On simple lax idempotent monads and weak factorization systems

Maria Manuel Clementino
Friday, July 26, 2013 - from 16:00 to 16:25
Cassidy, H\'{e}bert and Kelly show in [2] that, in a category with pullbacks, every simple idempotent monad induces a factorization system, constructed via a pullback lifting of the monad unit. In this talk we present a generalization of this result to lax idempotent (or Kock-Z\"oberlein) 2-monads. In a 2-category $\K$ with comma objects, a 2-monad $T$ defines 2-monads $\eT_Z$ on the slice 2-categories $\K/Z$ for every object $Z$, inducing a 2-monad $\eT$ on $[2,\K]$. The 2-monad $T$ is said to be simple if the components of the unit of $\eT$ are $T$-embeddings. Every simple 2-monad $T$ with $\eT$ lax idempotent induces a functorial weak factorization system so that every morphism factors as a $T$-embedding followed by a $\eT$-algebra, in a functorial way. Examples of this construction include filter monads on topological T0-spaces (as studied in [4,1]) and (relative) presheaf monads in (generalized) enriched $\V$-categories (as studied in [3]). \\ Joint work with Ignacio L\'{o}pez-Franco.