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Patrick Nicodemus (Dec 12 2021 at 02:06):In Goerss-Jardine they point out that there are two reasonable definitions of the (reduced) suspension of a simplicial set. One is the smash product with S1 and one is the join with S0, the "Kan suspension". They claim these two are homotopy equivalent, but I can't find a proof of this described in the book. I would like a reference for the homotopy equivalence of the definitions.
Likewise I am interested in a proof that their corresponding loop space adjoints are homotopy equivalent. Perhaps this follows by formal nonsense but even so I would like to see a reference for the formal nonsense.
I need explicit descriptions of the maps in both directions for computational reasons.
John Baez (Dec 12 2021 at 02:43):Do you want simplicial maps, not just continuous maps between the geometric realizations? I could probably cook up the latter but I'm afraid you want the former.
Patrick Nicodemus (Dec 12 2021 at 08:04):John Baez said:
Do you want simplicial maps, not just continuous maps between the geometric realizations? I could probably cook up the latter but I'm afraid you want the former.
Oh, yeah, thank you. It's easy for me to believe the associated spaces are homeomorphic, but no I really want the simplicial maps.
Maybe I should lead with my motivation : I'm really interested in simplicial Abelian groups rather than simplicial sets, I'm thinking about the Dold-Kan correspondence. But I figured a formula that worked for simplicial sets would also hold for Abelian groups.
Patrick Nicodemus (Dec 12 2021 at 08:07):Often people use Dold-Kan in a handwavy way to say "well everything should be translatable across the equivalence" but in computational practice it seems quite difficult to do this translation. For example, if C is a chain complex and is the loop space (just downshifting), there is a natural map given by the differential. Easy as pie. But what does this map look like when we cross to the category of simplicial Abelian groups. It's hard to give a nice description of it.
Patrick Nicodemus (Dec 12 2021 at 08:10):Similarly, the Alexander-Whitney map is of the form , so we should be able to transport it back across the equivalence and get a map . The right hand term should not be so complicated and pathological as it might look at first because the tensor product of chain complexes is very closely related to the Day convolution of simplicial Abelian groups (let's view simplicial Abelian groups as augmented simplicial groups with zero augmentation so we are working with presheaves over a monoidal category.) But I can't find a description of this map anywhere.
Patrick Nicodemus (Dec 12 2021 at 08:13):I'm going a little crazy because in spite of the pervasive influence of the Dold-Kan correspondence it still feels like the vast majority of homological algebra is done in the category of chain complexes and most of it hasn't been translated to the category of simplicial Abelian groups to see what it looks like in that setting. Sometimes I've gained a lot of intuition for a hom alg construction this way, because the simplicial Abelian group version of the construction is easier to understand (if not to compute with)
Reid Barton (Dec 12 2021 at 14:58):Patrick Nicodemus said:
I need explicit descriptions of the maps in both directions for computational reasons.
There generally won't be a direct map in either direction because neither side is fibrant.
I would guess the easiest way to see that they are homotopy equivalent is to express both as versions of the homotopy pushout that produces the suspension.