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July 6, 2009

Generalized Homotopy Theory

Posted by David Corfield

Over at nnLab we’re itching for some discussion as to whether there can be something which is to homotopy as Nonabelian (unstable) cohomology is to cohomology. Can we free things up so we don’t just map spheres into spaces? At the entry homotopy (as an operation) you can read the suggestion of a ‘homotopy with co-coefficients in BB’, rather than a sphere.

I suppose Moore spaces as domain would be a start as suggested here for spaces of type (A,2)(A, 2) and suspensions.

A couple of things to check out perhaps:

I have the feeling they’ll be a huge amount out there to learn, e.g., about cofibrant co-grouplike objects.

Posted at July 6, 2009 5:01 PM UTC

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Re: Generalized Homotopy Theory

In a 1996 discussion on the cat-list André Joyal writes

Peter Hilton and Beno Eckmann…proved that any cogroup in the category of groups is free.

I would like to say that the younger generation is not always aware that Eckmann and Hilton have fundamental contributions to category theory. They have provided basic examples of objects equipped with algebraic structures in categories. Consequently opening the road to further abstractions, like the concept of algebraic theory in the sense of Lawvere.

Among other things, Eckmann and Hilton were interested in identifying all cogroups in the homotopy category hTop *hTop_* of pointed topological spaces. A basic example of such cogroup is the circle S 1S^1. It explains why the functor π 1()=[S 1,]:hTop *Sets\pi_1(-)=[S^1,-]: hTop_* \to Sets has a natural group structure. If GG is a cogroup in hTop *hTop_* then so is the smash product XGX \wedge G for any pointed topological space XX. This is because X()X \wedge (-) preserves coproduct since it has a right adjoint (here we are supposing that Top is a convenient category of topological space). In particular, the spheres S n+1=S nS 1S^{n+1} = S^n \wedge S^1 have a cogroup structure. Any wedge (topologists call the coproduct the wedge) of cogroups is obviously a cogroup. In particular, any wedge of spheres of dimension n1n \geq 1 has a co-group structure.

All the known examples of cogroups in hTop *hTop_* are obtained by taking a smash XS 1X \wedge S^1. It was conjectured by Eckmann and Hilton that all cogroups in hTop *hTop^* are of the form XS 1X \wedge S^1. They observe that π 1(G)\pi_1(G) is a cogroup in GroupsGroups when GG is a cogroup in hTop *hTop_*. This is because the functor π 1:hTop *Groups\pi_1: hTop_* \to Groups preserves coproducts by Van Kampen theorem. In support to their conjecture they proved that any cogroup in GroupsGroups is free. It follows that all cogroups in GroupsGroups are of the form π 1(XS 1)\pi_1(X \wedge S^1) where XX is a pointed set.

Posted by: David Corfield on July 6, 2009 9:45 PM | Permalink | Reply to this

Re: Generalized Homotopy Theory

As an example of the sort of theory that is available for this, Baues in his book `Algebraic Homotopy’ (section II.6, starting on page 115), develops the notion of homotopy groups in a cofibration category, using suspensions π n A(U)=[Σ nA,U].\pi_n^A(U) = [\Sigma^n A,U]. These have all the usual elementary properties with relative forms, long exact sequences etc. AA has to be a based object for it to work.

The development is quite detailed and lengthy.

Posted by: Tim Porter on July 7, 2009 12:26 PM | Permalink | Reply to this

Re: Generalized Homotopy Theory

So these are abelian for n>1n \gt 1. Presumably stablization of homotopy groups with coefficients goes through.

Posted by: David Corfield on July 7, 2009 2:22 PM | Permalink | Reply to this

Re: Generalized Homotopy Theory

David you say: “Presumably stablization of homotopy groups with coefficients goes through.” I could not find a mention of stabilisation with in Baues’ book. (Perhaps he treats that in another text.)

I do not quite see what you mean by “homotopy groups with coefficients”. Are the AA somehow the coefficient? (I suppose the duality could lead either to (co)2^2efficients or worse `efficients’ in that case … I know I should have resisted the obvious pun.)

Posted by: Tim Porter on July 7, 2009 2:39 PM | Permalink | Reply to this

Re: Generalized Homotopy Theory

Urs is using the phrase ‘homotopy of XX with co-coefficients in BB’ at the speculative page homotopy (as an operation), and is also tempted by that bad pun.

Hatcher in section 4.H talks about ‘Homotopy Groups with Coefficients’, using Moore spaces as domain. Presumably in this case [Σ n(M(k,G)),Σ n(X)][\Sigma^n (M(k, G)), \Sigma^n (X)] stabilizes as nn \to \infty.

Posted by: David Corfield on July 7, 2009 3:06 PM | Permalink | Reply to this

Re: Generalized Homotopy Theory

the notion of homotopy groups [with co-co-efficients] in a cofibration category, using suspensions

π n A(U)=[Σ nA,U] \pi_n^A(U) = [\Sigma^n A, U]

Hey, that’s great. That’s pretty much what David was looking for!

We are gonna work this into [[homotopy]] to make it become more completely the abstract dual of [[cohomology]].

While I am looking at the book:

“cofibration category” is that [[cat of fib objects]] opposed or is it [[Waldhausen category]]?

Posted by: Urs Schreiber on July 7, 2009 7:12 PM | Permalink | Reply to this

Re: Generalized Homotopy Theory

From memory it’s a category of cofibrant objects, or the same with minor variations. More suited to sSetsSet than TopTop

Posted by: David Roberts on July 8, 2009 5:27 AM | Permalink | Reply to this

Re: Generalized Homotopy Theory

Yes, it is categories of cofibrant objects (in the Baues definition in which the axioms are the duals of Ken Brown’s axioms with some slight additions). Many of the arguments used are really just the Dold-Puppe fibration / cofibration long exact sequences suitably abstracted.

(I tried to post this yesterday but the system went AWOL!)

Posted by: Tim Porter on July 8, 2009 10:52 AM | Permalink | Reply to this

Re: Generalized Homotopy Theory

with some slight additions

Yes, these should be sorted out. It all looks almost trivially equivalent, but it seems one has to exercis a bit of care here and there for precisely relating these axiom systems.

I see that Baues in his remark (1a.6) comments briefly and roughly on the slight variations in the axioms of

-Brown cat of fibs / dually of cofibs

- Anderson cat of cofibs

- Heller and Shitanda who do something else

- Waldhausen of cofibs.

One easy thing is how the factorization lemma assumed by Baues implies the existence of (co)path objects assumed by Brown (as in his remark on p. 422), while Brown of course proves the converse. So that bit is clear.

I have yet to think about the other differences.

It would be good to eventually list the relations somewhere, it’s slightly annyong that with each author one needs to start with slightly new axioms.

Posted by: Urs Schreiber on July 8, 2009 1:06 PM | Permalink | Reply to this

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