# Lax distributive laws for topology, II

## Hongliang Lai, Lili Shen and Walter Tholen

For a small quantaloid Q we consider four fundamental 2-monads T on Q-Cat, given by the presheaf 2-monad P and the copresheaf 2-monad P^{\dagger}, as well as by their two composite 2-monads, and establish that they all laxly distribute over P. These four 2-monads therefore admit lax extensions to the category Q-Dist of Q-categories and their distributors. We characterize the corresponding (T,Q)-categories in each of the four cases, leading us to both known and novel categorical structures.

Keywords: quantaloid, monad, presheaf monad, copresheaf monad, double presheaf monad, double copresheaf monad, lax distributive law, lax $\lambda$-algebra, lax monad extension, Q-closure space, Q-interior space

2010 MSC: 18C15, 18C20, 18D99

Theory and Applications of Categories, Vol. 32, 2017, No. 21, pp 736-768.

Published 2017-07-11.

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