Injective spaces and the filter monad

Martín Hötzel Escardó


An injective space is a topological space with a strong extension property for continuous maps with values on it. A certain filter space construction embeds every T0 topological space into an injective space. The construction gives rise to a monad. We show that the monad is of the Kock-Zöberlein type and apply this to obtain a simple proof of the fact that the algebras are the continuous lattices (Alan Day, 1975). In previous work we established an injectivity theorem for monads of this type, which characterizes the injective objects over a certain class of embeddings as the algebras. For the filter monad, the class turns out to consist precisely of the subspace embeddings. We thus obtain as a corollary that the injective spaces over subspace embeddings are the continuous lattices endowed with the Scott topology (Dana Scott, 1972). Similar results are obtained for continuous Scott domains, which are characterized as the injective spaces over dense subspace embeddings.


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