Re: Topology on cohomology groups

[Michael Barr <barr@math.mcgill.ca>]

Very tentatively, I have a memory that Lefschetz in the 30s invented
linearly compact vector spaces (and proved that the "category" of them
was dual to the "category" of discrete vector space (this was before
categories) for the purpose of making cohomology more closely dual to
homology.

Michael

On Fri, 19 Jun 2009, Steve Vickers wrote:

> A cohomology group can easily be an infinite power of the coefficient
> group. But such a group has a natural non-discrete topology, namely the
> compact-open (which in this case is also the product topology).
>
> Are there approaches to cohomology that, as part of the process, also
> supply topologies on the cohomology groups?
>
> [I'm trying to understand the topos-theoretic account of cohomology as
> in Johnstone's "Topos Theory". But it looks heavily dependent on having
> a classical base topos, since it uses the classical proof of sufficiency
> of injectives (together with the existence of Barr covers) to deduce the
> same property internally in any Grothendieck topos. For a more fully
> constructive theory I wonder if one needs to take better care of the
> topologies.]
>
> Steve Vickers.
>


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