Last night arrived at the conclusion that probably a topological space is something like a special case of a T-multicategory where T is the monad 2^(2^-) --- and after a bit of googling this appears to be a known fact. See http://arxiv.org/pdf/1412.4628.pdf, third hit in the paper for the phrase "topological space".

That's pretty neat, though! I like seeing that the contravariance in the definition of topological continuity "comes from" the contravariance in (each half of) the CPS monad.