Let $\mathcal{S}$ be the $(\infty, 1)$-category of spaces. Then the compact objects of $\mathcal{S}$ are precisely the retracts of finite CW complexes. These are not the same as the finite CW complexes (i.e., the smallest subcategory of $\mathcal{S}$ containing $\ast$ and closed under finite homotopy colimits). Namely, given $X$ (let's say connected), the homology of the universal cover defines a perfect complex of $\mathbb{Z}[\pi_1 X]$-modules, which gives a class in $\widetilde{K}_0( \mathbb{Z}[\pi_1 X])$, which vanishes when $X$ is a finite CW complex. This class is the Wall finiteness obstruction; it is a classical result that this is the only obstruction for $X$ to be a finite CW complex. The reason for its existence is that taking the "image" of an idempotent is not a finite homotopy colimit (though it would be if one worked with $n$-categories for some $n$).

I'm curious about other examples of this type of finiteness obstruction. For instance, what happens if we do this in (for example) the $(\infty, 1)$-category of $E_\infty$-rings? The analogs of the finite CW complexes are the $E_\infty$-rings obtained from the free ones (free on a generator in some degree) via finite colimits, and the compact objects are these and the retracts. Is there an analog of the finiteness obstruction here? Other examples: algebras over some other operad, simplicial commutative rings, (pre)sheaves of spaces (resp. spectra), and equivariant spaces (resp. spectra). The only example that I am aware of considers the derived category of modules over a ring, where $\widetilde{K_0}$ measures the difference between "perfect complexes" and "those representable via finite complexes of frees."

  • $\begingroup$ That's an intriguing characterization of the finite CW complexes! Do you have a reference for it? $\endgroup$ – Qiaochu Yuan Aug 27 '13 at 6:49
  • $\begingroup$ @Qiaochu: Here's a link to Wall's paper: math.uchicago.edu/~shmuel/tom-readings/…. Andrew Ranicki gives more references below. I also blogged about it at one point, see amathew.wordpress.com/2012/09/22/…. $\endgroup$ – Akhil Mathew Aug 27 '13 at 13:50
  • $\begingroup$ Ah, sorry, by "that" I meant "the smallest subcategory of $S$ containing $\bullet$ and closed under finite homotopy colimits." $\endgroup$ – Qiaochu Yuan Aug 27 '13 at 21:55
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    $\begingroup$ @QiaochuYuan: well, the operation of building up a CW complex via attaching cells is essentially that of forming iterated (finite) homotopy pushouts, and vice versa. (This argument works for the derived category of modules over a ring, too.) $\endgroup$ – Akhil Mathew Aug 27 '13 at 23:13
  • $\begingroup$ Have you looked at Andy Baker's papers on TAQ? He approaches TAQ by thinking of it as a sort of cellular homology theory (which is an awesome perspective). See the paper by Baker Gilmour and Reinhard first, then the computational approach ones on the arxiv. $\endgroup$ – Sean Tilson Sep 4 '13 at 9:34

My 1985 Math. Scand. paper The algebraic theory of the finiteness obstruction has a chain complex treatment of the Wall finiteness obstruction.

Chapter VII of Hans-Joachim Baues' 1999 Springer Monograph Combinatorial Foundation of Homology and Homotopy has an abstract homotopy theory treatment of the Wall finiteness obstruction.

  • $\begingroup$ Thank you! I had seen your paper, but I was not aware that this was treated in Baues's book. I don't have access right now, but I'll have a look once I get back to school. $\endgroup$ – Akhil Mathew Aug 27 '13 at 13:51

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