Enriched Kleisli Objects for Pseudomonads

Abstract A pseudomonad on a 2-category whose underlying endomorphism is a 2-functor can be seen as a diagram $$\textbf{Psmnd} \rightarrow \textbf{Gray}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Psmnd</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Gray</mml:mi> </mml:mrow> </mml:math> for which weighted limits and colimits can be considered. The 2-category of pseudoalgebras, pseudomorphisms and 2-cells is such a Gray-enriched weighted limit [21], however neither the Kleisli bicategory nor the 2-category of free pseudoalgebras are the analogous weighted colimit [13]. In this paper we describe the actual weighted colimit via a presentation, and show that the comparison 2-functor induced by any other pseudoadjunction splitting the original pseudomonad is bi-fully faithful. As a consequence, we see that biessential surjectivity on objects characterises left pseudoadjoints whose codomains have an ‘up to biequivalence’ version of the universal property for Kleisli objects. This motivates a homotopical study of Kleisli objects for pseudomonads, and to this end we show that the weight for Kleisli objects is cofibrant in the projective model structure on the enriched functor category $$[\textbf{Psmnd}^\text {op} , \textbf{Gray}]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:msup> <mml:mi>Psmnd</mml:mi> <mml:mtext>op</mml:mtext> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>Gray</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> .