Gavin Seal

Lax algebras, monadic topology and tower extensions

ABSTRACT:
The category of topological spaces may be conveniently described as a
category of lax algebras, or more precisely, as a category of
(F,2)-algebras [1], where F denotes the filter monad and 2 the two-chain
{0,1}. The monad F can be replaced by a "fuzzy filter" monad to yield a
"convergence"-based description of monadic topologies [2], while various
tower extensions [3] are obtained when the two-chain is replaced by a
larger lattice. In turn, these interpretations open the way to a
"neighborhood"-based description of lax algebras.

[1] M.M. Clementino, D. Hofman and W. Tholen, "One setting for all:
metric, topology, uniformity, approach structure", Appl. Cat. Struct. 12
(2004) 127-154.

[2] W. G\"ahler, "Monadic topology - A new concept of generalized
topology", Math. Res. 67  (1992), 136-149.

[3] D. Zhang, "Tower extension of topological constructs", Commentat.
Math. Univ. Carol. 41 (2000), 41-51.