### Being Tentative on *n*Lab

#### Posted by David Corfield

I’m not completely convinced of $n$Lab acting as both reference wiki and as a place to work out ideas. Perhaps it can work by the method we’re using at the moment of flagging up tentative pages, but we have to be careful. The homotopy (as an operation) page is certainly tentative. It goes on to wonder whether we can dualize everything in sight at the cohomology page. Doing this threw up an interesting effect when it turned out that there was an already existing Cech homotopy as a candidate dual for Cech cohomology. As Tim Porter describes on the Cech homotopy page, there is work of long standing falling under that title, linked to (strong) shape theory. But is there a problem with its being linked to from a tentative page? Will anyone check the extent to which Cech homotopy is dual to Cech cohomology?

Elsewhere at sphere, we have Toby questioning the need for generalised homotopy theory. But we only know it’s Toby because he mentioned this at ‘Latest Changes’. It seems likely to me that an airing here would get wider attention, where views can be easily attributed to specific people.

Maybe the point is that discussing ideas we need to know identities. On ‘Latest Changes’ we read Urs saying that about twisted K-theory that he is

…feeling slightly uneasy about making this public, though, maybe later I get scared and remove that content again, or move it to my private web.

Expressing personal views on an anonymous wiki can be awkward. On the other hand, are people visiting each other’s private webs?

So let’s discuss Toby’s point in the old-fashioned Café style. Toby says that

… spheres, or rather their underlying topological spaces or simplicial sets, are fundamental in (ungeneralised) homotopy theory. In a sense, Whitehead’s theorem says that these are all that you need; no further generalised homotopy theory (in a sense dual to Eilenberg–Steenrod cohomology theory) is needed.

Whitehead’s theorem says that if a map between connected CW complexes induces isomorphisms of all homotopy groups, then the map is a homotopy equivalence. The Wikipedia entry mentions that this does not hold for topological spaces in general and that shape theory studies possible generalizations of Whitehead’s theorem. So is that why Tim’s Cech homotopy is necessary?

Could we say that there are some quasicategories where the homotopy side of the coin, the study of maps into a space, somehow involves a much simpler set of objects, than the cohomology side, the study of maps out of a space? Presumably then there must be cases where the latter is the simpler one and the former the complicated one, such as the opposite of the category of CW complexes.

Is it that $Top$ as a slight broadening of $CW-complexes$ requires a little more work on the homotopy side, but that still there’s much less need to involve a complicated array of objects as probes, than there is for coprobes for cohomology?

## Re: Being Tentative on nLab

David raises some good questions here. Perhaps those who are using the nLab as a means of trying out ideas to get reactions from others would find it useful to have another type of page, but one that was separately listed.. i.e. tentatively we might suggest a `tentative’ or `work in progress’ heading listed as well as latest changes. I know that I have put some stuff (draft project descriptions etc.) on my personal pages but have no way of knowing if my attempts to get reactions have been ignored or what. If that stuff was more centrally listed it might get more reaction. I don’t know.

On a mathematical point I would raise the possible meanings of duality between Cech homotopy and Cech cohomology. What is to be hoped for?