# Colocales

Colocales

Steve Vickers

Joint work with Matthew de Brecht and Paul Taylor (but this is my own take on it); also involved are Tatsuji Kawai and Francesco Ciraulo.

“Colocale” here means an Eilenberg-Moore algebra for the double powerlocale monad PP on the order-enriched category Loc of locales; and colocale maps are PP-algebra homomorphisms in the reverse direction. Hence -

``Coloc = (PP-Alg)^op``

The idea is that a PP-algebra is somehow a “localic frame”, a frame but one whose carrier is a locale instead of a set, and in Loc the PP-algebras do indeed have finite meets and all joins of locale maps.

In 2001 I noticed that Coloc too has a monad analogous to PP, and then Loc is dual to its E-M category, the category of “colocalic frames”. But for a long time I was puzzled as to what to make of this result.

PT gave a lab lunch on this topic a few weeks ago, presenting abstract category theory to model the result, in a dualized form that the Eilenberg-Moore adjunction for PP is both monadic (by definition) and comonadic. Paul’s approach used the fact that, defining Loc as the opposite of the category Fr of frames, Fr is itself the Eilenberg-Moore algebra category for a certain monad on Dcpo, and then exploited results on stabilisation of the zig-zag sequence of adjunctions got by alternately taking E-M and co-E-M categories.

I shall describe what happens if you don’t start from the hypothesis that Loc is got as dual of a pre-existing Fr, itself an E-M category. Then, in an argument I had around 2005, a simple application of Beck’s monadicity theorem provides some conditions on PP to be checked and they can be reduced to corresponding conditions for the upper and lower powerlocales (localic hyperspaces).

This then potentially allows Loc to be replaced by other order-enriched categories of “spaces” that are not initially defined as frames, but which do have well behaved hyperspaces. Then the comonadicity result shows that, in the end, the analogue of Loc is indeed the dual of an E-M category of “colocalic frames”, and it creates the colocales that can carry the frames.

I shall describe some intuitions by which we can think of colocales as “alternative spaces”, with their own points. Present investigations look at analogous phenomena for quasi-Polish spaces (de Brecht) and inductively generated formal topology (Kawai, Ciraulo).

This talk is part of the Lab Lunch series.