I recently read E. Dror’s paper “Acyclic spaces,” which studies the category of spaces with vanishing homology groups. It turns out that this category has a fair bit of structure; in particular, it has a theory resembling the theory of Postnikov systems. In this post and the next, I’d like to explain how the results in Dror’s paper show that the decomposition is really a special case of the notion of a Postnikov system, valid in a general {\infty}-category. Dror didn’t have this language available, but his results fit neatly into it.

Let {\mathcal{S}} be the {\infty}-category of pointed spaces. We have a functor

\displaystyle \widetilde{C}_*: \mathcal{S} \rightarrow D( \mathrm{Ab})

into the derived category of abelian groups, which sends a pointed space into the reduced chain complex. This functor preserves colimits, and it is in fact uniquely determined by this condition and the fact that {\widetilde{C}_*(S^0)} is {\mathbb{Z}[0]}. We can look at the subcategory {\mathcal{AC} \subset \mathcal{S}} consisting of spaces sent by {\widetilde{C}_*} to zero (that is, to a contractible complex).


Definition 1 Spaces in {\mathcal{AC}} are called acyclic spaces.


The subcategory {\mathcal{AC} \subset \mathcal{S}} is closed under colimits (as {\widetilde{C}_*} is colimit-preserving). It is in fact a very good candidate for a homotopy theory: that is, it is a presentable {\infty}-category. In other words, it is a homotopy theory that one might expect to describe by a sufficiently nice model category. I am not familiar with the details, but I believe that the process of right Bousfield localization (with respect to the class of acyclic spaces), can be used to construct such a model category. (more…)