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Strong homotopy equivalences are the weak equivalences of the Strøm model structure on topological spaces. Hm, but you are looking for something Quillen equivalent but on cominatorial structures? Is that right?
It really depends on what you mean by ‘algebraic’? You can build abstract homotopy theory in various ways, and there are model category structures that do correspond more or less to what you seem to be asking for,( as in Urs’ reply above), but from that to algebraic models of strong homotopy types is still a distance. Strong shape theory goes some way in another direction, and it is possible to combine the singular homotopy type and the strong shape of a space into a single invariant and to look at that. I do not know if anyone else can think of other interpretations of your query. You do not make precise the type of spaces you are considering (e.g. compact? paracompact, etc.) and again that is likely to influence the answer.
As most researchers end up being frustrated at certain times of their life, it would be nice to know which of the thousands of such we are chatting to, so do say who you are, when you feel like it.
By just a gut feeling, I’d be surprised if there were a combinatorial model for strong homotopy type. Because all cominatorial models we know present – almost by definition – a topological space as a cell complex.
Or in other words, you’d need a notion of “combinatorial presentation of a topological space” which is not by cell complexes. Are there any candidate constructions in the literature?
(I am really just asking. I may just be ignorant. This is just expressing a gut feeling. Will be happy to be proven wrong.)
My thought about a combination of singular and strong shape may in fact help. It will not give complete models for homotopy types but should give both ’left’ and ’right’ approximations to such. For instance there is the ordinary fundamental group of a pointed space and also the shape one and there is a homomorphism from the ordinary to the shape. Perhaps this could be thought of as a model for the whole space. I thik this would mean replacing say the singular complex or the homotopy limit of the Cech complex of a compact space by map from the first to the second and studying that. More generally, if there were some ‘good categories’ of spaces that gave nice theories, you can look at the interrelationships between them as an invariant. (This is a bit like the situation in Sullivan’s genetics of homotopy types, where at each prime you get invariants and you combine them to get really good invariants for the homotopy type. That is the idea but of course life is not as simple as that.)
My previous reply was to see if something on those lines was what you wanted. There is a notion of categorical shape theory in which you have a category C of nice objects sitting inside a category D of (less nice) objects. For each object d in D you look at the comma category (d,C) of C-objects under d, and use this to give a model of what you want. For instance take C to be the finite groups and D all groups then (d,C) consists in part of all finite quotients of the group d so we end up working with the profinite completion of d (which is the limit of all finite quotients). The shape category looks at all the objects of D and has (certain) functors between these comma categories as the morphisms. Now go over to the infity category setting and you get strong shape. The archetypal example is with compact Hausdorff spaces as D and compact polyhedra as C (more or less) and is a sort of Cech homotopy theory. (Say this is made of C-approximations to d from the right)
If you now look at the dual construction using the other commas (C,d) then you get the singular homotopy theory in that ’classical’ case. This gives strong coshape! (and approximations from the left.)
Now put the two together, a morphism from d_1 to d_2 will be a pair of functors as if induced by composition on the two types of comma categories, so preserving the approximating objects. (This gives a notion of paired shape or some term like that.) Again this should really be in an infinity category context. My point was the approximations are via (C,d)x(d, C) so come with a composite to the Arrow category of C. In the classical case of spaces, this would mean assigning a system of map (of simplicial sets or CW-spaces) to each space, composing a left approximation with a right approximation .
As an example take a Warsaw circle (sin 1/x curve plus y axis) and spin it around the y axis. You get a disc with a singularity in the middle. The singularhomotopy type /coshape of this is a circle plus a point. The shape is a disc. There is a map from the coshape to the shape, which measures some facts about the difference of the two. Now apply your favorite algebraic models to these MAPS, i.e. do not just take the approximating spaces but consider the maps. If you do this in general you might need to replace the system of right approximations by their homotopy limit as being more amenable object.
There are phantom homotopy classes around in some of these limiting situations that can make life ’interesting’. There are also spaces that are compactifications of open manifolds that may be of interet (see items on Proper Homotopy Theory for these.)
As this is another tack I will start a new box! You ask what homotopy limits really mean. The cunning answer is to ask you what you really mean by really mean! Look at them from various directions and choose one that suites you best as a starting point. Homotopy pullbacks are a useful point of entry, then generalise (I like the homotopy coherent picture as being fairly concrete but others prefer the derived functor one or whatever. Their geometric picture that comes from Proper Homotopy Theory may be a good one to glance at as well.)
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