If $\mathcal C$ and $\mathcal D$ are two categories each equipped with a class of morphisms called weak equivalences, then by an equivalence of homotopy theories I will mean functors $F:\mathcal C\to \mathcal D$ and $G:\mathcal D\to\mathcal C$ such that each of the compositions $G\circ F$ and $F\circ G$ is related to the relevant identity functor by a zigzag of natural weak equivalences. This is much stronger than just requiring $F$ and $G$ to induce inverse equivalences of homotopy categories: For an example, let $\mathcal C$ be the category of chain complexes over a field with quasi-isomorphisms as weak equivalences and $\mathcal D$ the subcategory of complexes with zero boundary map. ($F$ is the homology functor and $G$ is the inclusion.) You can see that this is not giving an equivalence of homotopy categories, because if it did then it would also give an equivalence of homotopy categories of $I$-diagrams for any $I$. Of course, if $\mathcal C$ and $\mathcal D$ are model categories then a Quillen equivalence gives an equivalence in the sense I mean (after restricting to categories of fibrant-cofibrant objects). But let's look at this simpler notion.

This is meant to be an improvement on my previous question When do basepoints matter in homotopy theory?. I was tempted to delete the latter, but decided to let it stand for its possible educational value and as a hairshirt for myself. I seriously suggest that nobody should upvote this question without downvoting that one.

Consider the category of simply-connected spaces (a.k.a. $1$-connected spaces, i.e those having exactly one path component and trivial fundamental group) and continuous maps. We can universally invert the weak homotopy equivalences and get what might be called the homotopy category of $1$-connected spaces. This can be described as a full subcategory of the homotopy category associated to the Quillen model structure on $Top$. Alternatively, it can be described as the category whose objects are $1$-connected $CW$ complexes and whose morphisms are homotopy classes of maps. Note that I am using unbased spaces and unbased maps here.

QUESTION 1: Is there an equivalence of homotopy theories in the sense defined above between $1$-connected spaces and (the fibrant-cofibrant objects of) some model category.

Another good example of a functor inducing an equivalence of homotopy categories but NOT an equivalence of homotopy theories is the forgetful functor from based $1$-connected spaces to unbased $1$-connected spaces. The positive statement just means that every unbased map between based $1$-connected spaces is homotopic to a based map, and that two such based maps are based homotopic if they are homotopic. The negative statement can be proved by using what I said above about categories of diagrams; for example, you can get a contradiction by picking an action of a group on a $1$-connected space such that the homotopy fixed point set is empty.

The category of $1$-connected based spaces is equivalent in this sense to a model category. You can colocalize the category of based spaces. The main idea is this: In the category of based spaces every object has a universal $1$-connected object over it. As some kind of evidence for a "no" answer to the question above, note that the statement in the last sentence is false for unbased $1$-connected spaces. More evidence: a "yes" answer would seem to lead to a reasonable theory of homotopy limits and colimits within unbased $1$-connected spaces. How could that go? In the based setting, hocolim is usual hocolim and holim is universal covering space of basepoint component of usual holim.

All of the above statements, questions, and suggestions about $1$-connected spaces have analogues in which weak homotopy equivalence is replaced by rational equivalence. Sullivan's version of rational homotopy theory shows that the theory of unbased $1$-connected spaces of finite type is equivalent to that of the opposite of $1$-connected commutative DGAs (differential graded algebras) of finite type. I presume that this can be used to see that the theory of (unbased) $1$-connected spaces is equivalent to that of $1$-connected commutative DGCs (differential graded coalgebras). Quillen's version gives an alternative route, but with basepoints: a chain of Quillen equivalences between based $1$-connected spaces and $1$-connected based (i.e. coaugmented) commutative DGCs (differential graded coalgebras).

QUESTION 2: Assuming the answer to Q1 is "no", do you have a good point of view to offer regarding this tradeoff between the artificiality of working with based objects and the disturbing lack of homotopy (co)limits in the unbased setting?

  • $\begingroup$ Here's a random thought that occurred to me: wouldn't it make sense to consider spaces which are disjoint unions of simply-connected spaces? The content of homotopy theory should be essentially the same, but maybe the category of such spaces would be better behaved (for one thing it has coproducts)? $\endgroup$ Commented Mar 9, 2012 at 14:39
  • $\begingroup$ Sure, but it doesn't have homotopy colimits: you can make a circle by gluing homotopically discrete spaces. $\endgroup$ Commented Mar 9, 2012 at 14:41

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It seems to me that the answer to (1) is "no", for essentially the reason you've given. If $\mathcal{C}$ is a category with weak equivalences $W$, you can extract a simplicially enriched category $\mathcal{C}[W^{-1}]$ using Dwyer-Kan localization, say. If this simplicially enriched category comes from a model category, then it has homotopy coproducts. That is, for every pair of objects $X$ and $Y$, there is another object $Z$ and a pair of maps $X \rightarrow Z \leftarrow Y$ with the following property: for every object $W$, the induced map Hom(Z,W) $\rightarrow$ Hom(X,W) $\times$ Hom(Y,W) is a weak homotopy equivalence of simplicial sets. The simplicially enriched category of simply connected CW complexes doesn't have these: for example, there is no homotopy coproduct of a point with itself.

As for $(2)$: you don't have arbitrary homotopy colimits, but you do have lots of them. For example, you have all homotopy colimits indexed by diagrams with simply connected nerve. This includes filtered colimits, geometric realizations of simplicial objects, and the formation of pushouts. This is enough to allow for some useful techniques: for example, you can write every simply connected space as a geometric realization of a simplicial simply connected space, each term of which is a bouquet of $2$-spheres.

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    $\begingroup$ Thanks for the systematic answer. I realized also that, more simply, you can just write down a diagram of simply connected spaces whose usual holim is not simply-connected, apply the hypothetical $F$ and $G$ to it, and get a contradiction. $\endgroup$ Commented Mar 8, 2012 at 12:28
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    $\begingroup$ Jacob, Tom, your arguments also show that the homotopy category of 1-connected spaces cannot be enhanced to a right pointed derivator. I just wanted to point this out in case somebody finds it interesting. $\endgroup$ Commented Mar 8, 2012 at 15:32

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