2016/09/11 by Hongliang Lai, Lai, Hongliang, Walter Tholen +1
Mathematics · #06F99 #18A40 #18C15 #18D20 #Category Theory (math.CT) #FOS: Mathematics #math.CT #msc:06F99 #msc:18A40 #msc:18C15 #msc:18D20
paper · pdf · doi:10.48550/arxiv.1609.03214
arxiv created 2016/09/11 · arxiv updated 2016/09/13
For a small quantaloid Q, we consider 2-monads on the 2-category Q-\bfCat and their lax extensions to the 2-category Q-\bfDist of small Q-categories and their distributors, in particular those lax extensions that are flat, in the sense that they map identity distributors to identity distributors. In fact, unlike in the discrete case, a 2-monad on Q-\bfCat may admit only one flat lax extension. Every ordinary monad on the comma category \bfSet/\rm obQ with a lax extension to Q-\bfRel gives rise to such a 2-monad on Q-\bfCat, and we describe this process globally as a coreflective embedding. The Q-presheaf and the double Q-presheaf monads are important examples of 2-monads on Q-\bfCat allowing flat lax extensions to Q-\bfDist, and so are their submonads, obtained by the restriction to conical (co)presheaves and known as the Q-Hausdorff and double Q-Hausdorff monads, which we define here in full generality, thus generalizing some previous work in the case when Q is a quantale, or just the "metric" quantale [0,∞]. Their discretization leads naturally to various lax extensions of the relevant \bfSet-monads used in monoidal topology.