up to levelwise weak equivalence.

Background

First, let's recall the key facts:

1. For any single finite CW complex , we can find a finite simplicial set such that (the geometric r...