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Edit Originally the question was whether an arbitrary diagram of finite CW complexes can be approximated by a diagram of finite simplicial sets. In view of Tyler's comment, this was clearly asking for too much. I restricted the question from arbitrary diagrams to simplicial diagrams.

It is well-known that for any finite CW complex $K$ one can construct a finite simplicial set whose geometric realization is equivalent to $K$. But the construction is not functorial.

Suppose we have a simplicial object in finite CW complexes. Can one always construct a simplicial object of finite simplicial sets, whose levelwise geometric realization is connected to the original diagram by a zig-zag of levelwise weak equivalences?

One may ask a similar question in a more general context. Suppose we have an $\infty$-category $\mathcal C$. Suppose $A$ is a collection of (say compact) objects of $\mathcal C$. Let $CW^f(A)$ be the category of finite cellular objects generated by $A$. Roughly speaking, $CW^f(A)$ consists of objects that can be built as a finite homotopy colimit of objects of $A$. I am leaving the definition of $CW^f(A)$ a little vague - feel free to use any reasonable notion. Let $S_\bullet^f(A)$ be the category of simplicial objects in $\mathcal C$ that in each simplicial degree are a finite sum of elements of $A$, and are degenerate above some dimension. Geometric realization gives a functor $S_\bullet^f(A)\to CW^f(A)$. Now we have an obvious generalization of the question:

Suppose we have a simplicial object in $CW^f(A)$. Can we find a simplicial object in $S_\bullet^f(A)$ whose levelwise geometric realization is levelwise (weakly) equivalent to it? If it is not true in general, is there a reasonable sufficient assumption?

Improve this question
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    $\begingroup$ Hi Greg, I suspect that the map $B\Bbb N \to CW$, representing $S^1$ with the endomorphisms $2^k$ for $k > 0$, can't be realized by a diagram of finite simplicial sets, but to prove it I'd want to come up with some explicit growth condition on the number of edges needed to represent elements of $\pi_1$ for a fixed finite model. This may be a little harder for s.sets than for $\infty$-categories because the mapping complexes aren't Kan complexes. $\endgroup$
    – Tyler Lawson
    Commented Aug 19, 2019 at 14:07
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    $\begingroup$ In Tyler's example, can't we just observe that for any finite simplicial set $K$, only finitely many maps on $\pi_1$ are induced by endomorphisms $K\to K$? This is because there are only finitely many maps $K\to K$ in total. $\endgroup$
    – Achim Krause
    Commented Aug 19, 2019 at 16:20
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    $\begingroup$ @AchimKrause That is, indeed, a great deal simpler than what I had in mind. $\endgroup$
    – Tyler Lawson
    Commented Aug 19, 2019 at 16:28
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    $\begingroup$ A possible more general finiteness condition could be the assumption that every object in the diagram has finitely many maps out of it. (This has popped up in some inductive constructions I ran into; examples include finite diagrams and semisimplicial objects.) $\endgroup$
    – R. van Dobben de Bruyn
    Commented Aug 19, 2019 at 17:32
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    $\begingroup$ If the "more general" version is purely homotopical, Tyler's example isn't a counterexample, is it? (a counterexample to an even more general version) And a nitpick, just a problem for objects, if a "finite homotopy colimit" includes retracts, then you run into Wall finiteness problems, but @R.vanDobbendeBruyn's version doesn't. $\endgroup$
    – Ben Wieland
    Commented Aug 19, 2019 at 21:28

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