What is the universal property of the 2-category of monads?

arXiv2211.02210MaRDI QIDQ6416121

Stephen Lack, Adrian Miranda

Publication date: 3 November 2022

Abstract: For a 2-category $mathcalK$, we consider Street's 2-category Mnd($mathcalK$) of monads in $mathcalK$, along with Lack and Street's 2-category EM($mathcalK$) and the identity-on-objects-and-1-cells 2-functor Mnd($mathcalK$) $o$ EM($mathcalK$) between them. We show that this 2-functor can be obtained as a ``free completion of the 2-functor $1colonmathcalKomathcalK$. We do this by regarding 2-functors which act as the identity on both objects and 1-cells as categories enriched a cartesian closed category $mathbfBO$ whose objects are identity-on-objects functors. We also develop some of the theory of $mathbfBO$-enriched categories.

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