Topology on cohomology groups

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

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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