For a category $C$, there is a category Pro-$C$ whose objects are cofiltered diagrams $I \to C$ and whose morphisms are given by $$ {\rm Hom}(\{x_s\},\{y_t\}) = \varprojlim_t\ \varinjlim_s\ {\rm Hom}(x_s,y_t). $$ Generally, this category is fairly hard to work with. This is especially true because several types of maps of pro-objects are defined in terms of the existence of a representing map of diagrams with certain properties, and it can be very difficult to rectify several distinct properties at once.

One way to describe the category of pro-objects is by inverting morphisms. Specifically, we can form a more restricted category of pairs $(I,F)$ of a cofiltered index category $I$ and a diagram $F: I \to C$, with morphisms defined as pairs of a functor and a natural transformation of diagrams. Certain maps of cofiltered diagrams become isomorphisms of pro-objects (the most important ones being reindexing along a final subcategory). Inverting them gives us the pro-category.

In some sense, this automatically provides us with a "category with weak equivalences", but it's intrinsically very large and it's not necessarily clear if the "homotopy theory" is tractable.

Are there any circumstances under which the category of diagrams in $C$ automatically has the structure of a model category, with weak equivalences being pro-isomorphisms? In these cases, does the category Pro-$C$ have an interesting homotopy theory or are the mapping spaces essentially discrete?

Obviously being complete and cocomplete is going to be an obstacle to this kind of structure. Failing that, is there any further possibility of gaining control on the homotopy theory?

Having said all this, I've been a little bit vague about what I mean by a "map of diagrams" because I'd be open to the idea of having slightly restricted classes of maps in the definition.

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    $\begingroup$ Another way to think about pro-objects in a small category $C$ is as left exact functors $C\to Set$. I usually find this description more convenient, but I don't see any obvious homotopical structure associated to it. $\endgroup$ – Marc Hoyois Mar 3 '12 at 3:30
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    $\begingroup$ Two examples and a question: The category of pro-finite sets is equivalent to the category of compact spaces that are Hausdorff and totally disconnnectd. The bigger category of pro-sets is equivalent to the category of complete uniform spaces that are Hausdorff and totatlly disonnected. When can Pro $C$ be described as a category of "topological" objects in $C$? $\endgroup$ – Jeff Smith Mar 3 '12 at 21:06
  • $\begingroup$ @Jeff It depends what you mean by 'topological'? There are several categories that occur in pro-finite theory that have a topological side, e.g. pseudocompact modules over a pseudocompact ring, but they are not that far from the profinite set case. $\endgroup$ – Tim Porter Mar 4 '12 at 16:40
  • $\begingroup$ @Tyler: A paper by Barnea--Schlank: arxiv.org/pdf/1109.5477v6.pdf might be relevant. You probably can in usual circumstances define a weak fibration category structure on your category with isomorphisms as weak equivalences; their machinery gives you then a model structure on the pro category. $\endgroup$ – Lennart Meier Dec 9 '13 at 20:13

There are some answers to this way back. There is a lovely answer to several of your questions in

D. A. Edwards and H. M. Hastings, 1976, ˇCech and Steenrod homotopy theories with applications to geometric topology , volume 542 of Lecture Notes in Maths , Springer-Verlag.

The category of prosimplicial sets has a model category structure that corresponds to a geometrically defined notion of strong shape theory (i.e. a homotopy coherent version of Borsuk's shape theory). Edwards and Hastings extended a result of Chapman and showed this model category theory also to be a form of proper homotopy theory. (There is also some discussion of this in my article:

T. Porter, 1995, Proper homotopy theory , in Handbook of Algebraic Topology , 127–167, North-Holland, Amsterdam. )

The story does not end there. Because of the connection with étale homotopy theory (Artin and Mazur), there was a revival of interest in pro-categories in the last few years and there is a good discussion in

H. Fausk and D. Isaksen, Model structures on pro-categories , Homology, Homotopy and Applications, 9, (2007), 367 – 398.

I suggest that you also look at others of Dan Isaksen's papers on this area as they answer more of the quetions that you have asked.

On another point that you mention, the rectification process for properties is reasonably well understood due to what is known as the reindexing lemma (the simplest case is in Artin and Mazur's lecture notes but there are much fuller versions some of which are discussed in another of Isaksen's papers

D. C. Isaksen, Completions of pro-spaces , Math. Z., 250, (2005), 113 – 143. )

If you read these papers carefully you will come to the conclusion that certain problems are still not fully understood especially when pro-finite simplicial sets are concerned, and the applications of those beasties are again very important so that is a good area to explore!!!

(See also work by Quick (Profinite homotopy theory , Documenta Mathematica, 13, (2008), 585–612.) and Pridham (Pro-algebraic homotopy types , Proc. Lond. Math. Soc. (3), 97, (2008), 273 – 338. ) They show some of the more recent stuff on this with some good applications. There are copies on the ArXiv.)

  • $\begingroup$ If you only care about having an $\infty$-category rather than a model structure, things are much easier: pro-objects in an $\infty$-category $C$ are simply left exact (accessible) functors $C\to \infty Grpd$, which naturally form an $\infty$-category. The étale homotopy type is also easily defined in this language (see Higher Topos Theory, section 7.1.6). But this doesn't help for putting a nontrivial homotopy structure on pro-objects if the category $C$ did not have one in the first place. $\endgroup$ – Marc Hoyois Mar 4 '12 at 22:10
  • $\begingroup$ Tim, thanks for the references. I haven't looked into Edwards and Hastings, but part of my motivation for asking the question was working with some of the model structures that Fausk and Isaksen work with. My understanding is that most of these references still work with honest pro-objects, rather than having a "weak equivalence" structure on diagrams, correct? $\endgroup$ – Tyler Lawson Mar 5 '12 at 5:56
  • $\begingroup$ @ Tyler I don't know if this helps but I tried out a version of doing pro-homotopy in three papers on coherent pro-homotopy. (The references are on my nLab personal home page: ncatlab.org/timporter/show/HomePage but I let you look down the list for strong shape.) The idea was to take the diagram categories with a good homotopy structure and paste them together. My method was motivated by some of Vogt's work (his theorem on categories of homotopy coherent categories). Perhaps the methods would now be considered a bit 'crude' but they could be revisited using more modern viewpoints. $\endgroup$ – Tim Porter Mar 5 '12 at 8:11
  • $\begingroup$ @Marc Batanin also did some work on strong shape and infinity category structures and, if you have the time, that would be well worth looking at, again with a view to pushing through a modern view of it. His results might be compared with Lurie's approach. $\endgroup$ – Tim Porter Mar 5 '12 at 8:15

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