nLab pro-space




Sometimes systems of ‘spaces’ are used to approximate a space which does not have ‘nice’ properties. This can be a general compact space (as in shape theory, or an object of interest from algebraic geometry such as a scheme or algebraic stack. The use of Čech methods, for instance, allows for the extraction of invariants (and other information) from an approximating pro-space using methods from adapted standard homotopy theory, where standard method cannot be directly applied to the original ‘spatial object’.


A pro-space is a pro-object in a category of topological spaces.

In many applications, the spaces considered will have the homotopy type of CW-complexes. This is the case in strong shape theory, but also in the important applications to algebraic geometry.

Pro-simplicial sets are often called pro-spaces following the tradition in certain schools of homotopy theory of referring to simplicial sets as ‘spaces’.

There are several useful model category structures on categories of pro-spaces. These are described in Isaksen’s paper (listed below).


  • D. C. Isaksen, A model structure on the category of pro-simplicial sets,

    Trans. Amer. Math. Soc., 353, (2001), 2805–2841

Last revised on August 26, 2012 at 12:30:41. See the history of this page for a list of all contributions to it.